6656:
36:
1872:
1100:
2400:
6903:
6923:
6913:
3437:
3253:
3242:
4879:
4659:
4149:
3432:{\displaystyle {\begin{aligned}s&={\bigl (}F(f)\circ \pi _{\operatorname {dom} (f)}{\bigr )}_{f\in \operatorname {Hom} (J)}\\t&={\bigl (}\pi _{\operatorname {cod} (f)}{\bigr )}_{f\in \operatorname {Hom} (J)}.\end{aligned}}}
4060:
4322:
1944:
Limits and colimits can also be defined for collections of objects and morphisms without the use of diagrams. The definitions are the same (note that in definitions above we never needed to use composition of morphisms in
3943:
4446:
3543:
3825:
5174:
is that every right adjoint functor is continuous and every left adjoint functor is cocontinuous. Since adjoint functors exist in abundance, this gives numerous examples of continuous and cocontinuous functors.
3123:
3034:
1324:
of limits and cones are colimits and co-cones. Although it is straightforward to obtain the definitions of these by inverting all morphisms in the above definitions, we will explicitly state them here:
2679:
4779:
3258:
2996:
4765:
5967:
Older terminology referred to limits as "inverse limits" or "projective limits", and to colimits as "direct limits" or "inductive limits". This has been the source of a lot of confusion.
1605:
829:
5262:
5345:
1826:
1054:
6042:
6022:
2765:
1455:
682:
4590:
4083:
1728:
1676:
1169:
1137:
952:
900:
332:
2062:
1780:
1644:
1530:
1364:
1008:
868:
754:
528:
2816:
2082:
2030:
2006:
1986:
1963:
1934:
1911:
1866:
1846:
1748:
1696:
1550:
1498:
1478:
1404:
1384:
1266:
1246:
1209:
1189:
1094:
1074:
972:
920:
774:
722:
702:
634:
614:
594:
568:
548:
482:
446:
423:
403:
379:
355:
294:
274:
250:
230:
204:
184:
155:, exist at a high level of abstraction. In order to understand them, it is helpful to first study the specific examples these concepts are meant to generalize.
4016:
4256:
6024:". Terms like "cohomology" and "cofibration" all have a slightly stronger association with the first variable, i.e., the contravariant variable, of the
1290:
It is possible that a diagram does not have a limit at all. However, if a diagram does have a limit then this limit is essentially unique: it is unique
2107:
The definition of limits is general enough to subsume several constructions useful in practical settings. In the following we will consider the limit (
3446:
in terms of coequalizers and coproducts. Both of these theorems give sufficient and necessary conditions for the existence of all (co)limits of shape
3883:
1228:(see below for more information). As with every universal property, the above definition describes a balanced state of generality: The limit object
4380:
3499:
3237:{\displaystyle s,t:\prod _{i\in \operatorname {Ob} (J)}F(i)\rightrightarrows \prod _{f\in \operatorname {Hom} (J)}F(\operatorname {cod} (f))}
3781:
3760:
Like all universal constructions, the formation of limits and colimits is functorial in nature. In other words, if every diagram of shape
6300:
3007:
2084:. The limit (or colimit) of this diagram is the same as the limit (or colimit) of the original collection of objects and morphisms.
6192:
6100:
6585:
6256:
2606:
4874:{\displaystyle \operatorname {colim} \limits _{J}\lim _{I}F(i,j)\rightarrow \lim _{I}\operatorname {colim} \limits _{J}F(i,j).}
2975:
1965:). This variation, however, adds no new information. Any collection of objects and morphisms defines a (possibly large)
6947:
6240:
6209:
6177:
6117:
79:
57:
4721:
50:
3972:(and nonempty) then the unit of the adjunction is an isomorphism so that lim is a left inverse of Ξ. This fails if
1555:
779:
5383:
the corresponding notions would be a functor that takes colimits to limits, or one that takes limits to colimits.
5216:
5302:
4184:
2289:
is a category with two objects and two parallel morphisms from one object to the other, then a diagram of shape
2094:
are defined like limits and colimits, except that the uniqueness property of the mediating morphism is dropped.
