3531:
2853:
16497:
11830:
5171:
11336:
11319:
16744:
16764:
16754:
4284:
15139:
11825:{\displaystyle {\begin{aligned}f=\Phi _{Y,X}^{-1}(g)&=\varepsilon _{X}\circ F(g)&\in &\,\,\mathrm {hom} _{C}(F(Y),X)\\g=\Phi _{Y,X}(f)&=G(f)\circ \eta _{Y}&\in &\,\,\mathrm {hom} _{D}(Y,G(X))\\\Phi _{GX,X}^{-1}(1_{GX})&=\varepsilon _{X}&\in &\,\,\mathrm {hom} _{C}(FG(X),X)\\\Phi _{Y,FY}(1_{FY})&=\eta _{Y}&\in &\,\,\mathrm {hom} _{D}(Y,GF(Y))\\\end{aligned}}}
7367:, by formally adding an additive inverse for each bundle (or equivalence class). Alternatively one can observe that the functor that for each group takes the underlying monoid (ignoring inverses) has a left adjoint. This is a once-for-all construction, in line with the third section discussion above. That is, one can imitate the construction of
13303:
13085:
6465:. Any limit functor is right adjoint to a corresponding diagonal functor (provided the category has the type of limits in question), and the counit of the adjunction provides the defining maps from the limit object (i.e. from the diagonal functor on the limit, in the functor category). Below are some specific examples.
14740:
2314:
The equivalency of these definitions is quite useful. Adjoint functors arise everywhere, in all areas of mathematics. Since the structure in any of these definitions gives rise to the structures in the others, switching between them makes implicit use of many details that would otherwise have to be
1853:
Common mathematical constructions are very often adjoint functors. Consequently, general theorems about left/right adjoint functors encode the details of many useful and otherwise non-trivial results. Such general theorems include the equivalence of the various definitions of adjoint functors, the
12625:
A similar argument allows one to construct a hom-set adjunction from the terminal morphisms to a left adjoint functor. (The construction that starts with a right adjoint is slightly more common, since the right adjoint in many adjoint pairs is a trivially defined inclusion or forgetful functor.)
4172:
This definition is a logical compromise in that it is more difficult to satisfy than the universal morphism definitions, and has fewer immediate implications than the counitβunit definition. It is useful because of its obvious symmetry, and as a stepping-stone between the other definitions.
3852:
These definitions via universal morphisms are often useful for establishing that a given functor is left or right adjoint, because they are minimalistic in their requirements. They are also intuitively meaningful in that finding a universal morphism is like solving an optimization problem.
14940:
11977:
5043:
14001:
12869:
4505:
4882:
6797:. Any colimit functor is left adjoint to a corresponding diagonal functor (provided the category has the type of colimits in question), and the unit of the adjunction provides the defining maps into the colimit object. Below are some specific examples.
13131:
12913:
1201:
15392:
5830:
is fully determined by its action on generators, another restatement of the universal property of free groups. One can verify directly that this correspondence is a natural transformation, which means it is a hom-set adjunction for the pair
6311:
6040:
9385:
6768:
A suitable variation of this example also shows that the kernel functors for vector spaces and for modules are right adjoints. Analogously, one can show that the cokernel functors for abelian groups, vector spaces and modules are left
415:
13531:
8569:, quantifiers are identified with adjoints to the pullback functor. Such a realization can be seen in analogy to the discussion of propositional logic using set theory but the general definition make for a richer range of logics.
2299:
The definitions via universal morphisms are easy to state, and require minimal verifications when constructing an adjoint functor or proving two functors are adjoint. They are also the most analogous to our intuition involving
15484:
However, universal constructions are more general than adjoint functors: a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of
14621:
4656:
4589:
608:
535:
10093:
4125:
10305:
15134:{\displaystyle {\begin{aligned}&1_{\mathcal {E}}{\xrightarrow {\eta '}}G'F'{\xrightarrow {G'\eta F'}}G'GFF'\\&FF'G'G{\xrightarrow {F\varepsilon 'G}}FG{\xrightarrow {\varepsilon }}1_{\mathcal {C}}.\end{aligned}}}
13724:
10189:
8401:
6755:, which expresses the universal property of kernels. The counit of this adjunction is the defining embedding of a homomorphism's kernel into the homomorphism's domain, and the unit is the morphism identifying a group
4007:
9458:
11843:
4909:
2985:
784:
718:
8687:
3663:
14239:
14138:
8211:
221:
181:
1827:
13863:
1056:, and indeed every equivalence is an adjunction. In many situations, an adjunction can be "upgraded" to an equivalence, by a suitable natural modification of the involved categories and functors.
10970:
14945:
14626:
13868:
13804:
13611:
13136:
12918:
12744:
11848:
11341:
4914:
4799:
4424:
1346:
15702:
12739:
9579:
10879:
4419:
3530:
6208:
5938:
4794:
13405:
5887:
4746:
2398:). If the arrows for the left adjoint functor F were drawn they would be pointing to the left; if the arrows for the right adjoint functor G were drawn they would be pointing to the right.
14869:
1678:
1621:
10925:
6592:
etc. follow the same pattern; it can also be extended in a straightforward manner to more than just two factors. More generally, any type of limit is right adjoint to a diagonal functor.
15757:
10998:
2670:
69:. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain
5782:
5671:
5220:
2840:
8072:
10819:
14921:
13298:{\displaystyle {\begin{aligned}\Phi \Psi g&=G(\varepsilon _{X})\circ GF(g)\circ \eta _{Y}\\&=G(\varepsilon _{X})\circ \eta _{GX}\circ g\\&=1_{GX}\circ g=g\end{aligned}}}
13080:{\displaystyle {\begin{aligned}\Psi \Phi f&=\varepsilon _{X}\circ FG(f)\circ F(\eta _{Y})\\&=f\circ \varepsilon _{FY}\circ F(\eta _{Y})\\&=f\circ 1_{FY}=f\end{aligned}}}
3355:
5721:
3525:
9106:
9057:
8222:
in predicate logics is in forming propositions and also in expressing sophisticated predicates by closing formulas with possibly more variables. For example, consider a predicate
6973:-bilinear product with (r,0)(0,1) = (0,1)(r,0) = (r,0), (r,0)(s,0) = (rs,0), (0,1)(0,1) = (0,1). This constructs a left adjoint to the functor taking a ring to the underlying rng.
6397:
5589:
5524:
15803:
9730:
9271:
6141:
1094:
5076:
2310:
The definition via counitβunit adjunction is convenient for proofs about functors that are known to be adjoint, because they provide formulas that can be directly manipulated.
15282:
10738:
10685:
10651:
8910:
8879:
5100:
3700:
3022:
1054:
1030:
1002:
978:
828:
652:
312:
268:
134:
110:
10339:
9009:
8794:
11042:
11020:
10707:
10534:
9910:
9839:
9765:
9500:
9300:
8848:
8821:
6087:
9650:
7280:
be the category of pointed commutative rings with unity (pairs (A,a) where A is a ring, a β A and morphisms preserve the distinguished elements). The forgetful functor G:
3477:
3436:
2792:
2751:
8936:
8098:
6353:
5329:
1721:
10782:
10578:
9685:
8519:
8492:
8247:
8018:
4772:
1369:
951:
14478:
10485:
10465:
8624:
3588:
3121:
2910:
2436:
1224:
12679:
11062:
7015:
14763:
10507:
The twin fact in probability can be understood as an adjunction: that expectation commutes with affine transform, and that the expectation is in some sense the best
9936:
9605:
8761:
8559:
6221:
5950:
5161:
1858:(which are also found in every area of mathematics), and the general adjoint functor theorems giving conditions under which a given functor is a left/right adjoint.
14783:
10405:
9792:
9307:
8539:
5127:
3278:
2593:
324:
10598:
10445:
10425:
10379:
10359:
9965:
9859:
9812:
9226:
9206:
9186:
9166:
9146:
9126:
9029:
8976:
8956:
8735:
8715:
8592:
8465:
8445:
8425:
8307:
8287:
8267:
8138:
8118:
3847:
3827:
3807:
3787:
3764:
3740:
3720:
3556:
3395:
3375:
3298:
3249:
3229:
3209:
3189:
3165:
3145:
3086:
3062:
3042:
2878:
2710:
2690:
2613:
2564:
2544:
2524:
2504:
2480:
2460:
1771:
1751:
1641:
1584:
1564:
1544:
1521:
1501:
1481:
1457:
1437:
1413:
1393:
1291:
1270:
1250:
924:
898:
877:
851:
804:
628:
462:
442:
288:
244:
57:
may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as
11072:
There are hence numerous functors and natural transformations associated with every adjunction, and only a small portion is sufficient to determine the rest.
6535:. The universal property of the product group shows that Ξ is right-adjoint to Ξ. The counit of this adjunction is the defining pair of projection maps from
14735:{\displaystyle {\begin{aligned}\eta '&=(\tau \ast \sigma )\circ \eta \\\varepsilon '&=\varepsilon \circ (\sigma ^{-1}\ast \tau ^{-1}).\end{aligned}}}
13418:
5379:. That is, however, something particular to the case of tensor product. In category theory the 'naturality' of the bijection is subsumed in the concept of a
2339:
that are under consideration. Therefore it can be helpful to choose letters in alphabetical order according to whether they live in the "lefthand" category
5265:, which was at the time devoted to computations. Those faced with giving tidy, systematic presentations of the subject would have noticed relations such as
1829:, which is formally similar to the above relation between hom-sets. The analogy to adjoint maps of Hilbert spaces can be made precise in certain contexts.
4595:
4528:
540:
467:
9972:
7668:
that is known as soberification. Notably, the article also contains a detailed description of another adjunction that prepares the way for the famous
4019:
15894:
10199:
7021:
and 1 is an identity element. This construction gives a functor that is a left adjoint to the functor taking a monoid to the underlying semigroup.
15839:
Every monad arises from some adjunctionβin fact, typically from many adjunctionsβin the above fashion. Two constructions, called the category of
13619:
10102:
8314:
3914:
9396:
11972:{\displaystyle {\begin{aligned}1_{FY}&=\varepsilon _{FY}\circ F(\eta _{Y})\\1_{GX}&=G(\varepsilon _{X})\circ \eta _{GX}\end{aligned}}}
5233:
Note: The use of the prefix "co" in counit here is not consistent with the terminology of limits and colimits, because a colimit satisfies an
5038:{\displaystyle {\begin{aligned}1_{FY}&=\varepsilon _{FY}\circ F(\eta _{Y})\\1_{GX}&=G(\varepsilon _{X})\circ \eta _{GX}\end{aligned}}}
2852:
1956:
is also formulaic in this construction, since it is always the category of elements of the functor to which one is constructing an adjoint.
7208:
to rings. This functor is left adjoint to the functor that associates to a given ring its underlying multiplicative monoid. Similarly, the
6868:, again a consequence of the universal property of direct sums. The unit of this adjoint pair is the defining pair of inclusion maps from
723:
657:
8631:
10536:, with objects being the real numbers, and the morphisms being "affine functions evaluated at a point". That is, for any affine function
15886:
1890:
way is to adjoin an element '1' to the rng, adjoin all (and only) the elements that are necessary for satisfying the ring axioms (e.g.
2915:
16141:
9794:
is given by the direct image. Here is a characterization of this result, which matches more the logical interpretation: The image of
16040:
12040:
In particular, the equations above allow one to define Ξ¦, Ξ΅, and Ξ· in terms of any one of the three. However, the adjoint functors
9059:, the category of sets and functions, the canonical subobjects are the subset (or rather their canonical injections). The pullback
1845:
5845:
One can also verify directly that Ξ΅ and Ξ· are natural. Then, a direct verification that they form a counitβunit adjunction
16426:
15517:
then it is the left adjoint in an adjoint equivalence of categories, i.e. an adjunction whose unit and counit are isomorphisms.
14176:
14075:
13996:{\displaystyle {\begin{aligned}\Phi _{Y,X}(f)=G(f)\circ \eta _{Y}\\\Phi _{Y,X}^{-1}(g)=\varepsilon _{X}\circ F(g)\end{aligned}}}
8143:
5178:
These equations are useful in reducing proofs about adjoint functors to algebraic manipulations. They are sometimes called the
186:
146:
1776:
7689:
can be viewed as a category (where the elements of the poset become the category's objects and we have a single morphism from
3593:
1898:
in the ring), and impose no relations in the newly formed ring that are not forced by axioms. Moreover, this construction is
12048:
alone are in general not sufficient to determine the adjunction. The equivalence of these situations is demonstrated below.
15194:
Since many common constructions in mathematics are limits or colimits, this provides a wealth of information. For example:
14256:
12864:{\displaystyle {\begin{aligned}\Phi _{Y,X}(f)=G(f)\circ \eta _{Y}\\\Psi _{Y,X}(g)=\varepsilon _{X}\circ F(g)\end{aligned}}}
10930:
13737:
13544:
4500:{\displaystyle {\begin{aligned}\varepsilon &:FG\to 1_{\mathcal {C}}\\\eta &:1_{\mathcal {D}}\to GF\end{aligned}}}
1296:
15667:
14294:
9505:
8020:
is a unary predicate expressing some property, then a sufficiently strong set theory may prove the existence of the set
4877:{\displaystyle {\begin{aligned}1_{F}&=\varepsilon F\circ F\eta \\1_{G}&=G\varepsilon \circ \eta G\end{aligned}}}
1854:
uniqueness of a right adjoint for a given left adjoint, the fact that left/right adjoint functors respectively preserve
16800:
10824:
1905:
This is rather vague, though suggestive, and can be made precise in the language of category theory: a construction is
7560:
6053:. (Think of these words as placed in parentheses to indicate that they are independent generators.) The arrow
78:
16053:
16017:
15984:
6168:
5898:
13366:
12068:; in the sense of initial morphisms, one may construct the induced hom-set adjunction by doing the following steps.
6155:
it corresponds to (so this map is "dropping parentheses"). The composition of these maps is indeed the identity on
5848:
4707:
12245:
The commuting diagram of that factorization implies the commuting diagram of natural transformations, so Ξ· : 1
1646:
1589:
15167:
The most important property of adjoints is their continuity: every functor that has a left adjoint (and therefore
10891:
15721:
17:
10975:
2618:
16851:
7564:
5742:
5631:
5193:
2797:
14831:
8023:
7739:
The partial order case collapses the adjunction definitions quite noticeably, but can provide several themes:
16134:
16083:
16045:
10787:
3303:
16338:
16293:
14445:
5684:
3482:
15847:
are two extremal solutions to the problem of constructing an adjunction that gives rise to a given monad.
9062:
9042:
1196:{\displaystyle \varphi _{XY}:\mathrm {hom} _{\mathcal {C}}(FY,X)\cong \mathrm {hom} _{\mathcal {D}}(Y,GX)}
16767:
16707:
15840:
6366:
5363:. It can be seen to be natural on the basis, firstly, that these are two alternative descriptions of the
16416:
15387:{\displaystyle \Phi _{Y,X}:\mathrm {hom} _{\mathcal {C}}(FY,X)\cong \mathrm {hom} _{\mathcal {D}}(Y,GX)}
5549:
5484:
1921:. Universal properties come in two types: initial properties and terminal properties. Since these are
16866:
16757:
16543:
16407:
16315:
15768:
14880:
9690:
9231:
7314:
6450:
6116:
6438:
have generally the same description as in the detailed description of the free group situation above.
5052:
16907:
16716:
16360:
16298:
16221:
15514:
15199:
10712:
10659:
10603:
8884:
8853:
7961:
7901:
7751:
7669:
6782:
6446:
6415:
it corresponds to ("dropping parentheses"). The composition of these maps is indeed the identity on
1005:
31:
5081:
1035:
1011:
983:
959:
809:
633:
293:
249:
115:
91:
16747:
16703:
16308:
16127:
10314:
8981:
8766:
2303:
The definition via hom-sets makes symmetry the most apparent, and is the reason for using the word
2213:
yet; it is an important and not altogether trivial algebraic fact that such a left adjoint functor
1928:
The idea of using an initial property is to set up the problem in terms of some auxiliary category
11025:
11003:
10690:
10517:
9864:
9817:
9735:
9470:
9279:
8826:
8799:
7422:
that associates to every topological space its underlying set (forgetting the topology, that is).
6056:
16303:
16285:
15650:
15188:
15176:
9610:
7747:
7743:
adjunctions may not be dualities or isomorphisms, but are candidates for upgrading to that status
7724:
As is the case for Galois groups, the real interest lies often in refining a correspondence to a
6794:
6462:
6454:
5246:
3441:
3400:
2756:
2715:
2061:
is a ring map (which preserves the identity). (Note that this is precisely the definition of the
1855:
10194:
The right adjoint to the inverse image functor is given (without doing the computation here) by
8915:
8077:
6328:
5308:
1697:
16793:
16510:
16276:
16256:
16179:
16109:
15241:
14542:
12262:
11158:
10743:
10539:
9655:
8497:
8470:
8225:
7996:
7868:
7851:
multiplication, but in situations where this is not possible, we often attempt to construct an
7844:
7394:
6306:{\displaystyle GX{\xrightarrow {\;\eta _{GX}\;}}GFGX{\xrightarrow {\;G(\varepsilon _{X})\,}}GX}
6035:{\displaystyle FY{\xrightarrow {\;F(\eta _{Y})\;}}FGFY{\xrightarrow {\;\varepsilon _{FY}\,}}FY}
4751:
4411:
3668:
2990:
2382:, and whenever possible such things will be referred to in order from left to right (a functor
1922:
1351:
930:
421:
15934:
14451:
10470:
10450:
9380:{\displaystyle {\operatorname {Hom} }(\exists _{f}S,T)\cong {\operatorname {Hom} }(S,f^{*}T),}
8597:
3561:
3094:
2883:
2409:
1209:
16881:
16392:
16231:
12664:
11047:
10511:
to the problem of finding a real-valued approximation to a distribution on the real numbers.
7725:
7686:
7635:
7472:
7356:
7111:
7000:
6920:
5348:
15607:
In a sense, then, adjoints are "generalized" inverses. Note however that a right inverse of
14748:
13307:
hence ΦΨ is the identity transformation. Thus Φ is a natural isomorphism with inverse Φ = Ψ.
9915:
9584:
8740:
8544:
7375:. For the case of finitary algebraic structures, the existence by itself can be referred to
5526: be the set map given by "inclusion of generators". This is an initial morphism from
5136:
410:{\displaystyle \mathrm {hom} _{\mathcal {C}}(FY,X)\cong \mathrm {hom} _{\mathcal {D}}(Y,GX)}
16831:
16204:
16199:
16113:
16077:
14768:
10384:
9770:
8524:
7934:
which assigns to a category its set of connected components is left-adjoint to the functor
7864:
7617:
7390:
7201:
6928:
6833:
5681:
they correspond to, which exists by the universal property of free groups. Then each
5359:; those two groups are not really identical but there is a way of identifying them that is
5105:
16063:
16027:
15179:
in the category theoretical sense); every functor that has a right adjoint (and therefore
12684:, we can construct a hom-set adjunction by finding the natural transformation Ξ¦ : hom
5600:
3534:
The existence of the unit, a universal morphism, can prove the existence of an adjunction.
3254:
2569:
1080:: one is taken directly from Latin, the other from Latin via French. In the classic text
8:
16548:
16496:
16422:
16226:
15421:
15220:
11131:
10885:
7980:
and uncurrying; in many special cases, they are also continuous and form a homeomorphism.
7872:
7604:
7232:
7221:
7213:
7209:
5380:
5262:
5261:
in 1958. Like many of the concepts in category theory, it was suggested by the needs of
4277:
3906:
2846:
2014:
464:. Naturality here means that there are natural isomorphisms between the pair of functors
15942:
16402:
16397:
16379:
16261:
16236:
15913:
15865:
15450:
15213:
14502:
14351:
13526:{\displaystyle \varepsilon _{X}=\Phi _{GX,X}^{-1}(1_{GX})\in \mathrm {hom} _{C}(FGX,X)}
12022:
11994:
10583:
10430:
10410:
10364:
10344:
9941:
9844:
9797:
9211:
9191:
9171:
9151:
9131:
9111:
9014:
8961:
8941:
8720:
8700:
8577:
8450:
8430:
8410:
8292:
8272:
8252:
8219:
8123:
8103:
7673:
7664:
describes an adjunction between the category of topological spaces and the category of
7540:
7457:
7364:
7318:
7264:
5415:
3832:
3812:
3792:
3772:
3749:
3725:
3705:
3541:
3380:
3360:
3283:
3234:
3214:
3194:
3174:
3168:
3150:
3130:
3071:
3047:
3027:
2863:
2695:
2675:
2598:
2549:
2529:
2509:
2489:
2483:
2465:
2445:
1910:
1883:
1756:
1736:
1626:
1569:
1549:
1529:
1506:
1486:
1466:
1442:
1422:
1398:
1378:
1276:
1255:
1235:
909:
883:
862:
836:
789:
613:
447:
427:
273:
229:
70:
15151:
Since there is also a natural way to define an identity adjunction between a category
7721:
and to inverse order-preserving bijections between the corresponding closed elements.
4283:
16861:
16786:
16711:
16648:
16636:
16538:
16463:
16458:
16412:
16194:
16189:
16049:
16035:
16013:
15980:
15972:
14278:
8562:
7706:
7593:
7503:
7502:. The suspension functor is therefore left adjoint to the loop space functor in the
7419:
7415:
7376:
7372:
7352:
7044:
6585:
6581:
6431:
5444:
5356:
4519:
1085:
137:
82:
10740:
as "affine functions evaluated at a distribution". That is, for any affine function
4291:
The vertical arrows in this diagram are those induced by composition. Formally, Hom(
2331:
at its foundation, and there are many components that live in one of two categories
1948:
solutionβmeans something rigorous and recognisable, rather like the attainment of a
16876:
16846:
16841:
16821:
16672:
16558:
16533:
16468:
16453:
16448:
16387:
16184:
16059:
16023:
15955:
15903:
15844:
15561:
15257:
7733:
7718:
7468:
7443:
6725:
6589:
6516:
6458:
5409:
5364:
4227:
2069:
over the inclusion of unitary rings into rng.) The existence of a morphism between
1731:
35:
15941:, 1969. The notation is different nowadays; an easier introduction by Peter Smith
13411:
which defines families of initial and terminal morphisms, in the following steps:
4651:{\displaystyle G{\xrightarrow {\;\eta G\;}}GFG{\xrightarrow {\;G\varepsilon \,}}G}
4584:{\displaystyle F{\xrightarrow {\;F\eta \;}}FGF{\xrightarrow {\;\varepsilon F\,}}F}
2271:
This gives the intuition behind the fact that adjoint functors occur in pairs: if
16871:
16836:
16584:
16150:
15930:
7758:
7524:
7507:
7368:
7292:
2112:
can have more adjoined elements and/or more relations not imposed by axioms than
603:{\displaystyle {\mathcal {D}}(-,GX):{\mathcal {D}}\to \mathrm {Set^{\text{op}}} }
530:{\displaystyle {\mathcal {C}}(F-,X):{\mathcal {D}}\to \mathrm {Set^{\text{op}}} }
46:
10088:{\displaystyle \exists _{f}S=\{y\in Y\mid \exists (x\in f^{-1}).\,x\in S\;\}=f.}
7924:
form an adjoint pair. The unit and counit are natural isomorphisms in this case.
7017:{1} and defining a binary operation on it such that it extends the operation on
6882:
into the direct sum, and the counit is the additive map from the direct sum of (
4120:{\displaystyle \Phi _{Y,X}:\mathrm {hom} _{C}(FY,X)\to \mathrm {hom} _{D}(Y,GX)}
16886:
16621:
16616:
16600:
16563:
16553:
16473:
16004:
15156:
14286:
10447:. Note how the predicate determining the set is the same as above, except that
7856:
7651:
7578:
7431:
7322:
7217:
7056:
6992:
5332:
5187:
2062:
1933:
15428:), then any pair of adjoint functors between them are automatically additive.
10300:{\displaystyle \forall _{f}S=\{y\in Y\mid \forall (x\in f^{-1}).\,x\in S\;\}.}
7291:
has a left adjoint β it assigns to every ring R the pair (R,x) where R is the
16901:
16856:
16611:
16443:
16320:
16246:
16087:
7746:
closure operators may indicate the presence of adjunctions, as corresponding
7714:
7705:). A pair of adjoint functors between two partially ordered sets is called a
7661:
7521:
7360:
7348:
6952:
5298:
1879:
1727:
6434:, which assigns to an algebraic object its underlying set. These algebraic
5170:
1882:(which is like a ring that might not have a multiplicative identity) into a
16365:
16266:
15762:
is just the unit Ξ· of the adjunction and the multiplication transformation
13856:
The naturality of Ξ¦ implies the naturality of Ξ΅ and Ξ·, and the two formulas
13719:{\displaystyle \eta _{Y}=\Phi _{Y,FY}(1_{FY})\in \mathrm {hom} _{D}(Y,GFY)}
10184:{\displaystyle \{y\in Y\mid \exists x.\,\psi _{f}(x,y)\land \phi _{S}(x)\}}
8566:
8396:{\displaystyle \{y\in Y\mid \exists x.\,\psi _{f}(x,y)\land \phi _{S}(x)\}}
7713:
Galois connection). See that article for a number of examples: the case of
7380:
6932:
6924:
6435:
14441:
An analogous statement characterizes those functors with a right adjoint.
5190:. A way to remember them is to first write down the nonsensical equation
4002:{\displaystyle \Phi :\mathrm {hom} _{C}(F-,-)\to \mathrm {hom} _{D}(-,G-)}
1072:
1066:
16626:
16606:
16478:
16348:
16097:
15864:
Baez, John C. (1996). "Higher-Dimensional
Algebra II: 2-Hilbert Spaces".
9453:{\displaystyle \exists _{f}S\subseteq T\leftrightarrow S\subseteq f^{-1}}
7665:
7235:
instead of the category of rings, to get the monoid and group rings over
6957:
This example was discussed in the motivation section above. Given a rng
6786:
6427:
4231:
1875:
42:
16658:
16596:
16209:
15917:
11318:
7860:
7491:
7486:
is naturally isomorphic to the space of homotopy classes of maps from
7397:. This example foreshadowed the general theory by about half a century.
7263:
from fields has a left adjointβit assigns to every integral domain its
7216:
to rings, left adjoint to the functor that assigns to a given ring its
5424:
5397:
5258:
2237:, and pose the following (vague) question: is there a problem to which
74:
27:
Relationship between two functors abstracting many common constructions
16093:
15870:
7773:
the power set of the set of all mathematical structures. For a theory
7732:
order isomorphism). A treatment of Galois theory along these lines by
16652:
16343:
15425:
15206:
9228:. It therefore turns out to be (in bijection with) the inverse image
8694:
7938:
which assigns to a set the discrete category on that set. Moreover,
6978:
6778:
4209:
779:{\displaystyle {\mathcal {D}}(Y,G-):{\mathcal {C}}\to \mathrm {Set} }
713:{\displaystyle {\mathcal {C}}(FY,-):{\mathcal {C}}\to \mathrm {Set} }
315:
15908:
15105:
15076:
14999:
14966:
14793:
Adjunctions can be composed in a natural fashion. Specifically, if γ
12873:
The transformations Ξ¦ and Ξ¨ are natural because Ξ· and Ξ΅ are natural.
12629:
8682:{\displaystyle f^{*}:{\text{Sub}}(Y)\longrightarrow {\text{Sub}}(X)}
7252:
of integral domains with injective morphisms. The forgetful functor
6272:
6236:
6007:
5965:
4633:
4607:
4566:
4540:
2131:, that is, that there is a morphism from it to any other element of
1902:
in the sense that it works in essentially the same way for any rng.
16721:
16353:
16251:
16101:
14281:, then the functors with left adjoints can be characterized by the
10494:
7977:
7344:
6790:
6407:, which underlies the group homomorphism sending each generator of
5249:
where it looks like the insertion of the identity 1 into a monoid.
4235:
2182:
has an identity and considering it simply as a rng, so essentially
1949:
16119:
7736:
was influential in the recognition of the general structure here.
7383:; naturally there is also a proof adapted to category theory, too.
6049:
is the free group generated freely by the words of the free group
5230:
in one of the two simple ways that make the compositions defined.
16809:
16691:
16681:
16330:
16241:
16105:
15829:
14297:
and a certain smallness condition is satisfied: for every object
11092:
4384:
3877:
3863:
1918:
1077:
54:
16778:
15528:, Ξ΅, Ξ·γ extends an equivalence of certain subcategories. Define
15461:. Conversely, if there exists a universal morphism to a functor
12051:
8796:. If this functor has a left- or right adjoint, they are called
7717:
of course is a leading one. Any Galois connection gives rise to
2980:{\displaystyle \epsilon _{X}\circ F(G(f))=f\circ \epsilon _{X'}}
16686:
15155:
and itself, one can then form a category whose objects are all
13098:
is a functor, that Ξ· is natural, and the counitβunit equation 1
12882:
is a functor, that Ξ΅ is natural, and the counitβunit equation 1
7946:
which assigns to each category its set of objects, and finally
7205:
6988:
2347:, and also to write them down in this order whenever possible.
2295:
There are various equivalent definitions for adjoint functors:
2256:
is, in a certain rigorous sense, equivalent to the notion that
2162:
denote the above process of adjoining an identity to a rng, so
1643:
a right adjoint because it is applied to the right argument of
15212:
every right adjoint functor between two abelian categories is
13311:
7954:
which assigns to each set the indiscrete category on that set.
5673: be the group homomorphism that sends the generators of
2378:, Ξ· will consistently denote things that live in the category
2362:, Ξ΅ will consistently denote things that live in the category
16568:
15219:
every left adjoint functor between two abelian categories is
14934:
with unit and counit given respectively by the compositions:
14765:
denotes vertical composition of natural transformations, and
9767:. We conclude that left adjoint to the inverse image functor
7769:
to be the set of all logical theories (axiomatizations), and
6355: is the "inclusion of generators" set map from the set
5129:
denotes the identity natural transformation from the functor
2154:
can be expressed simultaneously by saying that it defines an
2146:
The two facts that this method of turning rngs into rings is
1586:
a left adjoint because it is applied to the left argument of
1503:
may have itself a right adjoint that is quite different from
11329:
The transformations Ξ΅, Ξ·, and Ξ¦ are related by the equations
8494:-related, and which itself is characterized by the property
7347:, the point of departure is to observe that the category of
6422:
16003:
AdΓ‘mek, JiΕΓ; Herrlich, Horst; Strecker, George E. (1990).
9912:
is non-empty. This works because it neglects exactly those
9302:, let us figure out the left adjoint, which is defined via
6961:, a multiplicative identity element can be added by taking
3091:
Similarly, we may define right-adjoint functors. A functor
14480:
is a functor between locally presentable categories, then
14234:{\displaystyle 1_{GX}=G(\varepsilon _{X})\circ \eta _{GX}}
14171:
in the first formula gives the second counitβunit equation
14133:{\displaystyle 1_{FY}=\varepsilon _{FY}\circ F(\eta _{Y})}
14068:
in the second formula gives the first counitβunit equation
11836:
The transformations Ξ΅, Ξ· satisfy the counitβunit equations
9168:
is characterized as the largest set which knows all about
8206:{\displaystyle \phi _{T}(y)=\phi _{Y}(y)\land \varphi (y)}
216:{\displaystyle G:{\mathcal {C}}\rightarrow {\mathcal {D}}}
176:{\displaystyle F:{\mathcal {D}}\rightarrow {\mathcal {C}}}
15397:
are, in fact, isomorphisms of abelian groups. Dually, if
7976:
has a right adjoint β. This pair is often referred to as
1822:{\displaystyle \langle Ty,x\rangle =\langle y,Ux\rangle }
15945:, which also attribute the concept to the article cited.
7650:. Here a more subtle point is that the left adjoint for
4366:
3658:{\displaystyle G(F(g))\circ \eta _{Y}=\eta _{Y'}\circ g}
2394:
can be thought of as "living" where its outputs are, in
1932:, so that the problem at hand corresponds to finding an
15604:
and yield inverse equivalences of these subcategories.
15276:
is also an additive functor and the hom-set bijections
7789:) be the set of all structures that satisfy the axioms
16002:
14580:, Ξ΅, Ξ·γ is an adjunction (with counitβunit (Ξ΅,Ξ·)) and
6724:
be the functor which assigns to each homomorphism its
4704:, and may indicate this relationship by writing
1925:
notions, it is only necessary to discuss one of them.
1866:
In a sense, an adjoint functor is a way of giving the
15771:
15724:
15670:
15285:
14943:
14883:
14834:
14771:
14751:
14624:
14454:
14179:
14078:
13866:
13809:
The bijectivity and naturality of Ξ¦ imply that each (
13740:
13622:
13547:
13421:
13369:
13134:
12916:
12742:
12667:
11846:
11339:
11050:
11028:
11006:
10978:
10965:{\displaystyle \mathbb {E} :\mu \mapsto \mathbb {E} }
10933:
10894:
10827:
10790:
10746:
10715:
10693:
10662:
10606:
10586:
10542:
10520:
10473:
10453:
10433:
10413:
10387:
10367:
10347:
10317:
10202:
10105:
9975:
9944:
9918:
9867:
9847:
9820:
9800:
9773:
9738:
9693:
9658:
9613:
9587:
9508:
9473:
9399:
9310:
9282:
9234:
9214:
9194:
9174:
9154:
9134:
9114:
9065:
9045:
9017:
8984:
8964:
8944:
8918:
8887:
8856:
8829:
8802:
8769:
8743:
8723:
8703:
8634:
8600:
8580:
8547:
8527:
8500:
8473:
8453:
8433:
8413:
8317:
8295:
8275:
8255:
8228:
8146:
8126:
8106:
8080:
8026:
7999:
7003:
6369:
6331:
6224:
6171:
6119:
6059:
5953:
5901:
5851:
5745:
5687:
5634:
5552:
5487:
5311:
5196:
5139:
5108:
5084:
5055:
4912:
4797:
4754:
4710:
4598:
4531:
4422:
4234:. Explicitly, the naturality of Ξ¦ means that for all
4022:
3917:
3835:
3815:
3795:
3775:
3752:
3728:
3708:
3671:
3596:
3564:
3544:
3485:
3444:
3403:
3383:
3363:
3306:
3286:
3257:
3237:
3217:
3197:
3177:
3153:
3133:
3097:
3074:
3050:
3030:
2993:
2918:
2886:
2866:
2800:
2759:
2718:
2698:
2678:
2621:
2601:
2572:
2552:
2532:
2512:
2492:
2468:
2448:
2412:
2401:
1861:
1779:
1759:
1739:
1700:
1649:
1629:
1592:
1572:
1552:
1532:
1509:
1489:
1469:
1445:
1425:
1401:
1381:
1354:
1299:
1279:
1258:
1238:
1212:
1097:
1038:
1014:
986:
962:
933:
912:
886:
865:
839:
812:
792:
726:
660:
636:
616:
543:
470:
450:
430:
327:
296:
276:
252:
232:
189:
149:
118:
94:
15441:
As stated earlier, an adjunction between categories
11262:
such that the diagrams below commute, and for every
8541:
of two sets directly corresponds to the conjunction
8074:
of terms that fulfill the property. A proper subset
6848:
is the functor which assigns to every abelian group
3856:
2224:
1088:
makes a distinction between the two. Given a family
13799:{\displaystyle 1_{FY}\in \mathrm {hom} _{C}(FY,FY)}
13606:{\displaystyle 1_{GX}\in \mathrm {hom} _{D}(GX,GX)}
12276:
on morphisms preserves compositions and identities.
5731:, because any group homomorphism from a free group
5237:property whereas the counit morphisms will satisfy
1341:{\displaystyle \mathrm {hom} _{\mathcal {C}}(FY,X)}
15797:
15751:
15697:{\displaystyle T:{\mathcal {D}}\to {\mathcal {D}}}
15696:
15386:
15133:
14915:
14863:
14777:
14757:
14734:
14472:
14233:
14132:
13995:
13798:
13718:
13605:
13525:
13399:
13297:
13079:
12863:
12673:
11971:
11824:
11056:
11036:
11014:
10992:
10964:
10919:
10873:
10813:
10776:
10732:
10701:
10679:
10645:
10592:
10572:
10528:
10479:
10459:
10439:
10419:
10399:
10373:
10353:
10333:
10299:
10183:
10087:
9959:
9930:
9904:
9853:
9833:
9806:
9786:
9759:
9724:
9679:
9644:
9599:
9574:{\displaystyle S\subseteq f^{-1}]\subseteq f^{-1}}
9573:
9494:
9452:
9379:
9294:
9265:
9220:
9200:
9180:
9160:
9140:
9120:
9100:
9051:
9023:
9003:
8970:
8950:
8930:
8904:
8873:
8842:
8815:
8788:
8755:
8729:
8709:
8681:
8618:
8586:
8553:
8533:
8513:
8486:
8459:
8439:
8419:
8395:
8301:
8281:
8261:
8241:
8205:
8132:
8112:
8092:
8066:
8012:
7672:of sober spaces and spatial locales, exploited in
7625:from the category of sheaves of abelian groups on
7009:
6391:
6347:
6305:
6202:
6135:
6081:
6034:
5932:
5881:
5776:
5715:
5665:
5583:
5518:
5323:
5214:
5155:
5121:
5094:
5070:
5037:
4876:
4766:
4740:
4650:
4583:
4499:
4119:
4001:
3841:
3821:
3801:
3781:
3758:
3734:
3714:
3694:
3657:
3582:
3550:
3519:
3471:
3430:
3389:
3369:
3349:
3292:
3272:
3243:
3223:
3203:
3183:
3159:
3139:
3115:
3080:
3056:
3036:
3016:
2979:
2904:
2872:
2845:The latter equation is expressed by the following
2834:
2786:
2745:
2704:
2684:
2664:
2607:
2587:
2558:
2538:
2518:
2498:
2474:
2454:
2430:
2209:Note however that we haven't actually constructed
2178:denote the process of βforgettingβ³ whether a ring
1838:The slogan is "Adjoint functors arise everywhere".
1821:
1765:
1745:
1715:
1672:
1635:
1615:
1578:
1558:
1538:
1515:
1495:
1475:
1451:
1431:
1407:
1387:
1363:
1340:
1285:
1264:
1244:
1218:
1195:
1048:
1024:
996:
972:
945:
918:
892:
871:
845:
822:
798:
778:
712:
646:
622:
602:
529:
456:
436:
409:
306:
282:
262:
238:
215:
175:
128:
104:
16006:Abstract and Concrete Categories. The joy of cats
15895:Transactions of the American Mathematical Society
12630:counitβunit adjunction induces hom-set adjunction
10874:{\displaystyle (\mu ,f):\mu \to \mu \circ f^{-1}}
8521:. Set theoretic operations like the intersection
6773:
16899:
8213:expressing a strictly more restrictive property.
7629:to the category of sheaves of abelian groups on
6793:are all examples of the categorical notion of a
6461:are all examples of the categorical notion of a
6045:should be the identity. The intermediate group
88:By definition, an adjunction between categories
16048:. Vol. 5 (2nd ed.). Springer-Verlag.
14926:More precisely, there is an adjunction between
13360:-), one can construct a counitβunit adjunction
10381:'s with the property that the inverse image of
6441:
6203:{\displaystyle 1_{G}=G\varepsilon \circ \eta G}
5933:{\displaystyle 1_{F}=\varepsilon F\circ F\eta }
5257:The idea of adjoint functors was introduced by
5186:because of the appearance of the corresponding
3211:. Spelled out, this means that for each object
2526:. Spelled out, this means that for each object
15209:of objects yields the coproduct of the images;
13400:{\displaystyle (\varepsilon ,\eta ):F\dashv G}
10927:, and the expectation defines another functor
6987:, we can add an identity element and obtain a
6317:should be the identity. The intermediate set
5882:{\displaystyle (\varepsilon ,\eta ):F\dashv G}
4741:{\displaystyle (\varepsilon ,\eta ):F\dashv G}
1870:solution to some problem via a method that is
1832:
16794:
16135:
16100:. Manipulation and visualization of objects,
15496:
15424:(i.e. preadditive categories with all finite
12658:, and a counitβunit adjunction (Ξ΅, Ξ·) :
12052:Universal morphisms induce hom-set adjunction
10709:with finite expectation. Define morphisms on
7607:(of sets, or abelian groups, or rings...) on
5473:is just the underlying set of the free group
5078:denotes the identify functor on the category
3769:It is true, as the terminology implies, that
1983:a ring having a multiplicative identity. The
1673:{\displaystyle \mathrm {hom} _{\mathcal {D}}}
1616:{\displaystyle \mathrm {hom} _{\mathcal {C}}}
1566:has a right adjoint" are equivalent. We call
15202:of objects yields the product of the images;
12281:Construct a natural isomorphism Ξ¦ : hom
10920:{\displaystyle \delta :x\mapsto \delta _{x}}
10394:
10388:
10291:
10268:
10262:
10219:
10178:
10106:
10064:
10041:
10035:
9992:
9890:
9884:
9011:and returns the thereby specified subset of
8390:
8318:
8061:
8033:
7654:will differ from that for sheaves (of sets).
7611:to the corresponding category of sheaves on
7539:be the inclusion functor to the category of
6935:of groups and by the disjoint union of sets.
6563:which define the limit, and the unit is the
5591: via a unique group homomorphism from
5163:denotes the identity morphism of the object
1816:
1801:
1795:
1780:
15752:{\displaystyle \eta :1_{\mathcal {D}}\to T}
15625:) need not be a right (or left) adjoint of
13312:Hom-set adjunction induces all of the above
11325:From this assertion, one can recover that:
8594:in a category with pullbacks. Any morphism
4777:In equation form, the above conditions on (
2315:repeated separately in every subject area.
1059:
16801:
16787:
16763:
16753:
16509:
16142:
16128:
15436:
12619:, and then Ξ¦ is natural in both arguments.
12268:Uniqueness of that factorization and that
12184:) is an initial morphism, then factorize Ξ·
10993:{\displaystyle \mathbb {E} \dashv \delta }
10290:
10063:
7408:A functor with a left and a right adjoint.
6480:be the functor that assigns to each pair (
6273:
6251:
6237:
6008:
5986:
5966:
5814:correspond precisely to maps from the set
4634:
4615:
4608:
4567:
4548:
4541:
2665:{\displaystyle \epsilon _{X}:F(G(X))\to X}
2119:. Therefore, the assertion that an object
30:For the construction in field theory, see
16086:β seven short lectures on adjunctions by
15907:
15869:
15794:
12143:on objects and the family of morphisms Ξ·.
11772:
11771:
11659:
11658:
11541:
11540:
11423:
11422:
11030:
11008:
10980:
10949:
10935:
10804:
10723:
10695:
10687:, the set of probability distribution on
10670:
10522:
10280:
10130:
10053:
8342:
7837:is right adjoint to the "syntax functor"
7563:. The unit of this adjoint pair yields a
7355:has a commutative monoid structure under
7087:is left adjoint to the forgetful functor
6423:Free constructions and forgetful functors
6293:
6105:to the corresponding word of length one (
6022:
5777:{\displaystyle \varepsilon _{X}:FGX\to X}
5666:{\displaystyle \varepsilon _{X}:FGX\to X}
5215:{\displaystyle 1=\varepsilon \circ \eta }
4641:
4574:
2835:{\displaystyle \epsilon _{X}\circ F(g)=f}
2350:In this article for example, the letters
1874:. For example, an elementary problem in
16041:Categories for the Working Mathematician
16034:
14864:{\displaystyle F\circ F':E\rightarrow C}
14293:has a left adjoint if and only if it is
13089:hence ΨΦ is the identity transformation.
8693:on the category that is the preorder of
8067:{\displaystyle Y=\{y\mid \phi _{Y}(y)\}}
6430:are all examples of a left adjoint to a
5806:Group homomorphisms from the free group
5601:universal property of the free group on
5169:
2856:Here the counit is a universal morphism.
1959:Back to our example: take the given rng
1846:Categories for the Working Mathematician
1082:Categories for the Working Mathematician
1004:is somewhat akin to a "weak form" of an
34:. For the construction in topology, see
13340:, and a hom-set adjunction Ξ¦ : hom
10814:{\displaystyle \mu \in M(\mathbb {R} )}
7793:; for a set of mathematical structures
6604:of homomorphisms of abelian groups. If
2098:is at least as efficient a solution as
420:such that this family of bijections is
77:in algebra, or the construction of the
14:
16900:
15198:applying a right adjoint functor to a
14545:. The same is true for left adjoints.
11067:
10099:Put this in analogy to our motivation
8737:(technically: monomorphism classes of
7942:is left-adjoint to the object functor
7904:, then we have an inverse equivalence
7371:; but there is the other option of an
7128:-module, then the tensor product with
6151:sending each generator to the word of
6143: is the group homomorphism from
6089: is the group homomorphism from
5620:is the free group generated freely by
5400:is a common and illuminating example.
4013:This specifies a family of bijections
3558:can be uniquely turned into a functor
3350:{\displaystyle \eta _{Y}:Y\to G(F(Y))}
1944:βthe sense that the process finds the
16782:
16508:
16161:
16123:
15205:applying a left adjoint functor to a
15162:
15159:and whose morphisms are adjunctions.
14444:An important special case is that of
12272:is a functor implies that the map of
11312:such that the diagrams below commute:
8938:to quantify a relation expressed via
7575:Direct and inverse images of sheaves.
7571:into its StoneβΔech compactification.
5716:{\displaystyle (GX,\varepsilon _{X})}
5419:be the functor assigning to each set
4367:Definition via counitβunit adjunction
3520:{\displaystyle G(f)\circ \eta _{Y}=g}
2860:In this situation, one can show that
2323:The theory of adjoints has the terms
2290:
318:between the respective morphism sets
136:is a pair of functors (assumed to be
15863:
15649:, Ξ΅, Ξ·γ gives rise to an associated
14817:β², Ξ΅β², Ξ·β²γ is an adjunction between
14257:Formal criteria for adjoint functors
10361:is characterized as the full set of
9101:{\displaystyle f^{*}T=X\times _{Y}T}
9052:{\displaystyle \operatorname {Set} }
8958:over, the functor/quantifier closes
7984:
7859:is adjoint to the multiplication by
6919:Analogous examples are given by the
6759:with the kernel of the homomorphism
6739:be the functor which maps the group
5245:here is borrowed from the theory of
4518:of the adjunction (terminology from
4208:as functors. In fact, they are both
16149:
15884:
14487:has a right adjoint if and only if
14305:there exists a family of morphisms
12395:is a functor, then for any objects
11177:A natural transformation Ξ· : 1
8850:, respectively. They both map from
7805:) be the minimal axiomatization of
7212:construction yields a functor from
6939:
6902:) of the direct sum to the element
6392:{\displaystyle G(\varepsilon _{X})}
5723: is a terminal morphism from
4230:). For details, see the article on
24:
15737:
15689:
15679:
15360:
15354:
15351:
15348:
15319:
15313:
15310:
15307:
15287:
15144:This new adjunction is called the
15118:
14955:
14615:β², Ξ΅β², Ξ·β²γ is an adjunction where
14497:has a left adjoint if and only if
13929:
13872:
13765:
13762:
13759:
13685:
13682:
13679:
13637:
13572:
13569:
13566:
13492:
13489:
13486:
13436:
13142:
13139:
12924:
12921:
12805:
12748:
11781:
11778:
11775:
11710:
11668:
11665:
11662:
11589:
11550:
11547:
11544:
11477:
11432:
11429:
11426:
11351:
11317:
10474:
10454:
10319:
10234:
10204:
10121:
10007:
9977:
9822:
9401:
9320:
8831:
8804:
8333:
7879:
7204:construction gives a functor from
5584:{\displaystyle \eta _{Y}:Y\to GFY}
5519:{\displaystyle \eta _{Y}:Y\to GFY}
5241:properties, and dually. The term
5087:
5062:
4661:are the identity transformations 1
4478:
4452:
4282:
4089:
4086:
4083:
4050:
4047:
4044:
4024:
3971:
3968:
3965:
3932:
3929:
3926:
3918:
3529:
2851:
2402:Definition via universal morphisms
1940:. This has an advantage that the
1862:Solutions to optimization problems
1664:
1658:
1655:
1652:
1607:
1601:
1598:
1595:
1314:
1308:
1305:
1302:
1169:
1163:
1160:
1157:
1128:
1122:
1119:
1116:
1041:
1017:
989:
965:
815:
772:
769:
766:
757:
729:
706:
703:
700:
691:
663:
639:
590:
586:
583:
574:
546:
517:
513:
510:
501:
473:
383:
377:
374:
371:
342:
336:
333:
330:
299:
255:
208:
198:
168:
158:
121:
97:
25:
16919:
16808:
16096:is a category theory package for
16071:
15798:{\displaystyle \mu :T^{2}\to T\,}
14916:{\displaystyle G'\circ G:C\to E.}
14801:, Ξ΅, Ξ·γ is an adjunction between
14501:preserves small limits and is an
11194:An equivalent formulation, where
11022:is the left adjoint, even though
9725:{\displaystyle S\subseteq f^{-1}}
9266:{\displaystyle f^{-1}\subseteq X}
7567:map from every topological space
7301:. Consider the inclusion functor
7198:From monoids and groups to rings.
6136:{\displaystyle \varepsilon _{FY}}
5784: via a unique set map from
3857:Definition via Hom-set adjunction
2225:Symmetry of optimization problems
956:An adjunction between categories
73:), such as the construction of a
16762:
16752:
16743:
16742:
16495:
16162:
15431:
15401:is additive with a left adjoint
14785:denotes horizontal composition.
13806: is the identity morphism.
13613: is the identity morphism.
12334:) is an initial morphism, then Ξ¦
12084:and a natural transformation Ξ·.
8249:with two open variables of sort
8140:is characterized by a predicate
8100:and the associated injection of
7709:(or, if it is contravariant, an
6210: says that for each group
5071:{\displaystyle 1_{\mathcal {C}}}
2753:there exists a unique morphism
2241:is the most efficient solution?
654:, and also the pair of functors
15489:(equivalently, every object of
14607:are natural isomorphisms then γ
14029:(which completely determine Ξ¦).
12391:Ξ· is a natural transformation,
10733:{\displaystyle M(\mathbb {R} )}
10680:{\displaystyle M(\mathbb {R} )}
10646:{\displaystyle (r,f):r\to f(r)}
9938:which are in the complement of
8912:. Very roughly, given a domain
8905:{\displaystyle {\text{Sub}}(Y)}
8874:{\displaystyle {\text{Sub}}(X)}
7847:is (in general) the attempt to
7391:representation theory of groups
7321:. It has a left adjoint called
6163:The second counitβunit equation
3438:there exists a unique morphism
1726:The terminology comes from the
1206:of hom-set bijections, we call
15966:
15948:
15924:
15878:
15857:
15788:
15743:
15684:
15553:is an isomorphism, and define
15381:
15366:
15340:
15325:
15148:of the two given adjunctions.
14904:
14855:
14788:
14722:
14690:
14656:
14644:
14464:
14446:locally presentable categories
14212:
14199:
14127:
14114:
13986:
13980:
13958:
13952:
13908:
13902:
13893:
13887:
13841:) is an initial morphism from
13819:) is a terminal morphism from
13793:
13775:
13713:
13695:
13671:
13655:
13600:
13582:
13520:
13502:
13478:
13462:
13382:
13370:
13231:
13218:
13189:
13183:
13171:
13158:
13035:
13022:
12984:
12971:
12962:
12956:
12854:
12848:
12826:
12820:
12784:
12778:
12769:
12763:
12095:, choose an initial morphism (
12056:Given a right adjoint functor
11946:
11933:
11903:
11890:
11815:
11812:
11806:
11791:
11744:
11728:
11702:
11693:
11687:
11678:
11631:
11615:
11581:
11578:
11572:
11560:
11517:
11511:
11498:
11492:
11463:
11454:
11448:
11442:
11412:
11406:
11380:
11374:
10959:
10953:
10945:
10904:
10849:
10840:
10828:
10808:
10800:
10756:
10750:
10727:
10719:
10674:
10666:
10640:
10634:
10628:
10619:
10607:
10552:
10546:
10502:
10274:
10271:
10259:
10237:
10175:
10169:
10153:
10141:
10079:
10073:
10047:
10044:
10032:
10010:
9954:
9948:
9893:
9881:
9748:
9742:
9719:
9713:
9668:
9662:
9639:
9633:
9568:
9562:
9543:
9540:
9534:
9528:
9483:
9477:
9447:
9441:
9419:
9371:
9349:
9338:
9316:
9254:
9248:
8899:
8893:
8868:
8862:
8747:
8676:
8670:
8662:
8659:
8653:
8610:
8387:
8381:
8365:
8353:
8289:. Using a quantifier to close
8200:
8194:
8185:
8179:
8163:
8157:
8058:
8052:
6774:Colimits and diagonal functors
6386:
6373:
6321:is just the underlying set of
6290:
6277:
6076:
6063:
5983:
5970:
5940: says that for each set
5893:The first counitβunit equation
5864:
5852:
5768:
5710:
5688:
5657:
5569:
5504:
5447:, which assigns to each group
5391:
5347:,β) (this is now known as the
5174:String diagram for adjunction.
5095:{\displaystyle {\mathcal {C}}}
5012:
4999:
4969:
4956:
4723:
4711:
4684:In this situation we say that
4522:), such that the compositions
4484:
4443:
4114:
4099:
4078:
4075:
4060:
3996:
3981:
3960:
3957:
3942:
3681:
3615:
3612:
3606:
3600:
3574:
3495:
3489:
3463:
3460:
3454:
3425:
3419:
3413:
3344:
3341:
3335:
3329:
3323:
3267:
3261:
3107:
3008:
2950:
2947:
2941:
2935:
2896:
2823:
2817:
2781:
2775:
2769:
2737:
2734:
2728:
2656:
2653:
2650:
2644:
2638:
2582:
2576:
2422:
2318:
1335:
1320:
1190:
1175:
1149:
1134:
1049:{\displaystyle {\mathcal {D}}}
1025:{\displaystyle {\mathcal {C}}}
997:{\displaystyle {\mathcal {D}}}
973:{\displaystyle {\mathcal {C}}}
823:{\displaystyle {\mathcal {D}}}
762:
749:
734:
696:
683:
668:
647:{\displaystyle {\mathcal {C}}}
579:
566:
551:
506:
493:
478:
404:
389:
363:
348:
307:{\displaystyle {\mathcal {D}}}
263:{\displaystyle {\mathcal {C}}}
203:
163:
129:{\displaystyle {\mathcal {D}}}
105:{\displaystyle {\mathcal {C}}}
13:
1:
16046:Graduate Texts in Mathematics
15996:
15977:Sheaves in Geometry and Logic
15227:
14508:
14245:
11000:. (Somewhat disconcertingly,
10334:{\displaystyle \forall _{f}S}
9004:{\displaystyle X\times _{Y}T}
8789:{\displaystyle X\times _{Y}T}
7325:which assigns to every group
7227:and consider the category of
6983:Similarly, given a semigroup
6590:product of topological spaces
5427:generated by the elements of
4176:In order to interpret Ξ¦ as a
2880:can be turned into a functor
1523:; see below for an example.)