1785:
1013:
2588:
6293:
6197:
6105:
6497:
6452:
5911:
6027:
6007:
2712:
6926:
6866:
1871:
1412:
639:
6575:
6235:. Encyclopedia of mathematics and its applications 50-51, 53 . Vol. 1. Cambridge University Press.
2156:
is any object that is uniquely factored through by every other object. This is just the definition of a
6916:
6702:
6566:
6474:
2327:
109:
6875:
6519:
6457:
6380:
6260:
6140:
6071:
6053:
4166:
is connected (and nonempty) then the counit is an isomorphism, so that colim is a left inverse of Ξ.
2875:
2164:
208:
137:
117:
105:
5477:. One can therefore talk about lifting products, equalizers, pullbacks, etc. Finally, one says that
4654:{\displaystyle \operatorname {Hom} (\operatorname {colim} F,N)\cong \lim \operatorname {Hom} (F-,N)}
6906:
6862:
6467:
6286:
2309:
1099:
44:
4144:{\displaystyle \operatorname {Hom} (\operatorname {colim} F,N)\cong \operatorname {Cocone} (F,N).}
6462:
6059:
2281:
17:
6669:
6435:
6415:
6338:
5910:
creates limits but does not preserve coproducts (the coproduct of two abelian groups being the
5771:
There are examples of functors which lift limits uniquely but neither create nor reflect them.
5294:
3831:
3634:
3459:
1701:
1649:
1321:
1142:
1110:
925:
873:
572:
302:
164:
61:
2035:
1753:
1617:
1503:
1337:
981:
841:
727:
501:
6551:
6390:
5781:
5380:
4546:
4248:
2788:
6363:
6358:
6259:
which generates examples of limits and colimits in the category of finite sets. Written by
1890:
6219:
6127:
8:
6707:
6655:
6581:
6385:
5942:
4888:
commute with finite limits. It also holds that small colimits commute with small limits.
4771:
4504:
3701:. Moreover, the requirement that the cone's diagrams commute is true simply because this
3074:
2549:
2392:
5841:
does not preserve coproducts. This situation is typical of algebraic forgetful functors.
6561:
6556:
6538:
6420:
6395:
6229:
3969:
2067:
2015:
1991:
1971:
1948:
1919:
1896:
1851:
1831:
1733:
1681:
1535:
1483:
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1389:
1369:
1251:
1231:
1225:
1194:
1174:
1079:
1059:
957:
905:
759:
707:
687:
619:
599:
579:
553:
533:
467:
431:
408:
388:
364:
340:
279:
259:
235:
215:
189:
169:
148:
4055:{\displaystyle \operatorname {colim} :{\mathcal {C}}^{\mathcal {J}}\to {\mathcal {C}}}
6870:
6807:
6795:
6697:
6622:
6617:
6571:
6353:
6348:
6236:
6205:
6187:
6173:
6113:
6095:
5819:
5375:
5148:
4709:
4170:
3981:
3042:
2553:
2238:
2173:
1881:
1280:
4928:
then by composition (recall that a diagram is just a functor) one obtains a diagram
3728:
Therefore, the definitions of limits and colimits can then be restated in the form:
1248:
has to be general enough to allow any cone to factor through it; on the other hand,
6831:
6717:
6692:
6627:
6612:
6607:
6546:
6375:
6343:
6215:
6123:
5834:
5543:
5171:
5163:
is continuous if and only if it preserves (small) products and equalizers. Dually,
4317:{\displaystyle \operatorname {Hom} (N,\lim F)\cong \lim \operatorname {Hom} (N,F-)}
4202:
3490:
3474:
2252:
2234:
152:
6743:
6309:
6201:
6109:
5167:
is cocontinuous if and only if it preserves (small) coproducts and coequalizers.
2132:
1276:
93:
5791:
preserves limits (but not necessarily colimits). In particular, for any object
5484:
if it lifts all limits. There are dual definitions for the lifting of colimits.
6780:
6775:
6759:
6722:
6712:
6632:
6164:
5888:
2833:
2520:
2241:
and the projections are just the natural projections onto the various factors.