15619:is naturally isomorphic to 1
15568:consisting of those objects
15539:consisting of those objects
14250:
11037:{\displaystyle \mathbb {E} }
11015:{\displaystyle \mathbb {E} }
10702:{\displaystyle \mathbb {R} }
10529:{\displaystyle \mathbb {R} }
9905:{\displaystyle f^{-1}\cap S}
9834:{\displaystyle \exists _{f}}
9760:{\displaystyle f\subseteq T}
9495:{\displaystyle f\subseteq T}
9295:{\displaystyle S\subseteq X}
9108:of an injection of a subset
8843:{\displaystyle \forall _{f}}
8816:{\displaystyle \exists _{f}}
7621:. It also induces a functor
7514:StoneβΔech compactification.
7454:Suspensions and loop spaces.
7220:. One can also start with a
6588:, the product of rings, the
6519:that assigns to every group
6442:Diagonal functors and limits
6082:{\displaystyle F(\eta _{Y})}
2672:such that for every object
2343:or the "righthand" category
7:
16437:Constructions on categories
15535:as the full subcategory of
15457:and one for each object in
14560:is naturally isomorphic to
14355:, such that every morphism
12704:-) in the following steps:
10656:Define a category based on
10514:Define a category based on
9645:{\displaystyle x\in f^{-1}}
7561:StoneβΔech compactification
7401:
6977:Adjoining an identity to a
6951:Adjoining an identity to a
6399: is the set map from
5534:, because any set map from
5386:
3472:{\displaystyle f:F(Y)\to X}
3431:{\displaystyle g:Y\to G(X)}
3357:such that for every object
2912:in a unique way such that
2787:{\displaystyle g:Y\to G(X)}
2746:{\displaystyle f:F(Y)\to X}
1833:Introduction and Motivation
79:StoneβΔech compactification
53:is a relationship that two
10:
16924:
16544:Higher-dimensional algebra
15935:Adjointness in foundations
15497:Equivalences of categories
15449:gives rise to a family of
14273:admits a left adjoint. If
14254:
10427:is fully contained within
8931:{\displaystyle S\subset X}
8093:{\displaystyle T\subset Y}
7833:: the "semantics functor"
7757:a very general comment of
7315:category of abelian groups
7186:) for every abelian group
6944:
6832:) of abelian groups their
6531:) in the product category
6348:{\displaystyle \eta _{GX}}
5739:will factor through
5546:will factor through
5451:its underlying set. Then
5324:{\displaystyle -\otimes A}
5252:
3251:, there exists an object
2406:By definition, a functor
1716:{\displaystyle F\dashv G.}
1395:(p. 81). The functor
29:
16817:
16738:
16671:
16635:
16583:
16576:
16527:
16517:
16504:
16493:
16436:
16378:
16329:
16284:
16275:
16172:
16168:
16157:
16012:. John Wiley & Sons.
15636:
15515:equivalence of categories
15453:, one for each object in
10777:{\displaystyle f(x)=ax+b}
10573:{\displaystyle f(x)=ax+b}
9680:{\displaystyle f(x)\in T}
8514:{\displaystyle \phi _{S}}
8487:{\displaystyle \psi _{f}}
8242:{\displaystyle \psi _{f}}
8013:{\displaystyle \phi _{Y}}
7962:cartesian closed category
7902:equivalence of categories
7752:Kuratowski closure axioms
7680:
7295:with coefficients from R.
6682:) of morphisms such that
5822:: each homomorphism from
5599:. This is precisely the
4887:which mean that for each
4767:{\displaystyle F\dashv G}
3695:{\displaystyle g:Y\to Y'}
3017:{\displaystyle f:X'\to X}
2143:solution to our problem.
1526:In general, the phrases "
1364:{\displaystyle \varphi f}
946:{\displaystyle F\dashv G}
32:Adjunction (field theory)
15975:; Moerdijk, Ieke (1992)
15850:
15841:EilenbergβMoore algebras
15715:. The unit of the monad
15657:, Ξ·, ΞΌγ in the category
15582:is an isomorphism. Then
14568:is also left adjoint to
14491:preserves small colimits
14473:{\displaystyle F:C\to D}
10972:, and they are adjoint:
10480:{\displaystyle \forall }
10460:{\displaystyle \exists }
9581:. Conversely, If for an
8619:{\displaystyle f:X\to Y}
7928:A series of adjunctions.
7363:out of this monoid, the
7190:, is a right adjoint to
7051:can be seen as a (left)
7035:are rings, and Ο :
5222:and then fill in either
4748: , or simply
4510:respectively called the
3583:{\displaystyle F:D\to C}
3116:{\displaystyle G:C\to D}
2905:{\displaystyle G:C\to D}
2566:there exists an object
2431:{\displaystyle F:D\to C}
2252:to the problem posed by
2158:. More explicitly: Let
1546:is a left adjoint" and "
1219:{\displaystyle \varphi }
1060:Terminology and notation
16354:Cokernels and quotients
16277:Universal constructions
16110:natural transformations
15885:Kan, Daniel M. (1958).
15437:Universal constructions
14572:β². More generally, if γ
14525:has two right adjoints
14283:adjoint functor theorem
12674:{\displaystyle \dashv }
12344:is a bijection, where Ξ¦
12139:)). We have the map of
11057:{\displaystyle \delta }
7916:, and the two functors
7809:. We can then say that
7506:, an important fact in
7010:{\displaystyle \sqcup }
6818:assigns to every pair (
6650:, then a morphism from
6097:sending each generator
5843:counitβunit adjunction.
4412:natural transformations
4375:between two categories
3868:between two categories
2250:most efficient solution
2229:It is also possible to
1076:are both used, and are
16852:Essentially surjective
16511:Higher category theory
16257:Natural transformation
15943:in these lecture notes
15816:. Dually, the triple γ
15799:
15753:
15698:
15629:. Adjoints generalize
15388:
15242:preadditive categories
15135:
14917:
14865:
14779:
14759:
14758:{\displaystyle \circ }
14736:
14474:
14235:
14134:
13997:
13800:
13720:
13607:
13527:
13401:
13299:
13081:
12878:Using, in order, that
12865:
12675:
12263:natural transformation
12238:). This is the map of
11973:
11826:
11322:
11206:denotes any object of
11198:denotes any object of
11159:natural transformation
11058:
11038:
11016:
10994:
10966:
10921:
10875:
10815:
10778:
10734:
10703:
10681:
10647:
10594:
10574:
10530:
10481:
10461:
10441:
10421:
10401:
10375:
10355:
10335:
10301:
10185:
10089:
9961:
9932:
9931:{\displaystyle y\in Y}
9906:
9855:
9835:
9808:
9788:
9761:
9726:
9681:
9646:
9601:
9600:{\displaystyle x\in S}
9575:
9496:
9454:
9391:which here just means
9381:
9296:
9267:
9222:
9202:
9182:
9162:
9142:
9122:
9102:
9053:
9025:
9005:
8972:
8952:
8932:
8906:
8875:
8844:
8817:
8790:
8757:
8756:{\displaystyle T\to Y}
8731:
8711:
8683:
8620:
8588:
8574:So consider an object
8555:
8554:{\displaystyle \land }
8535:
8515:
8488:
8461:
8447:for which there is an
8441:
8421:
8397:
8309:, we can form the set
8303:
8283:
8263:
8243:
8207:
8134:
8114:
8094:
8068:
8014:
7438:, and a right adjoint
7395:induced representation
7341:The Grothendieck group
7245:Consider the category
7011:
6600:Consider the category
6393:
6349:
6307:
6204:
6137:
6083:
6036:
5934:
5889: is as follows:
5883:
5800:) is an adjoint pair.
5778:
5717:
5667:
5585:
5538:to the underlying set
5520:
5325:
5216:
5175:
5157:
5156:{\displaystyle 1_{FY}}
5123:
5096:
5072:
5039:
4878:
4768:
4742:
4652:
4585:
4501:
4373:counitβunit adjunction
4288:
4276:the following diagram
4121:
4003:
3843:
3823:
3803:
3783:
3760:
3736:
3716:
3696:
3659:
3584:
3552:
3535:
3521:
3473:
3432:
3391:
3371:
3351:
3294:
3274:
3245:
3225:
3205:
3185:
3161:
3141:
3117:
3082:
3058:
3038:
3018:
2981:
2906:
2874:
2857:
2836:
2788:
2747:
2706:
2686:
2666:
2609:
2589:
2560:
2540:
2520:
2500:
2476:
2456:
2432:
2262:most difficult problem
2135:, means that the ring
1971:are rng homomorphisms
1963:, and make a category
1851:
1823:
1767:
1747:
1717:
1674:
1637:
1617:
1580:
1560:
1540:
1517:
1497:
1477:
1453:
1433:
1409:
1389:
1365:
1342:
1287:
1266:
1246:
1220:
1197:
1050:
1026:
998:
974:
947:
920:
894:
873:
847:
824:
800:
780:
714:
648:
624:
604:
531:
458:
438:
411:
308:
284:
264:
240:
217:
177:
130:
106:
15956:"Indiscrete category"
15800:
15754:
15699:
15590:can be restricted to
15473:from every object of
15389:
15260:with a right adjoint
15136:
14918:
14866:
14780:
14778:{\displaystyle \ast }
14760:
14737:
14475:
14236:
14135:
13998:
13801:
13721:
13608:
13528:
13402:
13300:
13082:
12866:
12676:
11974:
11827:
11321:
11059:
11039:
11017:
10995:
10967:
10922:
10876:
10816:
10779:
10735:
10704:
10682:
10648:
10595:
10575:
10531:
10482:
10462:
10442:
10422:
10402:
10400:{\displaystyle \{y\}}
10376:
10356:
10336:
10302:
10186:
10090:
9962:
9933:
9907:
9856:
9836:
9809:
9789:
9787:{\displaystyle f^{*}}
9762:
9727:
9682:
9647:
9602:
9576:
9497:
9455:
9382:
9297:
9268:
9223:
9203:
9188:and the injection of
9183:
9163:
9143:
9123:
9103:
9054:
9026:
9006:
8973:
8953:
8933:
8907:
8876:
8845:
8818:
8791:
8758:
8732:
8712:
8697:. It maps subobjects
8684:
8621:
8589:
8556:
8536:
8534:{\displaystyle \cap }
8516:
8489:
8462:
8442:
8422:
8398:
8304:
8284:
8264:
8244:
8208:
8135:
8115:
8095:
8069:
8015:
7687:partially ordered set
7636:inverse image functor
7603:from the category of
7387:Frobenius reciprocity
7012:
6894:(sending an element (
6575:(mapping x to (x,x)).
6394:
6350:
6308:
6205:
6138:
6084:
6037:
5935:
5884:
5779:
5718:
5668:
5586:
5521:
5349:tensor-hom adjunction
5326:
5217:
5173:
5158:
5124:
5122:{\displaystyle 1_{F}}
5097:
5073:
5040:
4879:
4787:counitβunit equations
4769:
4743:
4653:
4586:
4502:
4286:
4180:, one must recognize
4122:
4004:
3844:
3824:
3804:
3784:
3761:
3737:
3717:
3697:
3660:
3585:
3553:
3533:
3522:
3474:
3433:
3392:
3372:
3352:
3295:
3275:
3246:
3226:
3206:
3186:
3162:
3142:
3125:right adjoint functor
3118:
3083:
3059:
3039:
3019:
2982:
2907:
2875:
2855:
2837:
2789:
2748:
2707:
2687:
2667:
2610:
2590:
2561:
2541:
2521:
2501:
2477:
2457:
2433:
2015:commutative triangles
1836:
1824:
1768:
1748:
1718:
1675:
1638:
1618:
1581:
1561:
1541:
1518:
1498:
1478:
1454:
1434:
1410:
1390:
1366:
1343:
1288:
1267:
1247:
1221:
1198:
1051:
1027:
999:
975:
948:
921:
902:right adjoint functor
895:
874:
848:
825:
801:
781:
715:
649:
625:
605:
532:
459:
439:
412:
309:
285:
265:
241:
226:and, for all objects
218:
178:
131:
107:
16380:Algebraic categories
16114:universal properties
15769:
15722:
15668:
15481:has a left adjoint.
15283:
15187:(i.e. commutes with
15175:(i.e. commutes with
15171:a right adjoint) is
14941:
14881:
14832:
14769:
14749:
14622:
14543:naturally isomorphic
14452:
14177:
14076:
13864:
13738:
13620:
13545:
13419:
13367:
13132:
12914:
12740:
12665:
12072:Construct a functor
11844:
11337:
11278:, there is a unique
11232:, there is a unique
11048:
11026:
11004:
10976:
10931:
10892:
10825:
10821:, define a morphism
10788:
10744:
10713:
10691:
10660:
10604:
10600:, define a morphism
10584:
10580:and any real number
10540:
10518:
10471:
10451:
10431:
10411:
10385:
10365:
10345:
10315:
10200:
10103:
9973:
9942:
9916:
9865:
9845:
9818:
9798:
9771:
9736:
9691:
9656:
9611:
9585:
9506:
9471:
9397:
9308:
9280:
9232:
9212:
9192:
9172:
9152:
9132:
9112:
9063:
9043:
9015:
8982:
8962:
8942:
8916:
8885:
8854:
8827:
8800:
8767:
8741:
8721:
8701:
8632:
8598:
8578:
8545:
8525:
8498:
8471:
8451:
8431:
8411:
8315:
8293:
8273:
8253:
8226:
8144:
8124:
8104:
8078:
8024:
7997:
7829:) logically implies
7763:syntax and semantics
7618:direct image functor
7414:be the functor from
7202:integral monoid ring
7001:
6751:is right adjoint to
6743:to the homomorphism
6494:) the product group
6367:
6329:
6222:
6169:
6117:
6109:) as a generator of
6057:
5951:
5899:
5849:
5792:. This means that (
5743:
5685:
5632:
5550:
5485:
5396:The construction of
5343:was the functor hom(
5309:
5194:
5137:
5106:
5082:
5053:
4910:
4795:
4752:
4708:
4699:is right adjoint to
4596:
4529:
4420:
4164:is right adjoint to
4020:
3915:
3833:
3829:is right adjoint to
3813:
3793:
3773:
3750:
3726:
3706:
3669:
3594:
3562:
3542:
3483:
3442:
3401:
3381:
3361:
3304:
3284:
3273:{\displaystyle F(Y)}
3255:
3235:
3215:
3195:
3175:
3151:
3131:
3095:
3072:
3048:
3028:
2991:
2916:
2884:
2864:
2798:
2757:
2716:
2696:
2676:
2619:
2599:
2588:{\displaystyle G(X)}
2570:
2550:
2530:
2510:
2490:
2466:
2446:
2440:left adjoint functor
2410:
2283:is right adjoint to
2200:left adjoint functor
1777:
1757:
1737:
1698:
1647:
1627:
1590:
1570:
1550:
1530:
1507:
1487:
1467:
1443:
1423:
1399:
1379:
1352:
1297:
1277:
1256:
1236:
1210:
1095:
1036:
1012:
984:
960:
931:
910:
884:
863:
855:left adjoint functor
837:
810:
790:
724:
658:
634:
614:
541:
468:
448:
428:
325:
294:
274:
250:
230:
187:
147:
116:
92:
16549:Homotopy hypothesis
16227:Commutative diagram
15979:, Springer-Verlag.
15931:Lawvere, F. William
15451:universal morphisms
15422:additive categories
15183:a left adjoint) is
15109:
15091:
15019:
14975:
14874:is left adjoint to
14552:is left adjoint to
13951:
13461:
13094:Dually, using that
11614:
11373:
11132:natural isomorphism
11079:between categories
11068:Adjunctions in full
11044:is "forgetful" and
10888:defines a functor:
10886:Dirac delta measure
9841:is the full set of
7950:is left-adjoint to
7873:logical conjunction
7869:propositional logic
7643:is left adjoint to
7547:has a left adjoint
7520:be the category of
7426:has a left adjoint
7329:the quotient group
7243:Field of fractions.
7210:integral group ring
6864:is left adjoint to
6646:are two objects of
6508:, and let Ξ :
6363:. The arrow
6325:. The arrow
6294:
6252:
6113:. The arrow
6023:
5987:
5804:Hom-set adjunction.
5677:to the elements of
5610:Terminal morphisms.
5455:is left adjoint to
5381:natural isomorphism
5297:in the category of
5263:homological algebra
5182:, or sometimes the
5180:triangle identities
4689:is left adjoint to
4642:
4616:
4575:
4549:
4178:natural isomorphism
4154:is left adjoint to
4149:In this situation,
3907:natural isomorphism
3789:is left adjoint to
3397:and every morphism
3127:if for each object
2847:commutative diagram
2712:and every morphism
2442:if for each object
2275:is left adjoint to
1843:Saunders Mac Lane,
1687:is left adjoint to
1232:adjunction between
75:free group on a set
16262:Universal property
16036:Mac Lane, Saunders
15973:Mac Lane, Saunders
15887:"Adjoint Functors"
15795:
15749:
15694:
15641:Every adjunction γ
15520:Every adjunction γ
15513:is one half of an
15412:Moreover, if both
15409:is also additive.
15384:
15163:Limit preservation
15131:
15129:
14913:
14861:
14775:
14755:
14732:
14730:
14503:accessible functor
14470:
14413:and some morphism
14378:can be written as
14338:where the indices
14261:Not every functor
14231:
14130:
13993:
13991:
13928:
13796:
13716:
13603:
13523:
13435:
13397:
13295:
13293:
13077:
13075:
12861:
12859:
12671:
11969:
11967:
11822:
11820:
11588:
11350:
11323:
11054:
11034:
11012:
10990:
10962:
10917:
10871:
10811:
10774:
10730:
10699:
10677:
10643:
10590:
10570:
10526:
10477:
10457:
10437:
10417:
10397:
10371:
10351:
10331:
10297:
10181:
10085:
9957:
9928:
9902:
9851:
9831:
9804:
9784:
9757:
9722:
9677:
9642:
9597:
9571:
9492:
9450:
9377:
9292:
9263:
9218:
9198:
9178:
9158:
9138:
9118:
9098:
9049:
9021:
9001:
8968:
8948:
8928:
8902:
8871:
8840:
8813:
8786:
8763:) to the pullback
8753:
8727:
8707:
8679:
8626:induces a functor
8616:
8584:
8561:of predicates. In
8551:
8531:
8511:
8484:
8457:
8437:
8417:
8393:
8299:
8279:
8259:
8239:
8203:
8130:
8110:
8090:
8064:
8010:
7958:Exponential object
7765:are adjoint: take
7674:pointless topology
7596:induces a functor
7594:topological spaces
7541:topological spaces
7458:topological spaces
7416:topological spaces
7365:Grothendieck group
7359:. One may make an
7319:category of groups
7265:field of fractions
7007:
6567:of a group X into
6565:diagonal inclusion
6411:to the element of
6389:
6345:
6303:
6200:
6133:
6079:
6032:
5930:
5879:
5774:
5713:
5663:
5624:, the elements of
5581:
5516:
5463:Initial morphisms.
5351:). The use of the
5321:
5212:
5176:
5153:
5119:
5092:
5068:
5035:
5033:
4874:
4872:
4764:
4738:
4648:
4581:
4497:
4495:
4289:
4255:and all morphisms
4117:
3999:
3839:
3819:
3799:
3779:
3756:
3732:
3712:
3692:
3655:
3580:
3548:
3536:
3517:
3469:
3428:
3387:
3367:
3347:
3290:
3270:
3241:
3221:
3201:
3181:
3169:universal morphism
3167:, there exists a
3157:
3137:
3113:
3078:
3054:
3034:
3014:
2987:for all morphisms
2977:
2902:
2870:
2858:
2832:
2784:
2743:
2702:
2682:
2662:
2605:
2585:
2556:
2536:
2516:
2496:
2484:universal morphism
2472:
2452:
2428:
2291:Formal definitions
1911:universal property
1909:if it satisfies a
1819:
1763:
1743:
1713:
1670:
1633:
1613:
1576:
1556:
1536:
1513:
1493:
1473:
1449:
1429:
1405:
1385:
1361:
1338:
1283:
1262:
1242:
1216:
1193:
1046:
1022:
994:
970:
943:
916:
890:
869:
843:
820:
796:
776:
710:
644:
620:
600:
527:
454:
434:
407:
304:
280:
260:
236:
213:
183: and
173:
126:
102:
71:universal property
65:and the other the
16895:
16894:
16867:Full and faithful
16776:
16775:
16734:
16733:
16730:
16729:
16712:monoidal category
16667:
16666:
16539:Enriched category
16491:
16490:
16487:
16486:
16464:Quotient category
16459:Opposite category
16374:
16373:
15110:
15092:
15020:
14976:
14825:then the functor
14279:complete category
14143:and substituting
13117:
12896:
12615:
12591:
12576:
12564:
12553:
12532:
12517:
12506:
12495:
12489:
12381:
12257:
12195:
11995:terminal morphism
11210:, is as follows:
10593:{\displaystyle r}
10440:{\displaystyle S}
10420:{\displaystyle f}
10374:{\displaystyle y}
10354:{\displaystyle Y}
9960:{\displaystyle f}
9854:{\displaystyle y}
9807:{\displaystyle S}
9221:{\displaystyle Y}
9201:{\displaystyle T}
9181:{\displaystyle f}
9161:{\displaystyle f}
9141:{\displaystyle Y}
9121:{\displaystyle T}
9024:{\displaystyle Y}
8971:{\displaystyle X}
8951:{\displaystyle f}
8891:
8860:
8730:{\displaystyle Y}
8710:{\displaystyle T}
8668:
8651:
8587:{\displaystyle Y}
8563:categorical logic
8460:{\displaystyle x}
8440:{\displaystyle Y}
8420:{\displaystyle y}
8302:{\displaystyle X}
8282:{\displaystyle Y}
8262:{\displaystyle X}
8133:{\displaystyle Y}
8113:{\displaystyle T}
7985:Categorical logic
7821:) if and only if
7719:closure operators
7707:Galois connection
7504:homotopy category
7471:of maps from the
7377:universal algebra
7373:existence theorem
7353:topological space
7132:yields a functor
7063:yields a functor
7055:-module, and the
7045:ring homomorphism
6783:fibred coproducts
6582:cartesian product
6432:forgetful functor
6295:
6253:
6024:
5988:
5445:forgetful functor
5365:bilinear mappings
5357:abuse of notation
5184:zig-zag equations
4643:
4617:
4576:
4550:
4520:universal algebra
4336:
4332:
3842:{\displaystyle F}
3822:{\displaystyle G}
3802:{\displaystyle G}
3782:{\displaystyle F}
3759:{\displaystyle F}
3742:is then called a
3735:{\displaystyle G}
3715:{\displaystyle D}
3551:{\displaystyle F}
3390:{\displaystyle C}
3370:{\displaystyle X}
3293:{\displaystyle C}
3244:{\displaystyle D}
3224:{\displaystyle Y}
3204:{\displaystyle G}
3184:{\displaystyle Y}
3160:{\displaystyle D}
3140:{\displaystyle Y}
3081:{\displaystyle G}
3064:is then called a
3057:{\displaystyle F}
3037:{\displaystyle C}
2873:{\displaystyle G}
2705:{\displaystyle D}
2685:{\displaystyle Y}
2608:{\displaystyle D}
2559:{\displaystyle C}
2539:{\displaystyle X}
2519:{\displaystyle X}
2499:{\displaystyle F}
2475:{\displaystyle C}
2455:{\displaystyle X}
2233:with the functor
2221:actually exists.
1878:is how to turn a
1766:{\displaystyle U}
1746:{\displaystyle T}
1732:adjoint operators
1636:{\displaystyle G}
1579:{\displaystyle F}
1559:{\displaystyle F}
1539:{\displaystyle F}
1516:{\displaystyle F}
1496:{\displaystyle G}
1476:{\displaystyle F}
1452:{\displaystyle G}
1432:{\displaystyle G}
1408:{\displaystyle F}
1388:{\displaystyle f}
1286:{\displaystyle f}
1265:{\displaystyle G}
1245:{\displaystyle F}
919:{\displaystyle F}
906:right adjoint to
893:{\displaystyle G}
872:{\displaystyle G}
846:{\displaystyle F}
799:{\displaystyle Y}
623:{\displaystyle X}
596:
523:
457:{\displaystyle Y}
437:{\displaystyle X}
283:{\displaystyle Y}
239:{\displaystyle X}
83:topological space
16:(Redirected from
16915:
16908:Adjoint functors
16803:
16796:
16789:
16780:
16779:
16766:
16765:
16756:
16755:
16746:
16745:
16581:
16580:
16559:Simplex category
16534:Categorification
16525:
16524:
16506:
16505:
16499:
16469:Product category
16454:Kleisli category
16449:Functor category
16294:Terminal objects
16282:
16281:
16217:Adjoint functors
16170:
16169:
16159:
16158:
16144:
16137:
16130:
16121:
16120:
16080:
16067:
16031:
16011:
15990:
15970:
15964:
15963:
15952:
15946:
15928:
15922:
15921:
15911:
15891:
15882:
15876:
15875:
15873:
15861:
15845:Kleisli category
15808:is given by ΞΌ =
15804:
15802:
15801:
15796:
15787:
15786:
15758:
15756:
15755:
15750:
15742:
15741:
15740:
15703:
15701:
15700:
15695:
15693:
15692:
15683:
15682:
15611:(i.e. a functor
15562:full subcategory
15393:
15391:
15390:
15385:
15365:
15364:
15363:
15357:
15324:
15323:
15322:
15316:
15301:
15300:
15258:additive functor
15157:small categories
15140:
15138:
15137:
15132:
15130:
15123:
15122:
15121:
15111:
15101:
15093:
15087:
15072:
15067:
15059:
15047:
15043:
15029:
15021:
15018:
15007:
14995:
14993:
14985:
14977:
14974:
14962:
14960:
14959:
14958:
14947:
14922:
14920:
14919:
14914:
14891:
14870:
14868:
14867:
14862:
14848:
14784:
14782:
14781:
14776:
14764:
14762:
14761:
14756:
14741:
14739:
14738:
14733:
14731:
14721:
14720:
14705:
14704:
14676:
14636:
14479:
14477:
14476:
14471:
14412:
14348:
14240:
14238:
14237:
14232:
14230:
14229:
14211:
14210:
14192:
14191:
14139:
14137:
14136:
14131:
14126:
14125:
14107:
14106:
14091:
14090:
14002:
14000:
13999:
13994:
13992:
13973:
13972:
13950:
13942:
13923:
13922:
13886:
13885:
13805:
13803:
13802:
13797:
13774:
13773:
13768:
13753:
13752:
13726: for each
13725:
13723:
13722:
13717:
13694:
13693:
13688:
13670:
13669:
13654:
13653:
13632:
13631:
13612:
13610:
13609:
13604:
13581:
13580:
13575:
13560:
13559:
13533: for each
13532:
13530:
13529:
13524:
13501:
13500:
13495:
13477:
13476:
13460:
13452:
13431:
13430:
13406:
13404:
13403:
13398:
13304:
13302:
13301:
13296:
13294:
13278:
13277:
13259:
13249:
13248:
13230:
13229:
13208:
13204:
13203:
13170:
13169:
13115:
13086:
13084:
13083:
13078:
13076:
13066:
13065:
13041:
13034:
13033:
13015:
13014:
12990:
12983:
12982:
12946:
12945:
12894:
12870:
12868:
12867:
12862:
12860:
12841:
12840:
12819:
12818:
12799:
12798:
12762:
12761:
12680:
12678:
12677:
12672:
12613:
12589:
12574:
12562:
12551:
12530:
12515:
12504:
12493:
12487:
12379:
12304:For each object
12255:
12193:
12087:For each object
12023:initial morphism
11978:
11976:
11975:
11970:
11968:
11964:
11963:
11945:
11944:
11922:
11921:
11902:
11901:
11883:
11882:
11863:
11862:
11831:
11829:
11828:
11823:
11821:
11790:
11789:
11784:
11763:
11762:
11743:
11742:
11727:
11726:
11677:
11676:
11671:
11650:
11649:
11630:
11629:
11613:
11605:
11559:
11558:
11553:
11532:
11531:
11491:
11490:
11441:
11440:
11435:
11399:
11398:
11372:
11364:
11063:
11061:
11060:
11055:
11043:
11041:
11040:
11035:
11033:
11021:
11019:
11018:
11013:
11011:
10999:
10997:
10996:
10991:
10983:
10971:
10969:
10968:
10963:
10952:
10938:
10926:
10924:
10923:
10918:
10916:
10915:
10880:
10878:
10877:
10872:
10870:
10869:
10820:
10818:
10817:
10812:
10807:
10783:
10781:
10780:
10775:
10739:
10737:
10736:
10731:
10726:
10708:
10706:
10705:
10700:
10698:
10686:
10684:
10683:
10678:
10673:
10652:
10650:
10649:
10644:
10599:
10597:
10596:
10591:
10579:
10577:
10576:
10571:
10535:
10533:
10532:
10527:
10525:
10486:
10484:
10483:
10478:
10466:
10464:
10463:
10458:
10446:
10444:
10443:
10438:
10426:
10424:
10423:
10418:
10407:with respect to
10406:
10404:
10403:
10398:
10380:
10378:
10377:
10372:
10360:
10358:
10357:
10352:
10340:
10338:
10337:
10332:
10327:
10326:
10306:
10304:
10303:
10298:
10258:
10257:
10212:
10211:
10190:
10188:
10187:
10182:
10168:
10167:
10140:
10139:
10094:
10092:
10091:
10086:
10031:
10030:
9985:
9984:
9966:
9964:
9963:
9958:
9937:
9935:
9934:
9929:
9911:
9909:
9908:
9903:
9880:
9879:
9860:
9858:
9857:
9852:
9840:
9838:
9837:
9832:
9830:
9829:
9813:
9811:
9810:
9805:
9793:
9791:
9790:
9785:
9783:
9782:
9766:
9764:
9763:
9758:
9731:
9729:
9728:
9723:
9712:
9711:
9686:
9684:
9683:
9678:
9651:
9649:
9648:
9643:
9632:
9631:
9606:
9604:
9603:
9598:
9580:
9578:
9577:
9572:
9561:
9560:
9527:
9526:
9501:
9499:
9498:
9493:
9459:
9457:
9456:
9451:
9440:
9439:
9409:
9408:
9386:
9384:
9383:
9378:
9367:
9366:
9348:
9328:
9327:
9315:
9301:
9299:
9298:
9293:
9272:
9270:
9269:
9264:
9247:
9246:
9227:
9225:
9224:
9219:
9207:
9205:
9204:
9199:
9187:
9185:
9184:
9179:
9167:
9165:
9164:
9159:
9147:
9145:
9144:
9139:
9127:
9125:
9124:
9119:
9107:
9105:
9104:
9099:
9094:
9093:
9075:
9074:
9058:
9056:
9055:
9050:
9030:
9028:
9027:
9022:
9010:
9008:
9007:
9002:
8997:
8996:
8977:
8975:
8974:
8969:
8957:
8955:
8954:
8949:
8937:
8935:
8934:
8929:
8911:
8909:
8908:
8903:
8892:
8889:
8880:
8878:
8877:
8872:
8861:
8858:
8849:
8847:
8846:
8841:
8839:
8838:
8822:
8820:
8819:
8814:
8812:
8811:
8795:
8793:
8792:
8787:
8782:
8781:
8762:
8760:
8759:
8754:
8736:
8734:
8733:
8728:
8716:
8714:
8713:
8708:
8688:
8686:
8685:
8680:
8669:
8666:
8652:
8649:
8644:
8643:
8625:
8623:
8622:
8617:
8593:
8591:
8590:
8585:
8565:, a subfield of
8560:
8558:
8557:
8552:
8540:
8538:
8537:
8532:
8520:
8518:
8517:
8512:
8510:
8509:
8493:
8491:
8490:
8485:
8483:
8482:
8466:
8464:
8463:
8458:
8446:
8444:
8443:
8438:
8426:
8424:
8423:
8418:
8407:of all elements
8402:
8400:
8399:
8394:
8380:
8379:
8352:
8351:
8308:
8306:
8305:
8300:
8288:
8286:
8285:
8280:
8268:
8266:
8265:
8260:
8248:
8246:
8245:
8240:
8238:
8237:
8212:
8210:
8209:
8204:
8178:
8177:
8156:
8155:
8139:
8137:
8136:
8131:
8119:
8117:
8116:
8111:
8099:
8097:
8096:
8091:
8073:
8071:
8070:
8065:
8051:
8050:
8019:
8017:
8016:
8011:
8009:
8008:
7964:the endofunctor
7652:coherent sheaves
7525:Hausdorff spaces
7469:homotopy classes
7467:, the space of
7444:trivial topology
7369:negative numbers
7271:Polynomial rings
7025:Ring extensions.
7016:
7014:
7013:
7008:
6940:Further examples
6517:diagonal functor
6398:
6396:
6395:
6390:
6385:
6384:
6354:
6352:
6351:
6346:
6344:
6343:
6312:
6310:
6309:
6304:
6296:
6289:
6288:
6268:
6254:
6250:
6249:
6232:
6214:the composition
6209:
6207:
6206:
6201:
6181:
6180:
6142:
6140:
6139:
6134:
6132:
6131:
6088:
6086:
6085:
6080:
6075:
6074:
6041:
6039:
6038:
6033:
6025:
6021:
6020:
6003:
5989:
5982:
5981:
5961:
5944:the composition
5939:
5937:
5936:
5931:
5911:
5910:
5888:
5886:
5885:
5880:
5783:
5781:
5780:
5775:
5755:
5754:
5722:
5720:
5719:
5714:
5709:
5708:
5672:
5670:
5669:
5664:
5644:
5643:
5590:
5588:
5587:
5582:
5562:
5561:
5525:
5523:
5522:
5517:
5497:
5496:
5330:
5328:
5327:
5322:
5305:was the functor
5221:
5219:
5218:
5213:
5162:
5160:
5159:
5154:
5152:
5151:
5128:
5126:
5125:
5120:
5118:
5117:
5101:
5099:
5098:
5093:
5091:
5090:
5077:
5075:
5074:
5069:
5067:
5066:
5065:
5044:
5042:
5041:
5036:
5034:
5030:
5029:
5011:
5010:
4988:
4987:
4968:
4967:
4949:
4948:
4929:
4928:
4883:
4881:
4880:
4875:
4873:
4847:
4846:
4811:
4810:
4773:
4771:
4770:
4765:
4747:
4745:
4744:
4739:
4657:
4655:
4654:
4649:
4644:
4629:
4618:
4603:
4590:
4588:
4587:
4582:
4577:
4562:
4551:
4536:
4506:
4504:
4503:
4498:
4496:
4483:
4482:
4481:
4457:
4456:
4455:
4383:consists of two
4334:
4330:
4271:
4250:
4228:category of sets
4221:
4207:
4193:
4130:for all objects
4126:
4124:
4123:
4118:
4098:
4097:
4092:
4059:
4058:
4053:
4038:
4037:
4008:
4006:
4005:
4000:
3980:
3979:
3974:
3941:
3940:
3935:
3904:
3876:consists of two
3848:
3846:
3845:
3840:
3828:
3826:
3825:
3820:
3808:
3806:
3805:
3800:
3788:
3786:
3785:
3780:
3765:
3763:
3762:
3757:
3741:
3739:
3738:
3733:
3721:
3719:
3718:
3713:
3701:
3699:
3698:
3693:
3691:
3664:
3662:
3661:
3656:
3648:
3647:
3646:
3630:
3629:
3589:
3587:
3586:
3581:
3557:
3555:
3554:
3549:
3526:
3524:
3523:
3518:
3510:
3509:
3478:
3476:
3475:
3470:
3437:
3435:
3434:
3429:
3396:
3394:
3393:
3388:
3376:
3374:
3373:
3368:
3356:
3354:
3353:
3348:
3316:
3315:
3299:
3297:
3296:
3291:
3279:
3277:
3276:
3271:
3250:
3248:
3247:
3242:
3230:
3228:
3227:
3222:
3210:
3208:
3207:
3202:
3190:
3188:
3187:
3182:
3166:
3164:
3163:
3158:
3146:
3144:
3143:
3138:
3122:
3120:
3119:
3114:
3087:
3085:
3084:
3079:
3063:
3061:
3060:
3055:
3043:
3041:
3040:
3035:
3023:
3021:
3020:
3015:
3007:
2986:
2984:
2983:
2978:
2976:
2975:
2974:
2928:
2927:
2911:
2909:
2908:
2903:
2879:
2877:
2876:
2871:
2841:
2839:
2838:
2833:
2810:
2809:
2793:
2791:
2790:
2785:
2752:
2750:
2749:
2744:
2711:
2709:
2708:
2703:
2691:
2689:
2688:
2683:
2671:
2669:
2668:
2663:
2631:
2630:
2614:
2612:
2611:
2606:
2594:
2592:
2591:
2586:
2565:
2563:
2562:
2557:
2545:
2543:
2542:
2537:
2525:
2523:
2522:
2517:
2505:
2503:
2502:
2497:
2481:
2479:
2478:
2473:
2461:
2459:
2458:
2453:
2437:
2435:
2434:
2429:
2244:The notion that
2105:to our problem:
1952:. The category
1917:if it defines a
1849:
1828:
1826:
1825:
1820:
1772:
1770:
1769:
1764:
1752:
1750:
1749:
1744:
1722:
1720:
1719:
1714:
1691:, we also write
1679:
1677:
1676:
1671:
1669:
1668:
1667:
1661:
1642:
1640:
1639:
1634:
1622:
1620:
1619:
1614:
1612:
1611:
1610:
1604:
1585:
1583:
1582:
1577:
1565:
1563:
1562:
1557:
1545:
1543:
1542:
1537:
1522:
1520:
1519:
1514:
1502:
1500:
1499:
1494:
1482:
1480:
1479:
1474:
1458:
1456:
1455:
1450:
1438:
1436:
1435:
1430:
1414:
1412:
1411:
1406:
1394:
1392:
1391:
1386:
1370:
1368:
1367:
1362:
1347:
1345:
1344:
1339:
1319:
1318:
1317:
1311:
1292:
1290:
1289:
1284:
1271:
1269:
1268:
1263:
1251:
1249:
1248:
1243:
1225:
1223:
1222:
1217:
1202:
1200:
1199:
1194:
1174:
1173:
1172:
1166:
1133:
1132:
1131:
1125:
1110:
1109:
1055:
1053:
1052:
1047:
1045:
1044:
1031:
1029:
1028:
1023:
1021:
1020:
1003:
1001:
1000:
995:
993:
992:
979:
977:
976:
971:
969:
968:
952:
950:
949:
944:
925:
923:
922:
917:
899:
897:
896:
891:
878:
876:
875:
870:
859:left adjoint to
852:
850:
849:
844:
829:
827:
826:
821:
819:
818:
805:
803:
802:
797:
785:
783:
782:
777:
775:
761:
760:
733:
732:
719:
717:
716:
711:
709:
695:
694:
667:
666:
653:
651:
650:
645:
643:
642:
629:
627:
626:
621:
609:
607:
606:
601:
599:
598:
597:
594:
578:
577:
550:
549:
536:
534:
533:
528:
526:
525:
524:
521:
505:
504:
477:
476:
463:
461:
460:
455:
443:
441:
440:
435:
416:
414:
413:
408:
388:
387:
386:
380:
347:
346:
345:
339:
313:
311:
310:
305:
303:
302:
289:
287:
286:
281:
269:
267:
266:
261:
259:
258:
245:
243:
242:
237:
222:
220:
219:
214:
212:
211:
202:
201:
182:
180:
179:
174:
172:
171:
162:
161:
135:
133:
132:
127:
125:
124:
111:
109:
108:
103:
101:
100:
61:, one being the
59:adjoint functors
36:Adjunction space
21:
16923:
16922:
16918:
16917:
16916:
16914:
16913:
16912:
16898:
16897:
16896:
16891:
16813:
16807:
16777:
16772:
16726:
16696:
16663:
16640:
16631:
16588:
16572:
16523:
16513:
16500:
16483:
16432:
16370:
16339:Initial objects
16325:
16271:
16164:
16153:
16151:Category theory
16148:
16090:of The Catsters
16078:
16074:
16056:
16020:
16009:
15999:
15994:
15993:
15971:
15967:
15954:
15953:
15949:
15929:
15925:
15909:10.2307/1993102
15889:
15883:
15879:
15862:
15858:
15853:
15782:
15778:
15770:
15767:
15766:
15736:
15735:
15731:
15723:
15720:
15719:
15688:
15687:
15678:
15677:
15669:
15666:
15665:
15639:
15624:
15603:
15596:
15581:
15559:
15552:
15534:
15499:
15439:
15434:
15359:
15358:
15347:
15346:
15318:
15317:
15306:
15305:
15290:
15286:
15284:
15281:
15280:
15230:
15165:
15128:
15127:
15117:
15116:
15112:
15100:
15080:
15071:
15060:
15052:
15045:
15044:
15036:
15022:
15011:
15000:
14994:
14986:
14978:
14967:
14961:
14954:
14953:
14949:
14944:
14942:
14939:
14938:
14884:
14882:
14879:
14878:
14841:
14833:
14830:
14829:
14791:
14770:
14767:
14766:
14750:
14747:
14746:
14729:
14728:
14713:
14709:
14697:
14693:
14677:
14669:
14666:
14665:
14637:
14629:
14625:
14623:
14620:
14619:
14548:Conversely, if
14513:If the functor
14511:
14453:
14450:
14449:
14428:
14410:
14401:
14346:
14333:
14316:
14259:
14253:
14248:
14222:
14218:
14206:
14202:
14184:
14180:
14178:
14175:
14174:
14166:
14160:
14154:
14121:
14117:
14099:
14095:
14083:
14079:
14077:
14074:
14073:
14063:
14057:
14047:
13990:
13989:
13968:
13964:
13943:
13932:
13925:
13924:
13918:
13914:
13875:
13871:
13867:
13865:
13862:
13861:
13840:
13818:
13769:
13758:
13757:
13745:
13741:
13739:
13736:
13735:
13689:
13678:
13677:
13662:
13658:
13640:
13636:
13627:
13623:
13621:
13618:
13617:
13576:
13565:
13564:
13552:
13548:
13546:
13543:
13542:
13496:
13485:
13484:
13469:
13465:
13453:
13439:
13426:
13422:
13420:
13417:
13416:
13368:
13365:
13364:
13355:
13345:
13316:Given functors
13314:
13292:
13291:
13270:
13266:
13257:
13256:
13241:
13237:
13225:
13221:
13206:
13205:
13199:
13195:
13165:
13161:
13148:
13135:
13133:
13130:
13129:
13123:
13113:
13103:
13074:
13073:
13058:
13054:
13039:
13038:
13029:
13025:
13007:
13003:
12988:
12987:
12978:
12974:
12941:
12937:
12930:
12917:
12915:
12912:
12911:
12905:
12893:
12887:
12858:
12857:
12836:
12832:
12808:
12804:
12801:
12800:
12794:
12790:
12751:
12747:
12743:
12741:
12738:
12737:
12699:
12689:
12666:
12663:
12662:
12634:Given functors
12632:
12607:
12606:
12599:
12573:
12572:
12541:
12540:
12482:
12481:
12474:
12466:
12459:
12448:
12441:
12426:
12419:
12408:
12401:
12387:
12353:
12343:
12333:
12296:
12286:
12250:
12237:
12226:
12207:
12206:
12192:
12191:
12183:
12182:
12174:
12163:
12156:
12122:
12108:
12054:
12020:
11992:
11966:
11965:
11956:
11952:
11940:
11936:
11923:
11914:
11910:
11907:
11906:
11897:
11893:
11875:
11871:
11864:
11855:
11851:
11847:
11845:
11842:
11841:
11819:
11818:
11785:
11774:
11773:
11769:
11764:
11758:
11754:
11747:
11735:
11731:
11713:
11709:
11706:
11705:
11672:
11661:
11660:
11656:
11651:
11645:
11641:
11634:
11622:
11618:
11606:
11592:
11585:
11584:
11554:
11543:
11542:
11538:
11533:
11527:
11523:
11501:
11480:
11476:
11467:
11466:
11436:
11425:
11424:
11420:
11415:
11394:
11390:
11383:
11365:
11354:
11340:
11338:
11335:
11334:
11291:
11245:
11182:
11170:
11149:
11139:
11070:
11049:
11046:
11045:
11029:
11027:
11024:
11023:
11007:
11005:
11002:
11001:
10979:
10977:
10974:
10973:
10948:
10934:
10932:
10929:
10928:
10911:
10907:
10893:
10890:
10889:
10862:
10858:
10826:
10823:
10822:
10803:
10789:
10786:
10785:
10745:
10742:
10741:
10722:
10714:
10711:
10710:
10694:
10692:
10689:
10688:
10669:
10661:
10658:
10657:
10605:
10602:
10601:
10585:
10582:
10581:
10541:
10538:
10537:
10521:
10519:
10516:
10515:
10505:
10472:
10469:
10468:
10467:is replaced by
10452:
10449:
10448:
10432:
10429:
10428:
10412:
10409:
10408:
10386:
10383:
10382:
10366:
10363:
10362:
10346:
10343:
10342:
10322:
10318:
10316:
10313:
10312:
10250:
10246:
10207:
10203:
10201:
10198:
10197:
10163:
10159:
10135:
10131:
10104:
10101:
10100:
10023:
10019:
9980:
9976:
9974:
9971:
9970:
9943:
9940:
9939:
9917:
9914:
9913:
9872:
9868:
9866:
9863:
9862:
9846:
9843:
9842:
9825:
9821:
9819:
9816:
9815:
9799:
9796:
9795:
9778:
9774:
9772:
9769:
9768:
9737:
9734:
9733:
9704:
9700:
9692:
9689:
9688:
9657:
9654:
9653:
9652:, then clearly
9624:
9620:
9612:
9609:
9608:
9586:
9583:
9582:
9553:
9549:
9519:
9515:
9507:
9504:
9503:
9472:
9469:
9468:
9432:
9428:
9404:
9400:
9398:
9395:
9394:
9362:
9358:
9344:
9323:
9319:
9311:
9309:
9306:
9305:
9281:
9278:
9277:
9239:
9235:
9233:
9230:
9229:
9213:
9210:
9209:
9193:
9190:
9189:
9173:
9170:
9169:
9153:
9150:
9149:
9133:
9130:
9129:
9113:
9110:
9109:
9089:
9085:
9070:
9066:
9064:
9061:
9060:
9044:
9041:
9040:
9016:
9013:
9012:
8992:
8988:
8983:
8980:
8979:
8963:
8960:
8959:
8943:
8940:
8939:
8917:
8914:
8913:
8888:
8886:
8883:
8882:
8857:
8855:
8852:
8851:
8834:
8830:
8828:
8825:
8824:
8807:
8803:
8801:
8798:
8797:
8777:
8773:
8768:
8765:
8764:
8742:
8739:
8738:
8722:
8719:
8718:
8702:
8699:
8698:
8665:
8648:
8639:
8635:
8633:
8630:
8629:
8599:
8596:
8595:
8579:
8576:
8575:
8546:
8543:
8542:
8526:
8523:
8522:
8505:
8501:
8499:
8496:
8495:
8478:
8474:
8472:
8469:
8468:
8467:to which it is
8452:
8449:
8448:
8432:
8429:
8428:
8412:
8409:
8408:
8375:
8371:
8347:
8343:
8316:
8313:
8312:
8294:
8291:
8290:
8274:
8271:
8270:
8254:
8251:
8250:
8233:
8229:
8227:
8224:
8223:
8173:
8169:
8151:
8147:
8145:
8142:
8141:
8125:
8122:
8121:
8105:
8102:
8101:
8079:
8076:
8075:
8046:
8042:
8025:
8022:
8021:
8004:
8000:
7998:
7995:
7994:
7991:Quantification.
7987:
7933:
7882:
7880:Category theory
7813:is a subset of
7759:William Lawvere
7697:if and only if
7683:
7660:The article on
7658:Soberification.
7649:
7602:
7508:homotopy theory
7430:, creating the
7404:
7293:polynomial ring
7286:
7279:
7262:
7251:
7177:
7112:Tensor products
7002:
6999:
6998:
6969:and defining a
6947:
6942:
6881:
6874:
6831:
6824:
6776:
6711:
6703:
6696:
6690:
6681:
6672:
6663:
6656:
6645:
6638:
6631:
6624:
6617:
6610:
6562:
6555:
6548:
6541:
6507:
6500:
6492:
6486:
6451:fibred products
6444:
6425:
6380:
6376:
6368:
6365:
6364:
6336:
6332:
6330:
6327:
6326:
6284:
6280:
6267:
6242:
6238:
6231:
6223:
6220:
6219:
6176:
6172:
6170:
6167:
6166:
6124:
6120:
6118:
6115:
6114:
6070:
6066:
6058:
6055:
6054:
6013:
6009:
6002:
5977:
5973:
5960:
5952:
5949:
5948:
5906:
5902:
5900:
5897:
5896:
5850:
5847:
5846:
5750:
5746:
5744:
5741:
5740:
5704:
5700:
5686:
5683:
5682:
5639:
5635:
5633:
5630:
5629:
5612:For each group
5557:
5553:
5551:
5548:
5547:
5492:
5488:
5486:
5483:
5482:
5394:
5389:
5331:(i.e. take the
5310:
5307:
5306:
5255:
5195:
5192:
5191:
5188:string diagrams
5144:
5140:
5138:
5135:
5134:
5133:to itself, and
5113:
5109:
5107:
5104:
5103:
5086:
5085:
5083:
5080:
5079:
5061:
5060:
5056:
5054:
5051:
5050:
5032:
5031:
5022:
5018:
5006:
5002:
4989:
4980:
4976:
4973:
4972:
4963:
4959:
4941:
4937:
4930:
4921:
4917:
4913:
4911:
4908:
4907:
4871:
4870:
4848:
4842:
4838:
4835:
4834:
4812:
4806:
4802:
4798:
4796:
4793:
4792:
4753:
4750:
4749:
4709:
4706:
4705:
4672:
4666:
4628:
4602:
4597:
4594:
4593:
4561:
4535:
4530:
4527:
4526:
4494:
4493:
4477:
4476:
4472:
4465:
4459:
4458:
4451:
4450:
4446:
4430:
4423:
4421:
4418:
4417:
4369:
4346:
4314:
4302:
4287:Naturality of Ξ¦
4256:
4238:
4213:
4201:
4195:
4187:
4181:
4093:
4082:
4081:
4054:
4043:
4042:
4027:
4023:
4021:
4018:
4017:
3975:
3964:
3963:
3936:
3925:
3924:
3916:
3913:
3912:
3892:
3859:
3834:
3831:
3830:
3814:
3811:
3810:
3809:if and only if
3794:
3791:
3790:
3774:
3771:
3770:
3751:
3748:
3747:
3727:
3724:
3723:
3707:
3704:
3703:
3684:
3670:
3667:
3666:
3639:
3638:
3634:
3625:
3621:
3595:
3592:
3591:
3563:
3560:
3559:
3543:
3540:
3539:
3505:
3501:
3484:
3481:
3480:
3443:
3440:
3439:
3402:
3399:
3398:
3382:
3379:
3378:
3362:
3359:
3358:
3311:
3307:
3305:
3302:
3301:
3300:and a morphism
3285:
3282:
3281:
3256:
3253:
3252:
3236:
3233:
3232:
3216:
3213:
3212:
3196:
3193:
3192:
3176:
3173:
3172:
3152:
3149:
3148:
3132:
3129:
3128:
3096:
3093:
3092:
3073:
3070:
3069:
3049:
3046:
3045:
3029:
3026:
3025:
3000:
2992:
2989:
2988:
2967:
2966:
2962:
2923:
2919:
2917:
2914:
2913:
2885:
2882:
2881:
2865:
2862:
2861:
2805:
2801:
2799:
2796:
2795:
2758:
2755:
2754:
2717:
2714:
2713:
2697:
2694:
2693:
2677:
2674:
2673:
2626:
2622:
2620:
2617:
2616:
2615:and a morphism
2600:
2597:
2596:
2571:
2568:
2567:
2551:
2548:
2547:
2531:
2528:
2527:
2511:
2508:
2507:
2491:
2488:
2487:
2482:there exists a
2467:
2464:
2463:
2447:
2444:
2443:
2411:
2408:
2407:
2404:
2321:
2293:
2227:
2156:adjoint functor
2118:
2111:
2104:
2097:
2090:
2079:
2060:
2056:
2052:
2045:
2038:
2027:
2012:
2001:
1864:
1856:colimits/limits
1850:
1842:
1835:
1778:
1775:
1774:
1758:
1755:
1754:
1738:
1735:
1734:
1699:
1696:
1695:
1663:
1662:
1651:
1650:
1648:
1645:
1644:
1628:
1625:
1624:
1606:
1605:
1594:
1593:
1591:
1588:
1587:
1571:
1568:
1567:
1551:
1548:
1547:
1531:
1528:
1527:
1508:
1505:
1504:
1488:
1485:
1484:
1468:
1465:
1464:
1444:
1441:
1440:
1424:
1421:
1420:
1400:
1397:
1396:
1380:
1377:
1376:
1353:
1350:
1349:
1313:
1312:
1301:
1300:
1298:
1295:
1294:
1293:is an arrow in
1278:
1275:
1274:
1257:
1254:
1253:
1237:
1234:
1233:
1211:
1208:
1207:
1168:
1167:
1156:
1155:
1127:
1126:
1115:
1114:
1102:
1098:
1096:
1093:
1092:
1062:
1040:
1039:
1037:
1034:
1033:
1016:
1015:
1013:
1010:
1009:
988:
987:
985:
982:
981:
964:
963:
961:
958:
957:
932:
929:
928:
911:
908:
907:
885:
882:
881:
864:
861:
860:
838:
835:
834:
814:
813:
811:
808:
807:
791:
788:
787:
765:
756:
755:
728:
727:
725:
722:
721:
699:
690:
689:
662:
661:
659:
656:
655:
638:
637:
635:
632:
631:
615:
612:
611:
593:
589:
582:
573:
572:
545:
544:
542:
539:
538:
520:
516:
509:
500:
499:
472:
471:
469:
466:
465:
449:
446:
445:
429:
426:
425:
382:
381:
370:
369:
341:
340:
329:
328:
326:
323:
322:
298:
297:
295:
292:
291:
275:
272:
271:
254:
253:
251:
248:
247:
231:
228:
227:
207:
206:
197:
196:
188:
185:
184:
167:
166:
157:
156:
148:
145:
144:
120:
119:
117:
114:
113:
96:
95:
93:
90:
89:
47:category theory
45:, specifically
39:
28:
23:
22:
18:Adjoint functor
15:
12:
11:
5:
16921:
16911:
16910:
16893:
16892:
16890:
16889:
16884:
16879:
16874:
16869:
16864:
16859:
16854:
16849:
16844:
16839:
16834:
16829:
16824:
16818:
16815:
16814:
16806:
16805:
16798:
16791:
16783:
16774:
16773:
16771:
16770:
16760:
16750:
16739:
16736:
16735:
16732:
16731:
16728:
16727:
16725:
16724:
16719:
16714:
16700:
16694:
16689:
16684:
16678:
16676:
16669:
16668:
16665:
16664:
16662:
16661:
16656:
16645:
16643:
16638:
16633:
16632:
16630:
16629:
16624:
16619:
16614:
16609:
16604:
16593:
16591:
16586:
16578:
16574:
16573:
16571:
16566:
16564:String diagram
16561:
16556:
16554:Model category
16551:
16546:
16541:
16536:
16531:
16529:
16522:
16521:
16518:
16515:
16514:
16502:
16501:
16494:
16492:
16489:
16488:
16485:
16484:
16482:
16481:
16476:
16474:Comma category
16471:
16466:
16461:
16456:
16451:
16446:
16440:
16438:
16434:
16433:
16431:
16430:
16420:
16410:
16408:Abelian groups
16405:
16400:
16395:
16390:
16384:
16382:
16376:
16375:
16372:
16371:
16369:
16368:
16363:
16358:
16357:
16356:
16346:
16341:
16335:
16333:
16327:
16326:
16324:
16323:
16318:
16313:
16312:
16311:
16301:
16296:
16290:
16288:
16279:
16273:
16272:
16270:
16269:
16264:
16259:
16254:
16249:
16244:
16239:
16234:
16229:
16224:
16219:
16214:
16213:
16212:
16207:
16202:
16197:
16192:
16187:
16176:
16174:
16166:
16165:
16155:
16154:
16147:
16146:
16139:
16132:
16124:
16118:
16117:
16104:, categories,
16091:
16073:
16072:External links
16070:
16069:
16068:
16054:
16032:
16018:
15998:
15995:
15992:
15991:
15965:
15947:
15923:
15902:(2): 294β329.