2181:
2145:
1966:
1886:
1220:
449:
358:
125:
3938:{\displaystyle \operatorname {Hom} (N,\lim F)\cong \operatorname {Cone} (N,F)}
6941:
6770:
6602:
6479:
6405:
6065:
5048:
3851:
2408:
2318:
2148:
in set theory). A cone to the empty diagram is essentially just an object of
2009:
493:
113:
6004:
are types of limits. Second, the prefix "co" implies "first variable of the
6524:
6425:
5938:
5876:
5845:
4441:{\displaystyle \lim \operatorname {Hom} (N,F-)=\operatorname {Cone} (N,F).}
4066:
2883:
2828:
Examples of colimits are given by the dual versions of the examples above:
2417:
1315:
141:
4224:. By duality, the contravariant Hom functor must take colimits to limits.
3862:. This adjunction gives a bijection between the set of all morphisms from
2471:, the category with a single object and morphism, then a diagram of shape
6785:
6765:
6637:
6507:
5970:
There are several ways to remember the modern terminology. First of all,
5800:
5096:
One can make analogous definitions for colimits. For instance, a functor
4190:
2857:
2512:
1295:
97:
6062: β Set of arguments where two or more functions have the same value
4347:. This isomorphism is the unique one which respects the limiting cones.
6817:
6755:
6368:
5864:
3538:{\displaystyle \Delta :{\mathcal {C}}\to {\mathcal {C}}^{\mathcal {J}}}
453:
129:
6266:
6162:
5373:
Preservation of limits and colimits is a concept that only applies to
5358:
if and only if Ο is a natural isomorphism. In this sense, the functor
4684:), this relationship can be used to define the colimit of the diagram
3956:. The counit of this adjunction is simply the universal cone from lim
2399:
1936:
has a colimit then this colimit is unique up to a unique isomorphism.
6811:
6502:
4713:
4205:. This follows, in part, from the fact the covariant Hom functor Hom(
3820:{\displaystyle \lim :{\mathcal {C}}^{\mathcal {J}}\to {\mathcal {C}}}
2841:
133:
104:
captures the essential properties of universal constructions such as
6880:
6512:
6410:
5549:
Lifting of limits is clearly related to preservation of limits. If
3036:(i.e. every pair of morphisms with common codomain has a pullback).
2865:
382:
6278:
3850:
commuting with the corresponding universal cones. This functor is
3046:
is a category that has all small limits (i.e. all limits of shape
2317:
is a special case of an equalizer where one of the morphisms is a
6850:
6840:
6489:
6400:
4925:
4358:. The first step is to observe that the limit of the functor Hom(
2693:
are given a small and thin category structure by adding an arrow
1330:
253:
4668:
and respects the colimiting cones. Identifying the limit of Hom(
6845:
3705:
is a natural transformation. (Dually, a natural transformation
3029:{\displaystyle \bullet \rightarrow \bullet \leftarrow \bullet }
2538:). Such an isomorphism uniquely determines a universal cone to
2552:, which are related to categorical limits as follows. Given a
6727:
5367:
5275:
if and only if this map is an isomorphism. If the categories
5151:, then, by the above existence theorem for limits, a functor
4695:
4541:). In fancy language, this amounts to saying that a limit of
1291:
5749:
Dually, one can define creation and reflection of colimits.
5542:
lifts limits uniquely if and only if it lifts limits and is
2869:
are coequalizers of a morphism and a parallel zero morphism.
2767:
becomes a functor and the following equivalence holds :
428:
One is most often interested in the case where the category
6270:
6144:
5752:
The following statements are easily seen to be equivalent:
5210:
have specified limits there is a unique canonical morphism
3980:
is a discrete category, the components of the unit are the
3480:
may be thought of as the category of all diagrams of shape
2851:
are colimits of constant diagrams from discrete categories.
5267:
which respects the corresponding limit cones. The functor
4366:–) can be identified with the set of all cones from
4350:
One can use the above relationship to define the limit of
3089:
has equalizers and all products indexed by the classes Ob(
2674:{\displaystyle F_{x,A}=\{G\in F\mid V(x)\cup A\subset G\}}
147:
Limits and colimits, like the strongly related notions of
6163:
AdΓ‘mek, JiΕΓ; Horst
Herrlich; George E. Strecker (1990).