15877:
15855:
15854:
15852:
15849:
15806:
15805:
15793:
15790:
15785:
15781:
15777:
15774:
15760:
15759:
15748:
15745:
15739:
15734:
15730:
15727:
15705:
15704:
15691:
15686:
15681:
15676:
15673:
15661:. The functor
15638:
15635:
15620:
15601:
15594:
15577:
15557:
15548:
15532:
15498:
15495:
15438:
15435:
15433:
15430:
15395:
15394:
15383:
15380:
15377:
15374:
15371:
15368:
15362:
15356:
15353:
15350:
15345:
15342:
15339:
15336:
15333:
15330:
15327:
15321:
15315:
15312:
15309:
15304:
15299:
15296:
15293:
15289:
15229:
15226:
15225:
15224:
15217:
15210:
15203:
15164:
15161:
15142:
15141:
15126:
15120:
15115:
15108:
15104:
15099:
15096:
15090:
15086:
15083:
15079:
15075:
15070:
15066:
15063:
15058:
15055:
15051:
15048:
15046:
15042:
15039:
15035:
15032:
15028:
15025:
15017:
15014:
15010:
15006:
15003:
14998:
14992:
14989:
14984:
14981:
14973:
14970:
14965:
14957:
14952:
14948:
14946:
14924:
14923:
14912:
14909:
14906:
14903:
14900:
14897:
14894:
14890:
14887:
14872:
14871:
14860:
14857:
14854:
14851:
14847:
14844:
14840:
14837:
14790:
14787:
14774:
14754:
14743:
14742:
14727:
14724:
14719:
14716:
14712:
14708:
14703:
14700:
14696:
14692:
14689:
14686:
14683:
14680:
14678:
14675:
14672:
14668:
14667:
14664:
14661:
14658:
14655:
14652:
14649:
14646:
14643:
14640:
14638:
14635:
14632:
14628:
14627:
14605:
14604:
14593:
14510:
14507:
14506:
14505:
14492:
14469:
14466:
14463:
14460:
14457:
14439:
14438:
14424:
14403:
14402:
14397:
14376:
14375:
14336:
14335:
14329:
14312:
14287:Peter J. Freyd
14252:
14249:
14247:
14244:
14243:
14242:
14228:
14225:
14221:
14217:
14214:
14209:
14205:
14201:
14198:
14195:
14190:
14187:
14183:
14172:
14162:
14156:
14152:
14141:
14129:
14124:
14120:
14116:
14113:
14110:
14105:
14102:
14098:
14094:
14089:
14086:
14082:
14070:
14069:
14059:
14049:
14043:
14031:
14030:
14003:
13988:
13985:
13982:
13979:
13976:
13971:
13967:
13963:
13960:
13957:
13954:
13949:
13946:
13941:
13938:
13935:
13931:
13927:
13926:
13921:
13917:
13913:
13910:
13907:
13904:
13901:
13898:
13895:
13892:
13889:
13884:
13881:
13878:
13874:
13870:
13869:
13858:
13857:
13854:
13836:
13814:
13807:
13795:
13792:
13789:
13786:
13783:
13780:
13777:
13772:
13767:
13764:
13761:
13756:
13751:
13748:
13744:
13734:, where
13715:
13712:
13709:
13706:
13703:
13700:
13697:
13692:
13687:
13684:
13681:
13676:
13673:
13668:
13665:
13661:
13657:
13652:
13649:
13646:
13643:
13639:
13635:
13630:
13626:
13614:
13602:
13599:
13596:
13593:
13590:
13587:
13584:
13579:
13574:
13571:
13568:
13563:
13558:
13555:
13551:
13541:, where
13522:
13519:
13516:
13513:
13510:
13507:
13504:
13499:
13494:
13491:
13488:
13483:
13480:
13475:
13472:
13468:
13464:
13459:
13456:
13451:
13448:
13445:
13442:
13438:
13434:
13429:
13425:
13409:
13408:
13396:
13393:
13390:
13387:
13384:
13381:
13378:
13375:
13372:
13351:
13341:
13313:
13310:
13309:
13308:
13305:
13290:
13287:
13284:
13281:
13276:
13273:
13269:
13265:
13262:
13260:
13258:
13255:
13252:
13247:
13244:
13240:
13236:
13233:
13228:
13224:
13220:
13217:
13214:
13211:
13209:
13207:
13202:
13198:
13194:
13191:
13188:
13185:
13182:
13179:
13176:
13173:
13168:
13164:
13160:
13157:
13154:
13151:
13149:
13147:
13144:
13141:
13138:
13137:
13126:
13125:
13119:
13109:
13099:
13091:
13090:
13087:
13072:
13069:
13064:
13061:
13057:
13053:
13050:
13047:
13044:
13042:
13040:
13037:
13032:
13028:
13024:
13021:
13018:
13013:
13010:
13006:
13002:
12999:
12996:
12993:
12991:
12989:
12986:
12981:
12977:
12973:
12970:
12967:
12964:
12961:
12958:
12955:
12952:
12949:
12944:
12940:
12936:
12933:
12931:
12929:
12926:
12923:
12920:
12919:
12908:
12907:
12901:
12889:
12883:
12875:
12874:
12871:
12856:
12853:
12850:
12847:
12844:
12839:
12835:
12831:
12828:
12825:
12822:
12817:
12814:
12811:
12807:
12803:
12802:
12797:
12793:
12789:
12786:
12783:
12780:
12777:
12774:
12771:
12768:
12765:
12760:
12757:
12754:
12750:
12746:
12745:
12734:
12733:
12695:
12685:
12670:
12631:
12628:
12623:
12622:
12621:
12620:
12604:
12597:
12593:
12570:
12566:
12538:
12534:
12479:
12472:
12468:
12464:
12457:
12446:
12439:
12424:
12417:
12413:, any objects
12406:
12399:
12389:
12383:
12345:
12335:
12329:
12312:, each object
12292:
12282:
12279:
12278:
12277:
12266:
12246:
12243:
12235:
12224:
12204:
12200:
12189:
12185:
12180:
12176:
12172:
12161:
12154:
12144:
12118:
12104:
12053:
12050:
12038:
12037:
12016:
12009:
11988:
11980:
11979:
11962:
11959:
11955:
11951:
11948:
11943:
11939:
11935:
11932:
11929:
11926:
11924:
11920:
11917:
11913:
11909:
11908:
11905:
11900:
11896:
11892:
11889:
11886:
11881:
11878:
11874:
11870:
11867:
11865:
11861:
11858:
11854:
11850:
11849:
11838:
11837:
11833:
11832:
11817:
11814:
11811:
11808:
11805:
11802:
11799:
11796:
11793:
11788:
11783:
11780:
11777:
11770:
11768:
11765:
11761:
11757:
11753:
11750:
11748:
11746:
11741:
11738:
11734:
11730:
11725:
11722:
11719:
11716:
11712:
11708:
11707:
11704:
11701:
11698:
11695:
11692:
11689:
11686:
11683:
11680:
11675:
11670:
11667:
11664:
11657:
11655:
11652:
11648:
11644:
11640:
11637:
11635:
11633:
11628:
11625:
11621:
11617:
11612:
11609:
11604:
11601:
11598:
11595:
11591:
11587:
11586:
11583:
11580:
11577:
11574:
11571:
11568:
11565:
11562:
11557:
11552:
11549:
11546:
11539:
11537:
11534:
11530:
11526:
11522:
11519:
11516:
11513:
11510:
11507:
11504:
11502:
11500:
11497:
11494:
11489:
11486:
11483:
11479:
11475:
11472:
11469:
11468:
11465:
11462:
11459:
11456:
11453:
11450:
11447:
11444:
11439:
11434:
11431:
11428:
11421:
11419:
11416:
11414:
11411:
11408:
11405:
11402:
11397:
11393:
11389:
11386:
11384:
11382:
11379:
11376:
11371:
11368:
11363:
11360:
11357:
11353:
11349:
11346:
11343:
11342:
11331:
11330:
11316:
11315:
11314:
11313:
11283:
11237:
11192:
11191:
11178:
11175:
11166:
11155:
11145:
11135:
11128:
11110:
11069:
11066:
11053:
11032:
11010:
10989:
10986:
10982:
10961:
10958:
10955:
10951:
10947:
10944:
10941:
10937:
10914:
10910:
10906:
10903:
10900:
10897:
10868:
10865:
10861:
10857:
10854:
10851:
10848:
10845:
10842:
10839:
10836:
10833:
10830:
10810:
10806:
10802:
10799:
10796:
10793:
10773:
10770:
10767:
10764:
10761:
10758:
10755:
10752:
10749:
10729:
10725:
10721:
10718:
10697:
10676:
10672:
10668:
10665:
10642:
10639:
10636:
10633:
10630:
10627:
10624:
10621:
10618:
10615:
10612:
10609:
10589:
10569:
10566:
10563:
10560:
10557:
10554:
10551:
10548:
10545:
10524:
10504:
10501:
10500:
10499:
10489:
10488:
10476:
10456:
10436:
10416:
10396:
10393:
10390:
10370:
10350:
10330:
10325:
10321:
10309:
10308:
10307:
10296:
10293:
10289:
10286:
10283:
10279:
10276:
10273:
10270:
10267:
10264:
10261:
10256:
10253:
10249:
10245:
10242:
10239:
10236:
10233:
10230:
10227:
10224:
10221:
10218:
10215:
10210:
10206:
10192:
10180:
10177:
10174:
10171:
10166:
10162:
10158:
10155:
10152:
10149:
10146:
10143:
10138:
10134:
10129:
10126:
10123:
10120:
10117:
10114:
10111:
10108:
10097:
10096:
10095:
10084:
10081:
10078:
10075:
10072:
10069:
10066:
10062:
10059:
10056:
10052:
10049:
10046:
10043:
10040:
10037:
10034:
10029:
10026:
10022:
10018:
10015:
10012:
10009:
10006:
10003:
10000:
9997:
9994:
9991:
9988:
9983:
9979:
9956:
9953:
9950:
9947:
9927:
9924:
9921:
9901:
9898:
9895:
9892:
9889:
9886:
9883:
9878:
9875:
9871:
9861:'s, such that
9850:
9828:
9824:
9803:
9781:
9777:
9756:
9753:
9750:
9747:
9744:
9741:
9721:
9718:
9715:
9710:
9707:
9703:
9699:
9696:
9676:
9673:
9670:
9667:
9664:
9661:
9641:
9638:
9635:
9630:
9627:
9623:
9619:
9616:
9596:
9593:
9590:
9570:
9567:
9564:
9559:
9556:
9552:
9548:
9545:
9542:
9539:
9536:
9533:
9530:
9525:
9522:
9518:
9514:
9511:
9491:
9488:
9485:
9482:
9479:
9476:
9464:
9463:
9462:
9461:
9449:
9446:
9443:
9438:
9435:
9431:
9427:
9424:
9421:
9418:
9415:
9412:
9407:
9403:
9389:
9388:
9387:
9376:
9373:
9370:
9365:
9361:
9357:
9354:
9351:
9347:
9343:
9340:
9337:
9334:
9331:
9326:
9322:
9318:
9314:
9291:
9288:
9285:
9274:
9262:
9259:
9256:
9253:
9250:
9245:
9242:
9238:
9217:
9197:
9177:
9157:
9137:
9117:
9097:
9092:
9088:
9084:
9081:
9078:
9073:
9069:
9048:
9033:
9032:
9020:
9000:
8995:
8991:
8987:
8967:
8947:
8927:
8924:
8921:
8901:
8898:
8895:
8870:
8867:
8864:
8837:
8833:
8810:
8806:
8785:
8780:
8776:
8772:
8752:
8749:
8746:
8726:
8706:
8691:
8690:
8689:
8678:
8675:
8672:
8664:
8661:
8658:
8655:
8647:
8642:
8638:
8615:
8612:
8609:
8606:
8603:
8583:
8571:
8570:
8550:
8530:
8508:
8504:
8481:
8477:
8456:
8436:
8416:
8405:
8404:
8403:
8392:
8389:
8386:
8383:
8378:
8374:
8370:
8367:
8364:
8361:
8358:
8355:
8350:
8346:
8341:
8338:
8335:
8332:
8329:
8326:
8323:
8320:
8298:
8278:
8258:
8236:
8232:
8215:
8214:
8202:
8199:
8196:
8193:
8190:
8187:
8184:
8181:
8176:
8172:
8168:
8165:
8162:
8159:
8154:
8150:
8129:
8109:
8089:
8086:
8083:
8063:
8060:
8057:
8054:
8049:
8045:
8041:
8038:
8035:
8032:
8029:
8007:
8003:
7986:
7983:
7982:
7981:
7955:
7931:
7925:
7881:
7878:
7877:
7876:
7871:is adjoint to
7857:ideal quotient
7842:
7755:
7744:
7682:
7679:
7678:
7677:
7655:
7647:
7600:
7579:continuous map
7572:
7511:
7451:
7432:discrete space
7403:
7400:
7399:
7398:
7384:
7349:vector bundles
7338:
7323:abelianization
7299:Abelianization
7296:
7284:
7277:
7268:
7260:
7249:
7240:
7218:group of units
7195:
7173:
7148:. The functor
7120:is a ring and
7108:
7057:tensor product
7022:
7006:
6993:disjoint union
6991:by taking the
6974:
6946:
6943:
6941:
6938:
6937:
6936:
6916:
6915:
6879:
6872:
6829:
6822:
6775:
6772:
6771:
6770:
6765:
6764:
6707:
6701:
6694:
6686:
6677:
6668:
6661:
6654:
6643:
6636:
6629:
6622:
6615:
6608:
6594:
6593:
6577:
6576:
6560:
6553:
6546:
6539:
6505:
6498:
6490:
6484:
6443:
6440:
6424:
6421:
6388:
6383:
6379:
6375:
6372:
6342:
6339:
6335:
6315:
6314:
6302:
6299:
6292:
6287:
6283:
6279:
6276:
6271:
6266:
6263:
6260:
6257:
6248:
6245:
6241:
6235:
6230:
6227:
6199:
6196:
6193:
6190:
6187:
6184:
6179:
6175:
6130:
6127:
6123:
6078:
6073:
6069:
6065:
6062:
6043:
6042:
6031:
6028:
6019:
6016:
6012:
6006:
6001:
5998:
5995:
5992:
5985:
5980:
5976:
5972:
5969:
5964:
5959:
5956:
5929:
5926:
5923:
5920:
5917:
5914:
5909:
5905:
5878:
5875:
5872:
5869:
5866:
5863:
5860:
5857:
5854:
5773:
5770:
5767:
5764:
5761:
5758:
5753:
5749:
5712:
5707:
5703:
5699:
5696:
5693:
5690:
5662:
5659:
5656:
5653:
5650:
5647:
5642:
5638:
5580:
5577:
5574:
5571:
5568:
5565:
5560:
5556:
5542:of some group
5515:
5512:
5509:
5506:
5503:
5500:
5495:
5491:
5393:
5390:
5388:
5385:
5333:tensor product
5320:
5317:
5314:
5299:abelian groups
5295:
5294:
5254:
5251:
5211:
5208:
5205:
5202:
5199:
5150:
5147:
5143:
5116:
5112:
5089:
5064:
5059:
5047:
5046:
5028:
5025:
5021:
5017:
5014:
5009:
5005:
5001:
4998:
4995:
4992:
4990:
4986:
4983:
4979:
4975:
4974:
4971:
4966:
4962:
4958:
4955:
4952:
4947:
4944:
4940:
4936:
4933:
4931:
4927:
4924:
4920:
4916:
4915:
4885:
4884:
4869:
4866:
4863:
4860:
4857:
4854:
4851:
4849:
4845:
4841:
4837:
4836:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4813:
4809:
4805:
4801:
4800:
4763:
4760:
4757:
4737:
4734:
4731:
4728:
4725:
4722:
4719:
4716:
4713:
4681:respectively.
4668:
4662:
4659:
4658:
4647:
4640:
4637:
4632:
4627:
4624:
4621:
4614:
4611:
4606:
4601:
4591:
4580:
4573:
4570:
4565:
4560:
4557:
4554:
4547:
4544:
4539:
4534:
4508:
4507:
4492:
4489:
4486:
4480:
4475:
4471:
4468:
4466:
4464:
4461:
4460:
4454:
4449:
4445:
4442:
4439:
4436:
4433:
4431:
4429:
4426:
4425:
4368:
4365:
4363:) is similar.
4344:
4323:) is given by
4312:
4300:
4197:
4183:
4128:
4127:
4116:
4113:
4110:
4107:
4104:
4101:
4096:
4091:
4088:
4085:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4057:
4052:
4049:
4046:
4041:
4036:
4033:
4030:
4026:
4011:
4010:
3998:
3995:
3992:
3989:
3986:
3983:
3978:
3973:
3970:
3967:
3962:
3959:
3956:
3953:
3950:
3947:
3944:
3939:
3934:
3931:
3928:
3923:
3920:
3858:
3855:
3838:
3818:
3798:
3778:
3755:
3731:
3711:
3702:a morphism in
3690:
3687:
3683:
3680:
3677:
3674:
3654:
3651:
3645:
3642:
3637:
3633:
3628:
3624:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3579:
3576:
3573:
3570:
3567:
3547:
3516:
3513:
3508:
3504:
3500:
3497:
3494:
3491:
3488:
3468:
3465:
3462:
3459:
3456:
3453:
3450:
3447:
3427:
3424:
3421:
3418:
3415:
3412:
3409:
3406:
3386:
3366:
3346:
3343:
3340:
3337:
3334:
3331:
3328:
3325:
3322:
3319:
3314:
3310:
3289:
3269:
3266:
3263:
3260:
3240:
3220:
3200:
3180:
3156:
3136:
3112:
3109:
3106:
3103:
3100:
3077:
3053:
3033:
3013:
3010:
3006:
3003:
2999:
2996:
2973:
2970:
2965:
2961:
2958:
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2926:
2922:
2901:
2898:
2895:
2892:
2889:
2869:
2831:
2828:
2825:
2822:
2819:
2816:
2813:
2808:
2804:
2783:
2780:
2777:
2774:
2771:
2768:
2765:
2762:
2742:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2701:
2681:
2661:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2629:
2625:
2604:
2584:
2581:
2578:
2575:
2555:
2535:
2515:
2495:
2471:
2451:
2427:
2424:
2421:
2418:
2415:
2403:
2400:
2366:, the letters
2320:
2317:
2312:
2311:
2308:
2301:
2300:optimizations.
2292:
2289:
2226:
2223:
2148:most efficient
2141:most efficient
2127:is initial in
2116:
2109:
2102:
2095:
2088:
2077:
2063:comma category
2058:
2054:
2050:
2043:
2036:
2025:
2010:
1999:
1946:most efficient
1934:initial object
1907:most efficient
1888:most efficient
1868:most efficient
1863:
1860:
1840:
1834:
1831:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1762:
1742:
1724:
1723:
1712:
1709:
1706:
1703:
1666:
1660:
1657:
1654:
1632:
1609:
1603:
1600:
1597:
1575:
1555:
1535:
1512:
1492:
1472:
1448:
1428:
1404:
1384:
1360:
1357:
1337:
1334:
1331:
1328:
1325:
1322:
1316:
1310:
1307:
1304:
1282:
1261:
1241:
1215:
1204:
1203:
1192:
1189:
1186:
1183:
1180:
1177:
1171:
1165:
1162:
1159:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1130:
1124:
1121:
1118:
1113:
1108:
1105:
1101:
1061:
1058:
1043:
1019:
991:
967:
942:
939:
936:
915:
889:
868:
842:
817:
795:
774:
771:
768:
764:
759:
754:
751:
748:
745:
742:
739:
736:
731:
708:
705:
702:
698:
693:
688:
685:
682:
679:
676:
673:
670:
665:
641:
619:
592:
588:
585:
581:
576:
571:
568:
565:
562:
559:
556:
553:
548:
519:
515:
512:
508:
503:
498:
495:
492:
489:
486:
483:
480:
475:
453:
433:
418:
417:
406:
403:
400:
397:
394:
391:
385:
379:
376:
373:
368:
365:
362:
359:
356:
353:
350:
344:
338:
335:
332:
301:
279:
257:
235:
224:
223:
210:
205:
200:
195:
192:
170:
165:
160:
155:
152:
123:
99:
26:
9:
6:
4:
3:
2:
16920:
16909:
16906:
16905:
16903:
16888:
16885:
16883:
16882:Representable
16880:
16878:
16875:
16873:
16870:
16868:
16865:
16863:
16860:
16858:
16855:
16853:
16850:
16848:
16845:
16843:
16840:
16838:
16835:
16833:
16830:
16828:
16825:
16823:
16820:
16819:
16816:
16811:
16804:
16799:
16797:
16792:
16790:
16785:
16784:
16781:
16769:
16761:
16759:
16751:
16749:
16741:
16740:
16737:
16723:
16720:
16718:
16715:
16713:
16709:
16705:
16701:
16699:
16697:
16690:
16688:
16685:
16683:
16680:
16679:
16677:
16674:
16670:
16660:
16657:
16654:
16650:
16647:
16646:
16644:
16642:
16634:
16628:
16625:
16623:
16620:
16618:
16615:
16613:
16612:Tetracategory
16610:
16608:
16605:
16602:
16601:pseudofunctor
16598:
16595:
16594:
16592:
16590:
16582:
16579:
16575:
16570:
16567:
16565:
16562:
16560:
16557:
16555:
16552:
16550:
16547:
16545:
16542:
16540:
16537:
16535:
16532:
16530:
16526:
16520:
16519:
16516:
16512:
16507:
16503:
16498:
16480:
16477:
16475:
16472:
16470:
16467:
16465:
16462:
16460:
16457:
16455:
16452:
16450:
16447:
16445:
16444:Free category
16442:
16441:
16439:
16435:
16428:
16427:Vector spaces
16424:
16421:
16418:
16414:
16411:
16409:
16406:
16404:
16401:
16399:
16396:
16394:
16391:
16389:
16386:
16385:
16383:
16381:
16377:
16367:
16364:
16362:
16359:
16355:
16352:
16351:
16350:
16347:
16345:
16342:
16340:
16337:
16336:
16334:
16332:
16328:
16322:
16321:Inverse limit
16319:
16317:
16314:
16310:
16307:
16306:
16305:
16302:
16300:
16297:
16295:
16292:
16291:
16289:
16287:
16283:
16280:
16278:
16274:
16268:
16265:
16263:
16260:
16258:
16255:
16253:
16250:
16248:
16247:Kan extension
16245:
16243:
16240:
16238:
16235:
16233:
16230:
16228:
16225:
16223:
16220:
16218:
16215:
16211:
16208:
16206:
16203:
16201:
16198:
16196:
16193:
16191:
16188:
16186:
16183:
16182:
16181:
16178:
16177:
16175:
16171:
16167:
16160:
16156:
16152:
16145:
16140:
16138:
16133:
16131:
16126:
16125:
16122:
16115:
16111:
16107:
16103:
16099:
16095:
16092:
16089:
16088:Eugenia Cheng
16085:
16081:
16076:
16075:
16065:
16061:
16057:
16055:0-387-98403-8
16051:
16047:
16043:
16042:
16037:
16033:
16029:
16025:
16021:
16019:0-471-60922-6
16015:
16008:
16007:
16001:
16000:
15989:
15986:
15985:0-387-97710-4
15982:
15978:
15974:
15969:
15961:
15957:
15951:
15944:
15940:
15936:
15932:
15927:
15919:
15915:
15910:
15905:
15901:
15897:
15896:
15888:
15881:
15872:
15871:q-alg/9609018
15867:
15860:
15856:
15848:
15846:
15842:
15837:
15835:
15831:
15827:
15823:
15819:
15815:
15811:
15791:
15783:
15779:
15775:
15772:
15765:
15764:
15763:
15746:
15732:
15728:
15725:
15718:
15717:
15716:
15714:
15710:
15674:
15671:
15664:
15663:
15662:
15660:
15656:
15652:
15648:
15644:
15634:
15632:
15628:
15623:
15618:
15614:
15610:
15605:
15600:
15593:
15589:
15585:
15580:
15575:
15571:
15567:
15563:
15556:
15551:
15546:
15542:
15538:
15531:
15527:
15523:
15518:
15516:
15512:
15508:
15504:
15501:If a functor
15494:
15492:
15488:
15482:
15480:
15476:
15472:
15468:
15464:
15460:
15456:
15452:
15448:
15444:
15432:Relationships
15429:
15427:
15423:
15419:
15415:
15410:
15408:
15404:
15400:
15378:
15375:
15372:
15369:
15343:
15337:
15334:
15331:
15328:
15302:
15297:
15294:
15291:
15279:
15278:
15277:
15275:
15271:
15267:
15263:
15259:
15255:
15251:
15247:
15243:
15239:
15235:
15222:
15218:
15215:
15211:
15208:
15204:
15201:
15197:
15196:
15195:
15192:
15190:
15186:
15182:
15178:
15174:
15170:
15160:
15158:
15154:
15149:
15147:
15124:
15113:
15106:
15102:
15097:
15094:
15088:
15084:
15081:
15077:
15073:
15068:
15064:
15061:
15056:
15053:
15049:
15040:
15037:
15033:
15030:
15026:
15023:
15015:
15012:
15008:
15004:
15001:
14996:
14990:
14987:
14982:
14979:
14971:
14968:
14963:
14950:
14937:
14936:
14935:
14933:
14929:
14910:
14907:
14901:
14898:
14895:
14892:
14888:
14885:
14877:
14876:
14875:
14858:
14852:
14849:
14845:
14842:
14838:
14835:
14828:
14827:
14826:
14824:
14820:
14816:
14812:
14808:
14804:
14800:
14796:
14786:
14772:
14752:
14725:
14717:
14714:
14710:
14706:
14701:
14698:
14694:
14687:
14684:
14681:
14679:
14673:
14670:
14662:
14659:
14653:
14650:
14647:
14641:
14639:
14633:
14630:
14618:
14617:
14616:
14614:
14610:
14602:
14598:
14594:
14591:
14587:
14583:
14582:
14581:
14579:
14575:
14571:
14567:
14563:
14559:
14555:
14551:
14546:
14544:
14540:
14536:
14532:
14528:
14524:
14520:
14516:
14504:
14500:
14496:
14493:
14490:
14486:
14483:
14482:
14481:
14467:
14461:
14458:
14455:
14447:
14442:
14436:
14432:
14427:
14423:
14419:
14416:
14415:
14414:
14408:
14400:
14396:
14392:
14388:
14384:
14381:
14380:
14379:
14373:
14369:
14365:
14361:
14358:
14357:
14356:
14354:
14353:
14345:
14341:
14332:
14328:
14324:
14320:
14315:
14311:
14308:
14307:
14306:
14304:
14300:
14296:
14292:
14288:
14284:
14280:
14276:
14272:
14268:
14264:
14258:
14226:
14223:
14219:
14215:
14207:
14203:
14196:
14193:
14188:
14185:
14181:
14173:
14170:
14165:
14159:
14150:
14146:
14142:
14122:
14118:
14111:
14108:
14103:
14100:
14096:
14092:
14087:
14084:
14080:
14072:
14071:
14067:
14062:
14056:
14052:
14046:
14041:
14037:
14034:Substituting
14033:
14032:
14028:
14024:
14020:
14016:
14012:
14008:
14004:
13983:
13977:
13974:
13969:
13965:
13961:
13955:
13947:
13944:
13939:
13936:
13933:
13919:
13915:
13911:
13905:
13899:
13896:
13890:
13882:
13879:
13876:
13860:
13859:
13855:
13852:
13848:
13844:
13839:
13834:
13830:
13826:
13822:
13817:
13812:
13808:
13790:
13787:
13784:
13781:
13778:
13770:
13754:
13749:
13746:
13742:
13733:
13729:
13710:
13707:
13704:
13701:
13698:
13690:
13674:
13666:
13663:
13659:
13650:
13647:
13644:
13641:
13633:
13628:
13624:
13615:
13597:
13594:
13591:
13588:
13585:
13577:
13561:
13556:
13553:
13549:
13540:
13536:
13517:
13514:
13511:
13508:
13505:
13497:
13481:
13473:
13470:
13466:
13457:
13454:
13449:
13446:
13443:
13440:
13432:
13427:
13423:
13414:
13413:
13412:
13394:
13391:
13388:
13385:
13379:
13376:
13373:
13363:
13362:
13361:
13359:
13354:
13349:
13344:
13339:
13335:
13331:
13327:
13323:
13319:
13306:
13288:
13285:
13282:
13279:
13274:
13271:
13267:
13263:
13261:
13253:
13250:
13245:
13242:
13238:
13234:
13226:
13222:
13215:
13212:
13210:
13200:
13196:
13192:
13186:
13180:
13177:
13174:
13166:
13162:
13155:
13152:
13150:
13145:
13128:
13127:
13122:
13112:
13107:
13102:
13097:
13093:
13092:
13088:
13070:
13067:
13062:
13059:
13055:
13051:
13048:
13045:
13043:
13030:
13026:
13019:
13016:
13011:
13008:
13004:
13000:
12997:
12994:
12992:
12979:
12975:
12968:
12965:
12959:
12953:
12950:
12947:
12942:
12938:
12934:
12932:
12927:
12910:
12909:
12904:
12899:
12892:
12886:
12881:
12877:
12876:
12872:
12851:
12845:
12842:
12837:
12833:
12829:
12823:
12815:
12812:
12809:
12795:
12791:
12787:
12781:
12775:
12772:
12766:
12758:
12755:
12752:
12736:
12735:
12731:
12727:
12723:
12719:
12715:
12711:
12707:
12706:
12705:
12703:
12698:
12693:
12688:
12683:
12668:
12661:
12657:
12653:
12649:
12645:
12641:
12637:
12627:
12618:
12611:
12603:
12596:
12587:
12583:
12579:
12569:
12560:
12556:
12549:
12545:
12537:
12528:
12524:
12520:
12513:
12509:
12502:
12498:
12492:
12486:
12478:
12471:
12463:
12456:
12452:
12445:
12438:
12434:
12430:
12423:
12416:
12412:
12405:
12398:
12394:
12390:
12386:
12377:
12373:
12369:
12365:
12361:
12357:
12352:
12348:
12342:
12338:
12332:
12327:
12323:
12319:
12315:
12311:
12307:
12303:
12302:
12300:
12295:
12290:
12285:
12280:
12275:
12271:
12267:
12264:
12260:
12254:
12249:
12244:
12242:on morphisms.
12241:
12234:
12230:
12223:
12219:
12215:
12211:
12203:
12198:
12188:
12179:
12171:
12167:
12160:
12153:
12149:
12145:
12142:
12138:
12134:
12130:
12126:
12121:
12116:
12112:
12107:
12102:
12098:
12094:
12090:
12086:
12085:
12083:
12079:
12075:
12071:
12070:
12069:
12067:
12063:
12059:
12049:
12047:
12043:
12036:
12032:
12028:
12024:
12019:
12014:
12010:
12008:
12004:
12000:
11996:
11991:
11986:
11982:
11981:
11960:
11957:
11953:
11949:
11941:
11937:
11930:
11927:
11925:
11918:
11915:
11911:
11898:
11894:
11887:
11884:
11879:
11876:
11872:
11868:
11866:
11859:
11856:
11852:
11840:
11839:
11835:
11834:
11809:
11803:
11800:
11797:
11794:
11786:
11766:
11759:
11755:
11751:
11749:
11739:
11736:
11732:
11723:
11720:
11717:
11714:
11699:
11696:
11690:
11684:
11681:
11673:
11653:
11646:
11642:
11638:
11636:
11626:
11623:
11619:
11610:
11607:
11602:
11599:
11596:
11593:
11575:
11569:
11566:
11563:
11555:
11535:
11528:
11524:
11520:
11514:
11508:
11505:
11503:
11495:
11487:
11484:
11481:
11473:
11470:
11460:
11457:
11451:
11445:
11437:
11417:
11409:
11403:
11400:
11395:
11391:
11387:
11385:
11377:
11369:
11366:
11361:
11358:
11355:
11347:
11344:
11333:
11332:
11328:
11327:
11326:
11320:
11311:
11307:
11303:
11299:
11295:
11290:
11286:
11281:
11277:
11273:
11269:
11265:
11261:
11257:
11253:
11249:
11244:
11240:
11235:
11231:
11227:
11223:
11219:
11215:
11214:
11213:
11212:
11211:
11209:
11205:
11201:
11197:
11190:
11186:
11181:
11176:
11174:
11169:
11164:
11160:
11156:
11153:
11148:
11143:
11138:
11133:
11129:
11127:
11126:right adjoint
11123:
11119:
11115:
11111:
11109:
11105:
11101:
11097:
11094:
11090:
11089:
11088:
11086:
11082:
11078:
11073:
11065:
11051:
10987:
10984:
10956:
10942:
10939:
10912:
10908:
10901:
10898:
10895:
10887:
10882:
10866:
10863:
10859:
10855:
10852:
10846:
10843:
10837:
10834:
10831:
10797:
10794:
10791:
10771:
10768:
10765:
10762:
10759:
10753:
10747:
10716:
10663:
10654:
10637:
10631:
10625:
10622:
10616:
10613:
10610:
10587:
10567:
10564:
10561:
10558:
10555:
10549:
10543:
10512:
10510:
10498:
10496:
10491:
10490:
10434:
10414:
10391:
10368:
10348:
10328:
10323:
10310:
10294:
10287:
10284:
10281:
10277:
10265:
10254:
10251:
10247:
10243:
10240:
10231:
10228:
10225:
10222:
10216:
10213:
10208:
10196:
10195:
10193:
10172:
10164:
10160:
10156:
10150:
10147:
10144:
10136:
10132:
10127:
10124:
10118:
10115:
10112:
10109:
10098:
10082:
10076:
10070:
10067:
10060:
10057:
10054:
10050:
10038:
10027:
10024:
10020:
10016:
10013:
10004:
10001:
9998:
9995:
9989:
9986:
9981:
9969:
9968:
9951:
9945:
9925:
9922:
9919:
9899:
9896:
9887:
9876:
9873:
9869:
9848:
9826:
9801:
9779:
9775:
9754:
9751:
9745:
9739:
9716:
9708:
9705:
9701:
9697:
9694:
9674:
9671:
9665:
9659:
9636:
9628:
9625:
9621:
9617:
9614:
9607:we also have
9594:
9591:
9588:
9565:
9557:
9554:
9550:
9546:
9537:
9531:
9523:
9520:
9516:
9512:
9509:
9489:
9486:
9480:
9474:
9466:
9465:
9444:
9436:
9433:
9429:
9425:
9422:
9416:
9413:
9410:
9405:
9393:
9392:
9390:
9374:
9368:
9363:
9359:
9355:
9352:
9345:
9341:
9335:
9332:
9329:
9324:
9312:
9304:
9303:
9289:
9286:
9283:
9275:
9260:
9257:
9251:
9243:
9240:
9236:
9215:
9195:
9175:
9155:
9135:
9115:
9095:
9090:
9086:
9082:
9079:
9076:
9071:
9067:
9046:
9038:
9035:
9034:
9018:
8998:
8993:
8989:
8985:
8965:
8945:
8925:
8922:
8919:
8896:
8865:
8835:
8808:
8783:
8778:
8774:
8770:
8750:
8744:
8724:
8704:
8696:
8692:
8673:
8656:
8645:
8640:
8636:
8628:
8627:
8613:
8607:
8604:
8601:
8581:
8573:
8572:
8568:
8564:
8548:
8528:
8506:
8502:
8479:
8475:
8454:
8434:
8414:
8406:
8384:
8376:
8372:
8368:
8362:
8359:
8356:
8348:
8344:
8339:
8336:
8330:
8327:
8324:
8321:
8311:
8310:
8296:
8276:
8256:
8234:
8230:
8221:
8217:
8216:
8197:
8191:
8188:
8182:
8174:
8170:
8166:
8160:
8152:
8148:
8127:
8107:
8087:
8084:
8081:
8055:
8047:
8043:
8039:
8036:
8030:
8027:
8005:
8001:
7992:
7989:
7988:
7979:
7975:
7971:
7967:
7963:
7959:
7956:
7953:
7949:
7945:
7941:
7937:
7930:The functor Ο
7929:
7926:
7923:
7919:
7915:
7911:
7907:
7903:
7899:
7895:
7891:
7887:
7886:Equivalences.
7884:
7883:
7874:
7870:
7866:
7862:
7858:
7855:instead: the
7854:
7850:
7846:
7843:
7840:
7836:
7832:
7828:
7824:
7820:
7816:
7812:
7808:
7804:
7800:
7796:
7792:
7788:
7784:
7780:
7776:
7772:
7768:
7764:
7760:
7756:
7753:
7749:
7745:
7742:
7741:
7740:
7737:
7735:
7731:
7727:
7722:
7720:
7716:
7715:Galois theory
7712:
7708:
7704:
7700:
7696:
7692:
7688:
7675:
7671:
7667:
7663:
7662:Stone duality
7659:
7656:
7653:
7646:
7642:
7638:
7637:
7632:
7628:
7624:
7620:
7619:
7614:
7610:
7606:
7599:
7595:
7591:
7587:
7583:
7580:
7576:
7573:
7570:
7566:
7562:
7558:
7554:
7550:
7546:
7542:
7538:
7534:
7530:
7526:
7523:
7519:
7515:
7512:
7509:
7505:
7501:
7497:
7493:
7489:
7485:
7481:
7477:
7474:
7470:
7466:
7462:
7459:
7455:
7452:
7449:
7445:
7442:creating the
7441:
7437:
7433:
7429:
7425:
7421:
7417:
7413:
7409:
7406:
7405:
7396:
7392:
7388:
7385:
7382:
7378:
7374:
7370:
7366:
7362:
7361:abelian group
7358:
7354:
7350:
7346:
7342:
7339:
7336:
7332:
7328:
7324:
7320:
7316:
7312:
7308:
7304:
7300:
7297:
7294:
7290:
7283:
7276:
7272:
7269:
7266:
7259:
7255:
7248:
7244:
7241:
7238:
7234:
7230:
7226:
7223:
7219:
7215:
7211:
7207:
7203:
7199:
7196:
7193:
7189:
7185:
7181:
7176:
7171:
7167:
7164:, defined by
7163:
7159:
7155:
7151:
7147:
7143:
7139:
7135:
7131:
7127:
7123:
7119:
7115:
7113:
7109:
7106:
7102:
7098:
7094:
7090:
7086:
7082:
7078:
7074:
7070:
7066:
7062:
7058:
7054:
7050:
7046:
7042:
7038:
7034:
7030:
7026:
7023:
7020:
7004:
6997:
6994:
6990:
6986:
6982:
6980:
6975:
6972:
6968:
6964:
6960:
6956:
6954:
6949:
6948:
6934:
6930:
6926:
6925:vector spaces
6922:
6918:
6917:
6913:
6909:
6905:
6901:
6897:
6893:
6890:) to back to
6889:
6885:
6878:
6871:
6867:
6863:
6859:
6855:
6851:
6847:
6843:
6839:
6835:
6828:
6821:
6817:
6814:
6811:
6807:
6803:
6800:
6799:
6798:
6796:
6792:
6788:
6784:
6780:
6767:
6766:
6762:
6758:
6754:
6750:
6746:
6742:
6738:
6735:
6731:
6727:
6723:
6719:
6715:
6710:
6706:
6700:
6693:
6689:
6685:
6680:
6676:
6671:
6667:
6660:
6653:
6649:
6642:
6635:
6628:
6621:
6614:
6607:
6603:
6599:
6596:
6595:
6591:
6587:
6583:
6579:
6578:
6574:
6570:
6566:
6559:
6552:
6545:
6538:
6534:
6530:
6526:
6522:
6518:
6514:
6511:
6504:
6497:
6493:
6483:
6479:
6475:
6472:Let Ξ :
6471:
6468:
6467:
6466:
6464:
6460:
6456:
6452:
6448:
6439:
6437:
6436:free functors
6433:
6429:
6420:
6418:
6414:
6410:
6406:
6402:
6381:
6377:
6370:
6362:
6358:
6340:
6337:
6333:
6324:
6320:
6300:
6297:
6285:
6281:
6274:
6269:
6264:
6261:
6258:
6255:
6246:
6243:
6239:
6233:
6228:
6225:
6217:
6216:
6215:
6213:
6197:
6194:
6191:
6188:
6185:
6182:
6177:
6173:
6164:
6160:
6158:
6154:
6150:
6146:
6128:
6125:
6121:
6112:
6108:
6104:
6100:
6096:
6092:
6071:
6067:
6060:
6052:
6048:
6029:
6026:
6017:
6014:
6010:
6004:
5999:
5996:
5993:
5990:
5978:
5974:
5967:
5962:
5957:
5954:
5947:
5946:
5945:
5943:
5927:
5924:
5921:
5918:
5915:
5912:
5907:
5903:
5894:
5890:
5876:
5873:
5870:
5867:
5861:
5858:
5855:
5844:
5840:
5838:
5834:
5829:
5825:
5821:
5817:
5813:
5809:
5805:
5801:
5799:
5795:
5791:
5787:
5771:
5765:
5762:
5759:
5756:
5751:
5747:
5738:
5734:
5730:
5726:
5705:
5701:
5697:
5694:
5691:
5680:
5676:
5660:
5654:
5651:
5648:
5645:
5640:
5636:
5627:
5623:
5619:
5615:
5611:
5607:
5605:
5604:
5598:
5594:
5578:
5575:
5572:
5566:
5563:
5558:
5554:
5545:
5541:
5537:
5533:
5529:
5513:
5510:
5507:
5501:
5498:
5493:
5489:
5481:. Let
5480:
5477:generated by
5476:
5472:
5468:
5465:For each set
5464:
5460:
5458:
5454:
5450:
5446:
5442:
5438:
5434:
5430:
5426:
5422:
5418:
5417:
5412:
5411:
5406:
5401:
5399:
5384:
5382:
5378:
5374:
5370:
5366:
5362:
5358:
5354:
5350:
5346:
5342:
5338:
5334:
5318:
5315:
5312:
5304:
5300:
5292:
5288:
5284:
5280:
5276:
5272:
5268:
5267:
5266:
5264:
5260:
5250:
5248:
5244:
5240:
5236:
5231:
5229:
5225:
5209:
5206:
5203:
5200:
5197:
5189:
5185:
5181:
5172:
5168:
5166:
5148:
5145:
5141:
5132:
5114:
5110:
5057:
5026:
5023:
5019:
5015:
5007:
5003:
4996:
4993:
4991:
4984:
4981:
4977:
4964:
4960:
4953:
4950:
4945:
4942:
4938:
4934:
4932:
4925:
4922:
4918:
4906:
4905:
4904:
4902:
4898:
4894:
4890:
4867:
4864:
4861:
4858:
4855:
4852:
4850:
4843:
4839:
4831:
4828:
4825:
4822:
4819:
4816:
4814:
4807:
4803:
4791:
4790:
4789:
4788:
4784:
4780:
4775:
4761:
4758:
4755:
4735:
4732:
4729:
4726:
4720:
4717:
4714:
4703:
4702:
4698:
4693:
4692:
4688:
4682:
4680:
4676:
4671:
4665:
4645:
4638:
4635:
4630:
4625:
4622:
4619:
4612:
4609:
4604:
4599:
4592:
4578:
4571:
4568:
4563:
4558:
4555:
4552:
4545:
4542:
4537:
4532:
4525:
4524:
4523:
4521:
4517:
4513:
4490:
4487:
4473:
4469:
4467:
4462:
4447:
4440:
4437:
4434:
4432:
4427:
4416:
4415:
4414:
4413:
4409:
4405:
4401:
4397:
4393:
4389:
4386:
4382:
4378:
4374:
4364:
4362:
4358:
4354:
4350:
4342:
4338:
4326:
4322:
4318:
4310:
4306:
4298:
4294:
4285:
4281:
4279:
4275:
4270:
4266:
4263:
4259:
4254:
4249:
4245:
4241:
4237:
4233:
4229:
4225:
4220:
4216:
4211:
4205:
4200:
4191:
4186:
4179:
4174:
4170:
4168:
4167:
4163:
4158:
4157:
4153:
4147:
4145:
4141:
4137:
4133:
4111:
4108:
4105:
4102:
4094:
4072:
4069:
4066:
4063:
4055:
4039:
4034:
4031:
4028:
4016:
4015:
4014:
3993:
3990:
3987:
3984:
3976:
3954:
3951:
3948:
3945:
3937:
3921:
3911:
3910:
3909:
3908:
3903:
3899:
3895:
3890:
3886:
3882:
3879:
3875:
3871:
3867:
3865:
3854:
3850:
3836:
3816:
3796:
3776:
3767:
3753:
3745:
3744:right adjoint
3729:
3709:
3688:
3685:
3678:
3675:
3672:
3652:
3649:
3643:
3640:
3635:
3631:
3626:
3622:
3618:
3609:
3603:
3597:
3577:
3571:
3568:
3565:
3545:
3532:
3528:
3514:
3511:
3506:
3502:
3498:
3492:
3486:
3466:
3457:
3451:
3448:
3445:
3422:
3416:
3410:
3407:
3404:
3384:
3364:
3338:
3332:
3326:
3320:
3317:
3312:
3308:
3287:
3264:
3258:
3238:
3218:
3198:
3178:
3170:
3154:
3134:
3126:
3110:
3104:
3101:
3098:
3089:
3075:
3067:
3051:
3031:
3011:
3004:
3001:
2997:
2994:
2971:
2968:
2963:
2959:
2956:
2953:
2944:
2938:
2932:
2929:
2924:
2920:
2899:
2893:
2890:
2887:
2867:
2854:
2850:
2848:
2843:
2829:
2826:
2820:
2814:
2811:
2806:
2802:
2778:
2772:
2766:
2763:
2760:
2740:
2731:
2725:
2722:
2719:
2699:
2679:
2659:
2647:
2641:
2635:
2632:
2627:
2623:
2602:
2579:
2573:
2553:
2533:
2513:
2493:
2485:
2469:
2449:
2441:
2425:
2419:
2416:
2413:
2399:
2397:
2393:
2389:
2385:
2381:
2377:
2373:
2369:
2365:
2361:
2357:
2353:
2348:
2346:
2342:
2338:
2334:
2330:
2326:
2316:
2309:
2306:
2302:
2298:
2297:
2296:
2288:
2286:
2282:
2278:
2274:
2269:
2267:
2263:
2259:
2255:
2251:
2247:
2242:
2240:
2236:
2232:
2222:
2220:
2216:
2212:
2207:
2205:
2201:
2197:
2193:
2189:
2185:
2181:
2177:
2173:
2169:
2165:
2161:
2157:
2153:
2149:
2144:
2142:
2138:
2134:
2130:
2126:
2122:
2115:
2108:
2101:
2094:
2091:implies that
2087:
2083:
2076:
2072:
2068:
2064:
2049:
2042:
2035:
2031:
2024:
2020:
2017:of the form (
2016:
2009:
2005:
1998:
1994:
1990:
1986:
1982:
1978:
1974:
1970:
1966:
1962:
1957:
1955:
1951:
1947:
1943:
1939:
1935:
1931:
1926:
1924:
1920:
1916:
1912:
1908:
1903:
1901:
1897:
1893:
1889:
1885:
1881:
1877:
1873:
1869:
1859:
1857:
1848:
1847:
1839:
1830:
1813:
1810:
1807:
1804:
1798:
1792:
1789:
1786:
1783:
1760:
1740:
1733:
1729:
1728:Hilbert space
1710:
1707:
1704:
1701:
1694:
1693:
1692:
1690:
1686:
1681:
1630:
1573:
1553:
1533:
1524:
1510:
1490:
1483:. (Note that
1470:
1462:
1461:right adjoint
1446:
1426:
1418:
1402:
1382:
1374:
1371:is the right
1358:
1355:
1332:
1329:
1326:
1323:
1280:
1272:
1259:
1239:
1229:
1213:
1187:
1184:
1181:
1178:
1152:
1146:
1143:
1140:
1137:
1111:
1106:
1103:
1099:
1091:
1090:
1089:
1087:
1083:
1079:
1075:
1074:
1069:
1068:
1057:
1007:
954:
940:
937:
934:
926:
913:
903:
887:
879:
866:
856:
840:
831:
793:
752:
746:
743:
740:
737:
686:
680:
677:
674:
671:
617:
569:
563:
560:
557:
554:
496:
490:
487:
484:
481:
451:
431:
423:
401:
398:
395:
392:
366:
360:
357:
354:
351:
321:
320:
319:
317:
277:
233:
193:
190:
153:
150:
143:
142:
141:
139:
86:
85:in topology.
84:
80:
76:
72:
68:
67:right adjoint
64:
60:
56:
52:
48:
44:
37:
33:
19:
16832:Conservative
16826:
16692:
16673:Categorified
16577:n-categories
16528:Key concepts
16366:Direct limit
16349:Coequalizers
16267:Yoneda lemma
16216:
16173:Key concepts
16163:Key concepts
16082:playlist on
16039:
16005:
15987:
15976:
15968:
15959:
15950:
15938:
15926:
15899:
15893:
15880:
15859:
15838:
15833:
15828:γ defines a
15825:
15821:
15817:
15813:
15809:
15807:
15761:
15712:
15708:
15707:is given by
15706:
15658:
15654:
15646:
15642:
15640:
15630:
15626:
15621:
15616:
15612:
15608:
15606:
15598:
15591:
15587:
15583:
15578:
15573:
15569:
15565:
15554:
15549:
15544:
15540:
15536:
15529:
15525:
15521:
15519:
15510:
15506:
15502:
15500:
15490:
15486:
15483:
15478:
15474:
15470:
15466:
15462:
15458:
15454:
15446:
15442:
15440:
15417:
15413:
15411:
15406:
15402:
15398:
15396:
15273:
15269:
15265:
15261:
15253:
15249:
15245:
15237:
15233:
15231:
15193:
15185:cocontinuous
15184:
15180:
15172:
15168:
15166:
15152:
15150:
15145:
15143:
14931:
14927:
14925:
14873:
14822:
14818:
14814:
14810:
14806:
14802:
14798:
14794:
14792:
14744:
14612:
14608:
14606:
14600:
14596:
14589:
14585:
14577:
14573:
14569:
14565:
14561:
14557:
14553:
14549:
14547:
14538:
14534:
14530:
14526:
14522:
14518:
14514:
14512:
14498:
14494:
14488:
14484:
14443:
14440:
14434:
14430:
14425:
14421:
14417:
14406:
14404:
14398:
14394:
14390:
14386:
14382:
14377:
14371:
14367:
14363:
14359:
14352:proper class
14350:
14343:
14342:come from a
14339:
14337:
14330:
14326:
14322:
14318:
14313:
14309:
14302:
14298:
14290:
14282:
14274:
14270:
14266:
14262:
14260:
14168:
14163:
14157:
14148:
14144:
14065:
14060:
14054:
14050:
14044:
14039:
14035:
14026:
14022:
14018:
14014:
14010:
14006:
13850:
13846:
13842:
13837:
13832:
13831:, and each (
13828:
13824:
13820:
13815:
13810:
13731:
13727:
13538:
13534:
13410:
13357:
13352:
13347:
13342:
13337:
13333:
13329:
13325:
13321:
13317:
13315:
13120:
13110:
13105:
13100:
13095:
12906:), we obtain
12902:
12897:
12890:
12884:
12879:
12729:
12725:
12721:
12717:
12713:
12709:
12701:
12696:
12691:
12686:
12681:
12659:
12655:
12651:
12647:
12643:
12639:
12635:
12633:
12624:
12616:
12609:
12601:
12594:
12585:
12581:
12577:
12567:
12558:
12554:
12547:
12543:
12535:
12526:
12522:
12518:
12511:
12507:
12500:
12496:
12490:
12484:
12476:
12469:
12461:
12454:
12450:
12443:
12436:
12432:
12428:
12421:
12414:
12410:
12403:
12396:
12392:
12384:
12375:
12371:
12367:
12363:
12359:
12355:
12350:
12346:
12340:
12336:
12330:
12325:
12321:
12317:
12313:
12309:
12305:
12298:
12293:
12288:
12283:
12273:
12269:
12258:
12252:
12247:
12239:
12232:
12228:
12221:
12217:
12213:
12209:
12201:
12196:
12186:
12177:
12169:
12165:
12158:
12151:
12147:
12140:
12136:
12132:
12128:
12124:
12119:
12114:
12110:
12105:
12100:
12096:
12092:
12088:
12081:
12077:
12073:
12065:
12061:
12057:
12055:
12045:
12041:
12039:
12034:
12030:
12026:
12017:
12012:
12006:
12002:
11998:
11989:
11984:
11324:
11309:
11305:
11301:
11297:
11293:
11288:
11284:
11279:
11275:
11271:
11267:
11263:
11259:
11255:
11251:
11247:
11242:
11238:
11233:
11229:
11225:
11221:
11217:
11207:
11203:
11199:
11195:
11193:
11188:
11184:
11179:
11172:
11167:
11162:
11151:
11146:
11141:
11136:
11134:Ξ¦ : hom
11125:
11121:
11117:
11113:
11108:left adjoint
11107:
11103:
11099:
11095:
11087:consists of
11084:
11080:
11076:
11074:
11071:
11064:is "free".)
10883:
10655:
10513:
10508:
10506:
10492:
9036:
8567:topos theory
8218:The role of
7990:
7973:
7969:
7965:
7957:
7951:
7947:
7943:
7939:
7935:
7927:
7921:
7917:
7913:
7909:
7905:
7897:
7893:
7889:
7885:
7852:
7848:
7838:
7834:
7830:
7826:
7822:
7818:
7814:
7810:
7806:
7802:
7798:
7794:
7790:
7786:
7782:
7778:
7774:
7770:
7766:
7762:
7738:
7729:
7723:
7710:
7702:
7698:
7694:
7690:
7684:
7666:sober spaces
7657:
7644:
7640:
7634:
7630:
7626:
7622:
7616:
7612:
7608:
7597:
7589:
7585:
7581:
7574:
7568:
7556:
7552:
7548:
7544:
7536:
7532:
7528:
7517:
7513:
7499:
7495:
7487:
7483:
7479:
7475:
7464:
7460:
7453:
7447:
7439:
7435:
7427:
7423:
7411:
7407:
7386:
7381:model theory
7340:
7334:
7330:
7326:
7310:
7306:
7302:
7298:
7288:
7281:
7274:
7270:
7257:
7253:
7246:
7242:
7236:
7228:
7224:
7197:
7191:
7187:
7183:
7179:
7174:
7169:
7165:
7161:
7157:
7153:
7149:
7145:
7141:
7137:
7133:
7129:
7125:
7121:
7117:
7110:
7104:
7100:
7096:
7092:
7088:
7084:
7080:
7076:
7072:
7068:
7064:
7060:
7052:
7048:
7040:
7036:
7032:
7028:
7024:
7018:
6995:
6984:
6976:
6970:
6966:
6962:
6958:
6950:
6933:free product
6911:
6907:
6903:
6899:
6895:
6891:
6887:
6883:
6876:
6869:
6865:
6861:
6857:
6853:
6849:
6845:
6841:
6837:
6826:
6819:
6815:
6812:
6809:
6805:
6801:
6787:coequalizers
6777:
6760:
6756:
6752:
6748:
6744:
6740:
6736:
6733:
6729:
6721:
6717:
6713:
6708:
6704:
6698:
6691:
6687:
6683:
6678:
6674:
6669:
6665:
6658:
6651:
6647:
6640:
6633:
6626:
6619:
6612:
6605:
6601:
6597:
6572:
6568:
6564:
6557:
6550:
6543:
6536:
6532:
6528:
6524:
6520:
6512:
6509:
6502:
6495:
6488:
6481:
6477:
6473:
6469:
6445:
6428:Free objects
6426:
6416:
6412:
6408:
6404:
6400:
6360:
6356:
6322:
6318:
6316:
6211:
6162:
6161:
6156:
6152:
6148:
6144:
6110:
6106:
6102:
6098:
6094:
6090:
6050:
6046:
6044:
5941:
5892:
5891:
5842:
5841:
5836:
5832:
5827:
5823:
5819:
5815:
5811:
5807:
5803:
5802:
5797:
5793:
5789:
5785:
5736:
5732:
5728:
5724:
5678:
5674:
5628:. Let
5625:
5621:
5617:
5616:, the group
5613:
5609:
5608:
5602:
5596:
5592:
5543:
5539:
5535:
5531:
5527:
5478:
5474:
5470:
5466:
5462:
5461:
5456:
5452:
5448:
5440:
5436:
5432:
5428:
5420:
5414:
5408:
5404:
5402:
5395:
5376:
5372:
5368:
5360:
5352:
5344:
5340:
5336:
5302:
5296:
5290:
5286:
5282:
5278:
5274:
5270:
5256:
5242:
5238:
5234:
5232:
5227:
5223:
5183:
5179:
5177:
5164:
5130:
5048:
4900:
4896:
4892:
4888:
4886:
4786:
4782:
4778:
4776:
4700:
4696:
4695:
4690:
4686:
4685:
4683:
4678:
4674:
4669:
4663:
4660:
4515:
4511:
4509:
4407:
4403:
4399:
4395:
4391:
4387:
4380:
4376:
4372:
4370:
4360:
4356:
4352:
4348:
4340:
4328:
4324:
4320:
4316:
4308:
4304:
4299:) : Hom
4296:
4292:
4290:
4273:
4268:
4265:
4261:
4257:
4252:
4247:
4243:
4239:
4232:hom functors
4223:
4218:
4214:
4203:
4198:
4189:
4184:
4177:
4175:
4171:
4165:
4161:
4160:
4155:
4151:
4150:
4148:
4143:
4139:
4135:
4131:
4129:
4012:
3901:
3897:
3893:
3888:
3884:
3880:
3873:
3869:
3862:
3860:
3851:
3768:
3743:
3538:Again, this
3537:
3124:
3090:
3066:left adjoint
3065:
2859:
2844:
2439:
2405:
2395:
2391:
2387:
2383:
2379:
2375:
2371:
2367:
2363:
2359:
2355:
2351:
2349:
2344:
2340:
2336:
2332:
2328:
2324:
2322:
2313:
2304:
2294:
2284:
2280:
2276:
2272:
2270:
2265:
2261:
2257:
2253:
2249:
2245:
2243:
2238:
2234:
2230:
2228:
2218:
2214:
2210:
2208:
2203:
2199:
2195:
2191:
2187:
2183:
2179:
2175:
2171:
2167:
2163:
2159:
2155:
2151:
2147:
2145:
2140:
2136:
2132:
2128:
2124:
2120:
2113:
2106:
2099:
2092:
2085:
2081:
2074:
2070:
2066:
2047:
2040:
2033:
2029:
2022:
2018:
2007:
2003:
1996:
1992:
1988:
1984:
1980:
1976:
1972:
1968:
1964:
1960:
1958:
1953:
1945:
1942:optimization
1941:
1937:
1929:
1927:
1914:
1906:
1904:
1899:
1895:
1894:+1 for each
1891:
1887:
1871:
1867:
1865:
1852:
1844:
1837:
1725:
1688:
1684:
1682:
1525:
1460:
1417:left adjoint
1416:
1372:
1231:
1227:
1205:
1081:
1071:
1065:
1063:
955:
905:
901:
900:is called a
858:
854:
853:is called a
833:The functor
832:
786:for a fixed
610:for a fixed
419:
225:
87:
66:
63:left adjoint
62:
58:
50:
40:
16641:-categories
16617:Kan complex
16607:Tricategory
16589:-categories
16479:Subcategory
16237:Exponential
16205:Preadditive
16200:Pre-abelian
16098:Mathematica
16079:Adjunctions
15988:See page 58
15576:for which Ξ·
15547:for which Ξ΅
15221:right exact
15146:composition
14789:Composition
13124:, we obtain
12467:, we have Ξ¦
12117:, so that Ξ·
12011:Each pair (
11983:Each pair (
11282:-morphism Ξ¦
11236:-morphism Ξ¦
11187:called the
11171:called the
11124:called the
11106:called the
10503:Probability
10311:The subset
8220:quantifiers
7972:given by βΓ
7865:implication
7861:ring ideals
7124:is a right
6802:Coproducts.
6664:is a pair (
6359:to the set
5818:to the set
5810:to a group
5398:free groups
5392:Free groups
5355:sign is an
2319:Conventions
1876:ring theory
1006:equivalence
927:. We write
43:mathematics
16659:3-category
16649:2-category
16622:β-groupoid
16597:Bicategory
16344:Coproducts
16304:Equalizers
16210:Bicategory
16064:0906.18001
16028:0695.18001
15997:References
15939:Dialectica
15633:inverses.
15615:such that
15426:biproducts
15228:Additivity
15214:left exact
15173:continuous
14509:Uniqueness
14295:continuous
14255:See also:
14246:Properties
13616:Let
13415:Let
13350:-,-) β hom
12694:-,-) β hom
12503:)) = G(x)
12291:-,-) β hom
11266:-morphism
11220:-morphism
11216:For every
11144:β,β) β hom
11112:A functor
11077:adjunction
10884:Then, the
8695:subobjects
7863:, and the
7565:continuous
7492:loop space
7473:suspension
7357:direct sum
6921:direct sum
6852:the pair (
6834:direct sum
6779:Coproducts
6747:β 0. Then
6523:the pair (
6455:equalizers
5469:, the set
5431:, and let
5425:free group
5259:Daniel Kan
5049:Note that
4785:) are the
4210:bifunctors
3866:adjunction
3590:such that
2260:poses the
1228:adjunction
1064:The terms
51:adjunction
16862:Forgetful
16708:Symmetric
16653:2-functor
16393:Relations
16316:Pullbacks
16102:morphisms
15789:→
15773:μ
15744:→
15726:η
15685:→
15631:two-sided
15344:≅
15288:Φ
15207:coproduct
15107:ε
15082:ε
15009:η
14969:η
14905:→
14893:∘
14856:→
14839:∘
14773:∗
14753:∘
14715:−
14711:τ
14707:∗
14699:−
14695:σ
14688:∘
14685:ε
14671:ε
14663:η
14660:∘
14654:σ
14651:∗
14648:τ
14631:η
14595:Ο :
14584:Ο :
14465:→
14405:for some
14251:Existence
14220:η
14216:∘
14204:ε
14119:η
14109:∘
14097:ε
14005:for each
13975:∘
13966:ε
13945:−
13930:Φ
13916:η
13912:∘
13873:Φ
13755:∈
13675:∈
13638:Φ
13625:η
13562:∈
13482:∈
13455:−
13437:Φ
13424:ε
13392:⊣
13380:η
13374:ε
13280:∘
13251:∘
13239:η
13235:∘
13223:ε
13197:η
13193:∘
13175:∘
13163:ε
13143:Ψ
13140:Φ
13052:∘
13027:η
13017:∘
13005:ε
13001:∘
12976:η
12966:∘
12948:∘
12939:ε
12925:Φ
12922:Ψ
12843:∘
12834:ε
12806:Ψ
12792:η
12788:∘
12749:Φ
12720:and each
12708:For each
12669:⊣
12216:) :
12146:For each
11954:η
11950:∘
11938:ε
11895:η
11885:∘
11873:ε
11767:∈
11756:η
11711:Φ
11654:∈
11643:ε
11608:−
11590:Φ
11536:∈
11525:η
11521:∘
11478:Φ
11418:∈
11401:∘
11392:ε
11367:−
11352:Φ
11161:Ξ΅ :
11052:δ
10988:δ
10985:⊣
10957:μ
10946:↦
10943:μ
10909:δ
10905:↦
10896:δ
10864:−
10856:∘
10853:μ
10850:→
10847:μ
10832:μ
10795:∈
10792:μ
10629:→
10493:See also
10475:∀
10455:∃
10320:∀
10285:∈
10252:−
10244:∈
10235:∀
10232:∣
10226:∈
10205:∀
10161:ϕ
10157:∧
10133:ψ
10122:∃
10119:∣
10113:∈
10058:∈
10025:−
10017:∈
10008:∃
10005:∣
9999:∈
9978:∃
9923:∈
9897:∩
9874:−
9823:∃
9780:∗
9752:⊆
9706:−
9698:⊆
9672:∈
9626:−
9618:∈
9592:∈
9555:−
9547:⊆
9521:−
9513:⊆
9502:. We see
9487:⊆
9467:Consider
9434:−
9426:⊆
9420:↔
9414:⊆
9402:∃
9364:∗
9342:≅
9321:∃
9287:⊆
9258:⊆
9241:−
9087:×
9072:∗
8990:×
8923:⊂
8832:∀
8805:∃
8775:×
8748:→
8663:⟶
8641:∗
8611:→
8549:∧
8529:∩
8503:ϕ
8476:ψ
8373:ϕ
8369:∧
8345:ψ
8334:∃
8331:∣
8325:∈
8231:ψ
8192:φ
8189:∧
8171:ϕ
8149:ϕ
8085:⊂
8044:ϕ
8040:∣
8002:ϕ
7750:(cf. the
7734:Kaplansky
7434:on a set
7313:from the
7005:⊔
6979:semigroup
6931:, by the
6836:, and if
6808: :
6791:cokernels
6769:adjoints.
6378:ε
6334:η
6282:ε
6240:η
6195:η
6192:∘
6189:ε
6122:ε
6068:η
6011:ε
5975:η
5928:η
5922:∘
5916:ε
5874:⊣
5862:η
5856:ε
5769:→
5748:ε
5702:ε
5658:→
5637:ε
5570:→
5555:η
5505:→
5490:η
5316:⊗
5313:−
5210:η
5207:∘
5204:ε
5020:η
5016:∘
5004:ε
4961:η
4951:∘
4939:ε
4895:and each
4865:η
4862:∘
4859:ε
4832:η
4826:∘
4820:ε
4759:⊣
4733:⊣
4721:η
4715:ε
4639:ε
4610:η
4569:ε
4546:η
4485:→
4463:η
4444:→
4428:ε
4339:for each
4236:morphisms
4079:→
4025:Φ
3994:−
3985:−
3961:→
3955:−
3949:−
3919:Φ
3682:→
3650:∘
3636:η
3623:η
3619:∘
3575:→
3503:η
3499:∘
3464:→
3414:→
3324:→
3309:η
3108:→
3009:→
2964:ϵ
2960:∘
2930:∘
2921:ϵ
2897:→
2812:∘
2803:ϵ
2770:→
2738:→
2657:→
2624:ϵ
2423:→
2152:formulaic
2053:) where S
1985:morphisms
1915:formulaic
1913:, and is
1900:formulaic
1872:formulaic
1817:⟩
1802:⟨
1796:⟩
1781:⟨
1705:⊣
1356:φ
1214:φ
1153:≅
1100:φ
938:⊣
763:→
747:−
697:→
681:−
580:→
555:−
507:→
485:−
367:≅
316:bijection
204:→
164:→
138:covariant
16902:Category
16877:Monoidal
16847:Enriched
16842:Diagonal
16822:Additive
16768:Glossary
16748:Category
16722:n-monoid
16675:concepts
16331:Colimits
16299:Products
16252:Morphism
16195:Concrete
16190:Additive
16180:Category
16106:functors
16094:WildCats
16038:(1998).
15843:and the
15505: :
15465: :
15264: :
15248: :
15189:colimits
15103:→
15085:′
15074:→
15065:′
15057:′
15041:′
15027:′
15016:′
15005:′
14997:→
14991:′
14983:′
14972:′
14964:→
14889:′
14846:′
14674:′
14634:′
14533:β², then
14517: :
14433:∈
14420: :
14362: :
14349:, not a
14317: :
14265: :
13332: :
13320: :
12732:, define
12724: :
12712: :
12650: :
12638: :
12453: :
12435: :
12358: :
12208:and get
12150: :
12123: :
12076: :
12060: :
12021:) is an
11300: :
11270: :
11254: :
11224: :
11116: :
11098: :
10784:and any
10509:solution
10495:powerset
9732:implies
8881:back to
7978:currying
7908: :
7892: :
7845:division
7761:is that
7730:antitone
7711:antitone
7592:between
7584: :
7551: :
7531: :
7402:Topology
7345:K-theory
7305: :
7233:algebras
7152: :
7136: :
7091: :
7067: :
7027:Suppose
6860:), then
6840: :
6732: :
6728:and let
6716: :
6632: :
6611: :
6598:Kernels.
6470:Products
6447:Products
6270:→
6234:→
6005:→
5963:→
5435: :
5407: :
5387:Examples
5301:, where
5281:) = hom(
5239:terminal
4774: .
4631:→
4605:→
4564:→
4538:→
4514:and the
4410:and two
4402: :
4390: :
4385:functors
4278:commutes
4260: :
4242: :
3896: :
3883: :
3878:functors
3689:′
3644:′
3005:′
2972:′
2386: :
2268:solves.
1991:between
1950:supremum
1841:β
1730:idea of
1086:Mac Lane
1078:cognates
1008:between
880:, while
55:functors
16872:Logical
16837:Derived
16827:Adjoint
16810:Functor
16758:Outline
16717:n-group
16682:2-group
16637:Strict
16627:β-topos
16423:Modules
16361:Pushout
16309:Kernels
16242:Functor
16185:Abelian
16084:YouTube
15918:1993102
15830:comonad
15560:as the
15477:, then
15405:, then
15272:, then
15200:product
14564:β² then
13407: ,
12109:) from
11993:) is a
11093:functor
9037:Example
7960:. In a
7853:adjoint
7726:duality
7670:duality
7605:sheaves
7543:. Then
7522:compact
7490:to the
7389:in the
7206:monoids
7172:) = hom
7083:. Then
7047:. Then
6945:Algebra
6929:modules
6795:colimit
6515:be the
6459:kernels
5443:be the
5361:natural
5339:), and
5253:History
5235:initial
4355:). Hom(
4311:) β Hom
3864:hom-set
2305:adjoint
2279:, then
2248:is the
2198:is the
2194:. Then
2174:. Let
2139:* is a
1979:, with
1969:objects
1919:functor
1886:. The
1373:adjunct
1073:adjunct
1067:adjoint
422:natural
16887:Smooth
16704:Traced
16687:2-ring
16417:Fields
16403:Groups
16398:Magmas
16286:Limits
16062:
16052:
16026:
16016:
15983:
15916:
15637:Monads
15256:is an
15177:limits
14556:, and
14541:β² are
14167:) for
14064:) for
12449:, any
12431:, any
12320:, as (
12199:with Ξ·
12164:, as (
11173:counit
9814:under
9148:along
7900:is an
7849:invert
7797:, let
7781:, let
7748:monads
7728:(i.e.
7685:Every
7681:Posets
7633:, the
7615:, the
7577:Every
7559:, the
7456:Given
7393:: see
7273:. Let
7214:groups
6989:monoid
6789:, and
6726:kernel
6712:. Let
6457:, and
6313:
6218:
6165:
5895:
5353:equals
5247:monads
4512:counit
4343:in Hom
3905:and a
1967:whose
1623:, and
1439:, and
1230:or an
16857:Exact
16812:types
16698:-ring
16585:Weak
16569:Topos
16413:Rings
16010:(PDF)
15914:JSTOR
15890:(PDF)
15866:arXiv
15851:Notes
15820:, Ξ΅,
15651:monad
14809:and γ
14745:Here
14448:. If
14277:is a
14158:GX, X
14151:and Ξ΅
14042:and Ξ·
12261:is a
12025:from
11997:from
9967:. So
9687:. So
9208:into
9128:into
9039:: In
8120:into
7557:KHaus
7533:KHaus
7518:KHaus
7379:, or
7351:on a
7343:. In
7254:Field
7222:field
7059:with
7043:is a
6510:Grp β
6463:limit
6093:into
5367:from
5335:with
4667:and 1
4226:(the
4212:from
4192:β, β)
3479:with
3171:from
3123:is a
2794:with
2486:from
2438:is a
2329:right
2264:that
2231:start
1773:with
1273:. If
81:of a
16388:Sets
16050:ISBN
16014:ISBN
15981:ISBN
15960:nLab
15597:and
15586:and
15445:and
15420:are
15416:and
15244:and
15240:are
15236:and
14932:G' G
14930:and
14928:F F'
14821:and
14805:and
14537:and
14529:and
14393:) β
14147:for
14038:for
14017:and
12370:) =
12366:) β
12328:), Ξ·
12301:-).
12227:) β
12175:), Ξ·
12103:), Ξ·
12044:and
11296:) =
11250:) =
11202:and
11189:unit
11083:and
9276:For
8823:and
8269:and
7920:and
7527:and
7516:Let
7463:and
7420:sets
7410:Let
7289:Ring
7282:Ring
7275:Ring
7200:The
7031:and
6927:and
6875:and
6763:β 0.
6734:Ab β
6625:and
6586:sets
6580:The
6556:and
6401:GFGX
6361:GFGX
6319:GFGX
6145:FGFY
6111:FGFY
6095:FGFY
6047:FGFY
5423:the
5403:Let
5269:hom(
5243:unit
4694:and
4677:and
4516:unit
4398:and
4379:and
4202:(β,
4194:and
4159:and
4138:and
3891:and
3872:and
3665:for
2335:and
2327:and
2325:left
2150:and
2080:and
2013:are
2002:and
1923:dual
1884:ring
1252:and
1070:and
1032:and
980:and
720:and
537:and
444:and
314:, a
270:and
112:and
16232:End
16222:CCC
16060:Zbl
16024:Zbl
15937:",
15933:, "
15904:doi
15832:in
15572:of
15564:of
15543:of
15493:).
15232:If
15191:).
14813:β²,
14611:β²,
14409:in
14344:set
14301:of
14285:of
14155:= Ξ¦
14048:= Ξ¦
13849:in
13845:to
13835:, Ξ·
13827:in
13823:to
13813:, Ξ΅
13730:in
13537:in
13356:(-,
12888:= Ξ΅
12700:(-,
12529:))
12427:in
12409:in
12316:in
12308:in
12297:(-,
12113:to
12091:in
12033:in
12029:to
12015:, Ξ·
12005:in
12001:to
11987:, Ξ΅
11308:in
11165:β 1
11150:(β,
11075:An
10341:of
9346:Hom
9313:Hom
9047:Set
8978:in
8890:Sub
8859:Sub
8717:of
8667:Sub
8650:Sub
8427:of
7993:If
7888:If
7867:in
7777:in
7693:to
7553:Top
7537:Top
7498:of
7482:to
7478:of
7446:on
7418:to
7317:to
7311:Grp
7258:Dom
7247:Dom
7162:Mod
7142:Mod
7116:If
7105:Mod
7097:Mod
7081:Mod
7073:Mod
6953:rng
6923:of
6910:of
6804:If
6657:to
6584:of
6549:to
6533:Grp
6513:Grp
6478:Grp
6474:Grp
6409:FGX
6403:to
6323:FGX
6147:to
6101:of
5839:).
5826:to
5788:to
5735:to
5727:to
5675:FGX
5618:FGX
5595:to
5530:to
5471:GFY
5441:Set
5437:Grp
5416:Grp
5410:Set
5375:to
5277:),
5226:or
4899:in
4891:in
4673:on
4317:FYβ²
4272:in
4251:in
4224:Set
4222:to
4196:hom
4182:hom
4142:in
4134:in
3746:to
3377:in
3280:in
3231:in
3191:to
3147:in
3068:to
3024:in
2692:in
2595:in
2546:in
2506:to
2462:in
2202:of
2065:of
2057:β S
1987:in
1936:of
1880:rng
1683:If
1463:to
1459:is
1419:to
1415:is
1375:of
1226:an
904:or
857:or
806:in
630:in
424:in
290:in
246:in
41:In
16904::
16710:)
16706:)(
16112:,
16108:,
16058:.
16044:.
16022:.
15958:.
15912:.
15900:87
15898:.
15892:.
15836:.
15818:FG
15713:GF
15711:=
15645:,
15617:FG
15524:,
15509:β
15469:β
15268:β
15252:β
15181:is
15169:is
14797:,
14599:β
14588:β
14576:,
14521:β
14429:β
14385:=
14366:β
14321:β
14289::
14269:β
14164:GX
14161:(1
14145:GX
14061:FY
14058:(1
14055:FY
14053:,
14036:FY
14027:GX
14025:β
14021::
14013:β
14011:FY
14009::
13833:FY
13811:GX
13336:β
13328:,
13324:β
13121:GX
13114:)
13108:(Ξ΅
13104:=
13101:GX
12900:(Ξ·
12891:FY
12885:FY
12730:GX
12728:β
12716:β
12714:FY
12654:β
12646:,
12642:β
12612:)
12600:,
12588:)
12580:=
12561:)
12550:)
12542:=
12514:)
12475:,
12460:β
12442:β
12420:,
12402:,
12378:)
12349:,
12339:,
12251:β
12157:β
12127:β
12080:β
12064:β
12013:FY
11985:GX
11304:β
11302:FY
11287:,
11276:GX
11274:β
11260:GX
11258:β
11241:,
11228:β
11226:FY
11185:GF
11183:β
11163:FG
11157:A
11154:β)
11130:A
11120:β
11102:β
11091:A
10881:.
10653:.
7968:β
7912:β
7896:β
7701:β€
7639:.
7588:β
7555:β
7535:β
7476:SX
7337:/.
7309:β
7307:Ab
7287:β
7256:β
7156:β
7154:Ab
7146:Ab
7144:β
7099:β
7075:β
7039:β
6914:).
6856:,
6846:Ab
6844:β
6842:Ab
6825:,
6816:Ab
6810:Ab
6785:,
6781:,
6722:Ab
6720:β
6697:=
6673:,
6639:β
6618:β
6527:,
6487:,
6476:β
6453:,
6449:,
6419:.
6417:GX
6405:GX
6357:GX
6159:.
6157:FY
6153:FY
6149:FY
6103:FY
6091:FY
6051:FY
5824:FY
5820:GX
5808:FY
5790:GX
5733:FZ
5622:GX
5606:.
5593:FY
5540:GW
5475:FY
5459::
5439:β
5413:β
5383:.
5371:Γ
5293:))
5285:,
5167:.
5165:FY
5102:,
4903:,
4406:β
4394:β
4371:A
4361:Gf
4359:,
4351:,
4349:FY
4337:Fg
4333:h
4329:f
4327:β
4321:Xβ²
4319:,
4307:,
4305:FY
4295:,
4293:Fg
4280::
4267:β
4248:Xβ²
4246:β
4217:Γ
4206:β)
4169:.
4146:.
3900:β
3887:β
3861:A
3849:.
3766:.
3722:;
3527:.
3088:.
3044:;
2849::
2842:.
2390:β
2374:,
2370:,
2358:,
2354:,
2287:.
2219:R*
2217:β
2211:R*
2206:.
2190:)=
2172:R*
2170:)=
2125:R*
2123:β
2084:β
2073:β
2046:β
2039:,
2032:β
2028:,
2021:β
2006:β
1995:β
1975:β
1753:,
1680:.
1348:,
1084:,
953:.
830:.
595:op
522:op
140:)
49:,
16802:e
16795:t
16788:v
16702:(
16695:n
16693:E
16655:)
16651:(
16639:n
16603:)
16599:(
16587:n
16429:)
16425:(
16419:)
16415:(
16143:e
16136:t
16129:v
16116:.
16066:.
16030:.
15962:.
15920:.
15906::
15874:.
15868::
15834:C
15826:G
15824:Ξ·
15822:F
15814:F
15812:Ξ΅
15810:G
15792:T
15784:2
15780:T
15776::
15747:T
15738:D
15733:1
15729::
15709:T
15690:D
15680:D
15675::
15672:T
15659:D
15655:T
15653:γ
15647:G
15643:F
15627:F
15622:D
15613:G
15609:F
15602:1
15599:C
15595:1
15592:D
15588:G
15584:F
15579:Y
15574:D
15570:Y
15566:D
15558:1
15555:D
15550:X
15545:C
15541:X
15537:C
15533:1
15530:C
15526:G
15522:F
15511:C
15507:D
15503:F
15491:C
15487:D
15479:G
15475:D
15471:D
15467:C
15463:G
15459:D
15455:C
15447:D
15443:C
15418:D
15414:C
15407:F
15403:F
15399:G
15382:)
15379:X
15376:G
15373:,
15370:Y
15367:(
15361:D
15355:m
15352:o
15349:h
15341:)
15338:X
15335:,
15332:Y
15329:F
15326:(
15320:C
15314:m
15311:o
15308:h
15303::
15298:X
15295:,
15292:Y
15274:G
15270:D
15266:C
15262:G
15254:C
15250:D
15246:F
15238:D
15234:C
15223:.
15216:;
15153:C
15125:.
15119:C
15114:1
15098:G
15095:F
15089:G
15078:F
15069:G
15062:G
15054:F
15050:F
15038:F
15034:F
15031:G
15024:G
15013:F
15002:G
14988:F
14980:G
14956:E
14951:1
14911:.
14908:E
14902:C
14899::
14896:G
14886:G
14859:C
14853:E
14850::
14843:F
14836:F
14823:E
14819:D
14815:G
14811:F
14807:D
14803:C
14799:G
14795:F
14726:.
14723:)
14718:1
14702:1
14691:(
14682:=
14657:)
14645:(
14642:=
14613:G
14609:F
14603:β²
14601:G
14597:G
14592:β²
14590:F
14586:F
14578:G
14574:F
14570:G
14566:F
14562:G
14558:G
14554:G
14550:F
14539:G
14535:G
14531:G
14527:G
14523:C
14519:D
14515:F
14499:F
14495:F
14489:F
14485:F
14468:D
14462:C
14459::
14456:F
14437:.
14435:C
14431:X
14426:i
14422:X
14418:t
14411:I
14407:i
14399:i
14395:f
14391:t
14389:(
14387:G
14383:h
14374:)
14372:X
14370:(
14368:G
14364:Y
14360:h
14347:I
14340:i
14334:)
14331:i
14327:X
14325:(
14323:G
14319:Y
14314:i
14310:f
14303:D
14299:Y
14291:G
14275:C
14271:D
14267:C
14263:G
14241:.
14227:X
14224:G
14213:)
14208:X
14200:(
14197:G
14194:=
14189:X
14186:G
14182:1
14169:f
14153:X
14149:Y
14140:,
14128:)
14123:Y
14115:(
14112:F
14104:Y
14101:F
14093:=
14088:Y
14085:F
14081:1
14066:g
14051:Y
14045:Y
14040:X
14023:Y
14019:g
14015:X
14007:f
13987:)
13984:g
13981:(
13978:F
13970:X
13962:=
13959:)
13956:g
13953:(
13948:1
13940:X
13937:,
13934:Y
13920:Y
13909:)
13906:f
13903:(
13900:G
13897:=
13894:)
13891:f
13888:(
13883:X
13880:,
13877:Y
13853:.
13851:D
13847:G
13843:Y
13838:Y
13829:C
13825:X
13821:F
13816:X
13794:)
13791:Y
13788:F
13785:,
13782:Y
13779:F
13776:(
13771:C
13766:m
13763:o
13760:h
13750:Y
13747:F
13743:1
13732:D
13728:Y
13714:)
13711:Y
13708:F
13705:G
13702:,
13699:Y
13696:(
13691:D
13686:m
13683:o
13680:h
13672:)
13667:Y
13664:F
13660:1
13656:(
13651:Y
13648:F
13645:,
13642:Y
13634:=
13629:Y
13601:)
13598:X
13595:G
13592:,
13589:X
13586:G
13583:(
13578:D
13573:m
13570:o
13567:h
13557:X
13554:G
13550:1
13539:C
13535:X
13521:)
13518:X
13515:,
13512:X
13509:G
13506:F
13503:(
13498:C
13493:m
13490:o
13487:h
13479:)
13474:X
13471:G
13467:1
13463:(
13458:1
13450:X
13447:,
13444:X
13441:G
13433:=
13428:X
13395:G
13389:F
13386::
13383:)
13377:,
13371:(
13358:G
13353:D
13348:F
13346:(
13343:C
13338:D
13334:C
13330:G
13326:C
13322:D
13318:F
13289:g
13286:=
13283:g
13275:X
13272:G
13268:1
13264:=
13254:g
13246:X
13243:G
13232:)
13227:X
13219:(
13216:G
13213:=
13201:Y
13190:)
13187:g
13184:(
13181:F
13178:G
13172:)
13167:X
13159:(
13156:G
13153:=
13146:g
13118:Ξ·
13116:o
13111:X
13106:G
13096:G
13071:f
13068:=
13063:Y
13060:F
13056:1
13049:f
13046:=
13036:)
13031:Y
13023:(
13020:F
13012:Y
13009:F
12998:f
12995:=
12985:)
12980:Y
12972:(
12969:F
12963:)
12960:f
12957:(
12954:G
12951:F
12943:X
12935:=
12928:f
12903:Y
12898:F
12895:o
12880:F
12855:)
12852:g
12849:(
12846:F
12838:X
12830:=
12827:)
12824:g
12821:(
12816:X
12813:,
12810:Y
12796:Y
12785:)
12782:f
12779:(
12776:G
12773:=
12770:)
12767:f
12764:(
12759:X
12756:,
12753:Y
12726:Y
12722:g
12718:X
12710:f
12702:G
12697:D
12692:F
12690:(
12687:C
12682:G
12660:F
12656:D
12652:C
12648:G
12644:C
12640:D
12636:F
12617:y
12614:o
12610:f
12608:(
12605:0
12602:X
12598:0
12595:Y
12592:Ξ¦
12590:o
12586:x
12584:(
12582:G
12578:y
12575:o
12571:0
12568:Y
12565:Ξ·
12563:o
12559:f
12557:(
12555:G
12552:o
12548:x
12546:(
12544:G
12539:1
12536:Y
12533:Ξ·
12531:o
12527:y
12525:(
12523:F
12521:(
12519:G
12516:o
12512:f
12510:(
12508:G
12505:o
12501:y
12499:(
12497:F
12494:o
12491:f
12488:o
12485:x
12483:(
12480:1
12477:X
12473:1
12470:Y
12465:0
12462:Y
12458:1
12455:Y
12451:y
12447:1
12444:X
12440:0
12437:X
12433:x
12429:D
12425:1
12422:Y
12418:0
12415:Y
12411:C
12407:1
12404:X
12400:0
12397:X
12393:G
12388:.
12385:Y
12382:Ξ·
12380:o
12376:f
12374:(
12372:G
12368:X
12364:Y
12362:(
12360:F
12356:f
12354:(
12351:X
12347:Y
12341:X
12337:Y
12331:Y
12326:Y
12324:(
12322:F
12318:D
12314:Y
12310:C
12306:X
12299:G
12294:D
12289:F
12287:(
12284:C
12274:F
12270:G
12265:.
12259:F
12256:o
12253:G
12248:D
12240:F
12236:1
12233:Y
12231:(
12229:F
12225:0
12222:Y
12220:(
12218:F
12214:f
12212:(
12210:F
12205:0
12202:Y
12197:f
12194:o
12190:1
12187:Y
12181:0
12178:Y
12173:0
12170:Y
12168:(
12166:F
12162:1
12159:Y
12155:0
12152:Y
12148:f
12141:F
12137:Y
12135:(
12133:F
12131:(
12129:G
12125:Y
12120:Y
12115:G
12111:Y
12106:Y
12101:Y
12099:(
12097:F
12093:D
12089:Y
12082:C
12078:D
12074:F
12066:D
12062:C
12058:G
12046:G
12042:F
12035:D
12031:G
12027:Y
12018:Y
12007:C
12003:X
11999:F
11990:X
11961:X
11958:G
11947:)
11942:X
11934:(
11931:G
11928:=
11919:X
11916:G
11912:1
11904:)
11899:Y
11891:(
11888:F
11880:Y
11877:F
11869:=
11860:Y
11857:F
11853:1
11816:)
11813:)
11810:Y
11807:(
11804:F
11801:G
11798:,
11795:Y
11792:(
11787:D
11782:m
11779:o
11776:h
11760:Y
11752:=
11745:)
11740:Y
11737:F
11733:1
11729:(
11724:Y
11721:F
11718:,
11715:Y
11703:)
11700:X
11697:,
11694:)
11691:X
11688:(
11685:G
11682:F
11679:(
11674:C
11669:m
11666:o
11663:h
11647:X
11639:=
11632:)
11627:X
11624:G
11620:1
11616:(
11611:1
11603:X
11600:,
11597:X
11594:G
11582:)
11579:)
11576:X
11573:(
11570:G
11567:,
11564:Y
11561:(
11556:D
11551:m
11548:o
11545:h
11529:Y
11518:)
11515:f
11512:(
11509:G
11506:=
11499:)
11496:f
11493:(
11488:X
11485:,
11482:Y
11474:=
11471:g
11464:)
11461:X
11458:,
11455:)
11452:Y
11449:(
11446:F
11443:(
11438:C
11433:m
11430:o
11427:h
11413:)
11410:g
11407:(
11404:F
11396:X
11388:=
11381:)
11378:g
11375:(
11370:1
11362:X
11359:,
11356:Y
11348:=
11345:f
11310:C
11306:X
11298:f
11294:g
11292:(
11289:X
11285:Y
11280:C
11272:Y
11268:g
11264:D
11256:Y
11252:g
11248:f
11246:(
11243:X
11239:Y
11234:D
11230:X
11222:f
11218:C
11208:D
11204:Y
11200:C
11196:X
11180:D
11168:C
11152:G
11147:D
11142:F
11140:(
11137:C
11122:D
11118:C
11114:G
11104:C
11100:D
11096:F
11085:D
11081:C
11031:E
11009:E
10981:E
10960:]
10954:[
10950:E
10940::
10936:E
10913:x
10902:x
10899::
10867:1
10860:f
10844::
10841:)
10838:f
10835:,
10829:(
10809:)
10805:R
10801:(
10798:M
10772:b
10769:+
10766:x
10763:a
10760:=
10757:)
10754:x
10751:(
10748:f
10728:)
10724:R
10720:(
10717:M
10696:R
10675:)
10671:R
10667:(
10664:M
10641:)
10638:r
10635:(
10632:f
10626:r
10623::
10620:)
10617:f
10614:,
10611:r
10608:(
10588:r
10568:b
10565:+
10562:x
10559:a
10556:=
10553:)
10550:x
10547:(
10544:f
10523:R
10497:.
10487:.
10435:S
10415:f
10395:}
10392:y
10389:{
10369:y
10349:Y
10329:S
10324:f
10295:.
10292:}
10288:S
10282:x
10278:.
10275:)
10272:]
10269:}
10266:y
10263:{
10260:[
10255:1
10248:f
10241:x
10238:(
10229:Y
10223:y
10220:{
10217:=
10214:S
10209:f
10191:.
10179:}
10176:)
10173:x
10170:(
10165:S
10154:)
10151:y
10148:,
10145:x
10142:(
10137:f
10128:.
10125:x
10116:Y
10110:y
10107:{
10083:.
10080:]
10077:S
10074:[
10071:f
10068:=
10065:}
10061:S
10055:x
10051:.
10048:)
10045:]
10042:}
10039:y
10036:{
10033:[
10028:1
10021:f
10014:x
10011:(
10002:Y
9996:y
9993:{
9990:=
9987:S
9982:f
9955:]
9952:S
9949:[
9946:f
9926:Y
9920:y
9900:S
9894:]
9891:}
9888:y
9885:{
9882:[
9877:1
9870:f
9849:y
9827:f
9802:S
9776:f
9755:T
9749:]
9746:S
9743:[
9740:f
9720:]
9717:T
9714:[
9709:1
9702:f
9695:S
9675:T
9669:)
9666:x
9663:(
9660:f
9640:]
9637:T
9634:[
9629:1
9622:f
9615:x
9595:S
9589:x
9569:]
9566:T
9563:[
9558:1
9551:f
9544:]
9541:]
9538:S
9535:[
9532:f
9529:[
9524:1
9517:f
9510:S
9490:T
9484:]
9481:S
9478:[
9475:f
9460:.
9448:]
9445:T
9442:[
9437:1
9430:f
9423:S
9417:T
9411:S
9406:f
9375:,
9372:)
9369:T
9360:f
9356:,
9353:S
9350:(
9339:)
9336:T
9333:,
9330:S
9325:f
9317:(
9290:X
9284:S
9273:.
9261:X
9255:]
9252:T
9249:[
9244:1
9237:f
9216:Y
9196:T
9176:f
9156:f
9136:Y
9116:T
9096:T
9091:Y
9083:X
9080:=
9077:T
9068:f
9031:.
9019:Y
8999:T
8994:Y
8986:X
8966:X
8946:f
8926:X
8920:S
8900:)
8897:Y
8894:(
8869:)
8866:X
8863:(
8836:f
8809:f
8784:T
8779:Y
8771:X
8751:Y
8745:T
8725:Y
8705:T
8677:)
8674:X
8671:(
8660:)
8657:Y
8654:(
8646::
8637:f
8614:Y
8608:X
8605::
8602:f
8582:Y
8507:S
8480:f
8455:x
8435:Y
8415:y
8391:}
8388:)
8385:x
8382:(
8377:S
8366:)
8363:y
8360:,
8357:x
8354:(
8349:f
8340:.
8337:x
8328:Y
8322:y
8319:{
8297:X
8277:Y
8257:X
8235:f
8201:)
8198:y
8195:(
8186:)
8183:y
8180:(
8175:Y
8167:=
8164:)
8161:y
8158:(
8153:T
8128:Y
8108:T
8088:Y
8082:T
8062:}
8059:)
8056:y
8053:(
8048:Y
8037:y
8034:{
8031:=
8028:Y
8006:Y
7974:A
7970:C
7966:C
7952:A
7948:U
7944:U
7940:D
7936:D
7932:0
7922:G
7918:F
7914:D
7910:C
7906:G
7898:C
7894:D
7890:F
7875:.
7841:.
7839:F
7835:G
7831:T
7827:S
7825:(
7823:F
7819:T
7817:(
7815:G
7811:S
7807:S
7803:S
7801:(
7799:F
7795:S
7791:T
7787:T
7785:(
7783:G
7779:C
7775:T
7771:D
7767:C
7754:)
7703:y
7699:x
7695:y
7691:x
7676:.
7648:β
7645:f
7641:f
7631:X
7627:Y
7623:f
7613:Y
7609:X
7601:β
7598:f
7590:Y
7586:X
7582:f
7569:X
7549:F
7545:G
7529:G
7510:.
7500:Y
7496:Y
7494:Ξ©
7488:X
7484:Y
7480:X
7465:Y
7461:X
7450:.
7448:Y
7440:H
7436:Y
7428:F
7424:G
7412:G
7335:G
7333:=
7331:G
7327:G
7303:G
7285:*
7278:*
7267:.
7261:m
7250:m
7239:.
7237:K
7231:-
7229:K
7225:K
7194:.
7192:F
7188:A
7184:A
7182:,
7180:M
7178:(
7175:Z
7170:A
7168:(
7166:G
7160:-
7158:R
7150:G
7140:-
7138:R
7134:F
7130:M
7126:R
7122:M
7118:R
7114:.
7107:.
7103:-
7101:R
7095:-
7093:S
7089:G
7085:F
7079:-
7077:S
7071:-
7069:R
7065:F
7061:S
7053:R
7049:S
7041:S
7037:R
7033:S
7029:R
7019:S
6996:S
6985:S
6981:.
6971:Z
6967:Z
6965:x
6963:R
6959:R
6955:.
6912:X
6908:b
6906:+
6904:a
6900:b
6898:,
6896:a
6892:X
6888:X
6886:,
6884:X
6880:2
6877:X
6873:1
6870:X
6866:G
6862:F
6858:Y
6854:Y
6850:Y
6838:G
6830:2
6827:X
6823:1
6820:X
6813:β
6806:F
6761:A
6757:A
6753:F
6749:G
6745:A
6741:A
6737:D
6730:F
6718:D
6714:G
6709:A
6705:g
6702:2
6699:f
6695:1
6692:f
6688:B
6684:g
6679:B
6675:g
6670:A
6666:g
6662:2
6659:f
6655:1
6652:f
6648:D
6644:2
6641:B
6637:2
6634:A
6630:2
6627:f
6623:1
6620:B
6616:1
6613:A
6609:1
6606:f
6602:D
6573:X
6571:Γ
6569:X
6561:2
6558:X
6554:1
6551:X
6547:2
6544:X
6542:Γ
6540:1
6537:X
6529:X
6525:X
6521:X
6506:2
6503:X
6501:Γ
6499:1
6496:X
6491:2
6489:X
6485:1
6482:X
6413:X
6387:)
6382:X
6374:(
6371:G
6341:X
6338:G
6301:X
6298:G
6291:)
6286:X
6278:(
6275:G
6265:X
6262:G
6259:F
6256:G
6247:X
6244:G
6229:X
6226:G
6212:X
6198:G
6186:G
6183:=
6178:G
6174:1
6129:Y
6126:F
6107:y
6099:y
6077:)
6072:Y
6064:(
6061:F
6030:Y
6027:F
6018:Y
6015:F
6000:Y
5997:F
5994:G
5991:F
5984:)
5979:Y
5971:(
5968:F
5958:Y
5955:F
5942:Y
5925:F
5919:F
5913:=
5908:F
5904:1
5877:G
5871:F
5868::
5865:)
5859:,
5853:(
5837:G
5835:,
5833:F
5831:(
5828:X
5816:Y
5812:X
5798:G
5796:,
5794:F
5786:Z
5772:X
5766:X
5763:G
5760:F
5757::
5752:X
5737:X
5729:X
5725:F
5711:)
5706:X
5698:,
5695:X
5692:G
5689:(
5679:X
5661:X
5655:X
5652:G
5649:F
5646::
5641:X
5626:X
5614:X
5603:Y
5597:W
5579:Y
5576:F
5573:G
5567:Y
5564::
5559:Y
5544:W
5536:Y
5532:G
5528:Y
5514:Y
5511:F
5508:G
5502:Y
5499::
5494:Y
5479:Y
5467:Y
5457:G
5453:F
5449:X
5433:G
5429:Y
5421:Y
5405:F
5377:Y
5373:A
5369:X
5345:A
5341:G
5337:A
5319:A
5303:F
5291:Y
5289:(
5287:G
5283:X
5279:Y
5275:X
5273:(
5271:F
5228:G
5224:F
5201:=
5198:1
5149:Y
5146:F
5142:1
5131:F
5115:F
5111:1
5088:C
5063:C
5058:1
5045:.
5027:X
5024:G
5013:)
5008:X
5000:(
4997:G
4994:=
4985:X
4982:G
4978:1
4970:)
4965:Y
4957:(
4954:F
4946:Y
4943:F
4935:=
4926:Y
4923:F
4919:1
4901:D
4897:Y
4893:C
4889:X
4868:G
4856:G
4853:=
4844:G
4840:1
4829:F
4823:F
4817:=
4808:F
4804:1
4783:Ξ·
4781:,
4779:Ξ΅
4762:G
4756:F
4736:G
4730:F
4727::
4724:)
4718:,
4712:(
4701:F
4697:G
4691:G
4687:F
4679:G
4675:F
4670:G
4664:F
4646:G
4636:G
4626:G
4623:F
4620:G
4613:G
4600:G
4579:F
4572:F
4559:F
4556:G
4553:F
4543:F
4533:F
4491:F
4488:G
4479:D
4474:1
4470::
4453:C
4448:1
4441:G
4438:F
4435::
4408:D
4404:C
4400:G
4396:C
4392:D
4388:F
4381:D
4377:C
4357:g
4353:X
4347:(
4345:C
4341:h
4335:o
4331:o
4325:h
4315:(
4313:C
4309:X
4303:(
4301:C
4297:f
4274:D
4269:Y
4264:β²
4262:Y
4258:g
4253:C
4244:X
4240:f
4219:C
4215:D
4204:G
4199:D
4190:F
4188:(
4185:C
4166:F
4162:G
4156:G
4152:F
4144:D
4140:Y
4136:C
4132:X
4115:)
4112:X
4109:G
4106:,
4103:Y
4100:(
4095:D
4090:m
4087:o
4084:h
4076:)
4073:X
4070:,
4067:Y
4064:F
4061:(
4056:C
4051:m
4048:o
4045:h
4040::
4035:X
4032:,
4029:Y
4009:.
3997:)
3991:G
3988:,
3982:(
3977:D
3972:m
3969:o
3966:h
3958:)
3952:,
3946:F
3943:(
3938:C
3933:m
3930:o
3927:h
3922::
3902:D
3898:C
3894:G
3889:C
3885:D
3881:F
3874:D
3870:C
3837:F
3817:G
3797:G
3777:F
3754:F
3730:G
3710:D
3686:Y
3679:Y
3676::
3673:g
3653:g
3641:Y
3632:=
3627:Y
3616:)
3613:)
3610:g
3607:(
3604:F
3601:(
3598:G
3578:C
3572:D
3569::
3566:F
3546:F
3515:g
3512:=
3507:Y
3496:)
3493:f
3490:(
3487:G
3467:X
3461:)
3458:Y
3455:(
3452:F
3449::
3446:f
3426:)
3423:X
3420:(
3417:G
3411:Y
3408::
3405:g
3385:C
3365:X
3345:)
3342:)
3339:Y
3336:(
3333:F
3330:(
3327:G
3321:Y
3318::
3313:Y
3288:C
3268:)
3265:Y
3262:(
3259:F
3239:D
3219:Y
3199:G
3179:Y
3155:D
3135:Y
3111:D
3105:C
3102::
3099:G
3076:G
3052:F
3032:C
3012:X
3002:X
2998::
2995:f
2969:X
2957:f
2954:=
2951:)
2948:)
2945:f
2942:(
2939:G
2936:(
2933:F
2925:X
2900:D
2894:C
2891::
2888:G
2868:G
2830:f
2827:=
2824:)
2821:g
2818:(
2815:F
2807:X
2782:)
2779:X
2776:(
2773:G
2767:Y
2764::
2761:g
2741:X
2735:)
2732:Y
2729:(
2726:F
2723::
2720:f
2700:D
2680:Y
2660:X
2654:)
2651:)
2648:X
2645:(
2642:G
2639:(
2636:F
2633::
2628:X
2603:D
2583:)
2580:X
2577:(
2574:G
2554:C
2534:X
2514:X
2494:F
2470:C
2450:X
2426:C
2420:D
2417::
2414:F
2396:C
2392:C
2388:D
2384:F
2380:D
2376:g
2372:G
2368:Y
2364:C
2360:f
2356:F
2352:X
2345:D
2341:C
2337:D
2333:C
2307:.
2285:F
2281:G
2277:G
2273:F
2266:F
2258:G
2254:G
2246:F
2239:F
2235:F
2215:R
2204:G
2196:F
2192:S
2188:S
2186:(
2184:G
2180:S
2176:G
2168:R
2166:(
2164:F
2160:F
2137:R
2133:E
2129:E
2121:R
2117:1
2114:S
2110:2
2107:S
2103:2
2100:S
2096:1
2093:S
2089:2
2086:S
2082:R
2078:1
2075:S
2071:R
2067:R
2059:2
2055:1
2051:2
2048:S
2044:1
2041:S
2037:2
2034:S
2030:R
2026:1
2023:S
2019:R
2011:2
2008:S
2004:R
2000:1
1997:S
1993:R
1989:E
1981:S
1977:S
1973:R
1965:E
1961:R
1954:E
1938:E
1930:E
1896:r
1892:r
1814:x
1811:U
1808:,
1805:y
1799:=
1793:x
1790:,
1787:y
1784:T
1761:U
1741:T
1711:.
1708:G
1702:F
1689:G
1685:F
1665:D
1659:m
1656:o
1653:h
1631:G
1608:C
1602:m
1599:o
1596:h
1574:F
1554:F
1534:F
1511:F
1491:G
1471:F
1447:G
1427:G
1403:F
1383:f
1359:f
1336:)
1333:X
1330:,
1327:Y
1324:F
1321:(
1315:C
1309:m
1306:o
1303:h
1281:f
1260:G
1240:F
1191:)
1188:X
1185:G
1182:,
1179:Y
1176:(
1170:D
1164:m
1161:o
1158:h
1150:)
1147:X
1144:,
1141:Y
1138:F
1135:(
1129:C
1123:m
1120:o
1117:h
1112::
1107:Y
1104:X
1042:D
1018:C
990:D
966:C
941:G
935:F
914:F
888:G
867:G
841:F
816:D
794:Y
773:t
770:e
767:S
758:C
753::
750:)
744:G
741:,
738:Y
735:(
730:D
707:t
704:e
701:S
692:C
687::
684:)
678:,
675:Y
672:F
669:(
664:C
640:C
618:X
591:t
587:e
584:S
575:D
570::
567:)
564:X
561:G
558:,
552:(
547:D
518:t
514:e
511:S
502:D
497::
494:)
491:X
488:,
482:F
479:(
474:C
452:Y
432:X
405:)
402:X
399:G
396:,
393:Y
390:(
384:D
378:m
375:o
372:h
364:)
361:X
358:,
355:Y
352:F
349:(
343:C
337:m
334:o
331:h
300:D
278:Y
256:C
234:X
209:D
199:C
194::
191:G
169:C
159:D
154::
151:F
122:D
98:C
38:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.