4154:
The unit of this adjunction is the universal cocone from
3117:
can be constructed as the equalizer of the two morphisms
2998:(i.e. every parallel pair of morphisms has an equalizer),
2845:
are colimits of diagrams indexed by discrete categories.
2140:
is the empty category there is only one diagram of shape
4065:
which assigns each diagram its colimit. This functor is
5925:
lifts limits and colimits uniquely but creates neither.
5875:
and is, therefore, cocontinuous. This explains why the
2879:
are colimits of a pair of morphisms with common domain.
4169:
Note that both the limit and the colimit functors are
381:
is thought of as indexing a collection of objects and
6030:
6010:
5305:
5219:
4782:
4724:
4593:
4383:
4259:
4086:
4019:
3886:
3784:
3502:
3256:
3126:
3010:
2978:
2791:
2715:
2609:
2070:
2038:
2018:
1994:
1974:
1951:
1922:
1899:
1854:
1834:
1788:
1756:
1736:
1704:
1684:
1652:
1620:
1558:
1538:
1506:
1486:
1466:
1415:
1392:
1372:
1340:
1254:
1234:
1197:
1177:
1145:
1113:
1082:
1062:
1016:
984:
960:
928:
908:
876:
844:
782:
762:
730:
710:
690:
642:
622:
602:
582:
556:
536:
504:
470:
434:
411:
391:
367:
343:
305:
282:
262:
238:
218:
192:
172:
5958:
lifts finite limits but does not lift them uniquely.
5631:
5628:
The dual statements for colimits are equally valid.
4339:–) is the composition of the Hom functor Hom(
3458:Limits and colimits are important special cases of
3057:One can also make the dual definitions. A category
6228:
6036:
6016:
5339:
5256:
5085:preserves products, equalizers, pullbacks, etc. A
4873:
4759:
4653:
4451:The limiting cone is given by the family of maps Ο
4440:
4316:
4143:
4054:
3937:
3819:
3537:
3431:
3236:
3028:
2991:{\displaystyle \bullet \rightrightarrows \bullet }
2990:
2887:are colimits of diagrams indexed by directed sets.
2810:
2759:
2673:
2247:. A special case of a product is when the diagram
2076:
2056:
2024:
2000:
1980:
1957:
1928:
1905:
1860:
1840:
1820:
1774:
1742:
1722:
1690:
1670:
1638:
1599:
1544:
1524:
1492:
1472:
1449:
1398:
1378:
1358:
1260:
1240:
1203:
1183:
1163:
1131:
1088:
1068:
1048:
1002:
966:
946:
914:
894:
862:
823:
768:
748:
716:
696:
676:
628:
608:
588:
562:
542:
522:
476:
440:
417:
397:
373:
349:
326:
288:
268:
244:
224:
198:
178:
4178:
2420:(considered as a small category by adding arrows
6939:
5321:
5315:
5245:
5236:
4828:
4797:
4624:
4384:
4287:
4275:
3902:
3785:
1272:such factorization is possible for every cone.
4760:{\displaystyle F:I\times J\to \mathbf {Set} ,}
3830:which assigns each diagram its limit and each
2861:are colimits of a parallel pair of morphisms.
1268:has to be sufficiently specific, so that only
6294:
6074: β Generalized object in category theory
5833:creates (and preserves) all small limits and
5473:if it lifts limits for all diagrams of shape
3393:
3363:
3318:
3273:
2530:has a limit, namely any object isomorphic to
1600:{\displaystyle \psi _{Y}\circ F(f)=\psi _{X}}
824:{\displaystyle F(f)\circ \psi _{X}=\psi _{Y}}
5715:), and furthermore, this cone is a limit of
5257:{\displaystyle \tau _{F}:G\lim F\to \lim GF}
4193:to relate limits and colimits in a category
2913:may or may not have a limit (or colimit) in
2668:
2629:
2548:. Limits of functions are a special case of
2351:, where the only non-identity morphisms are
6056: β Type of category in category theory
5340:{\displaystyle \tau :G\lim \to \lim G^{J}.}
5069:if it preserves the limits of all diagrams
4217:
6922:
6912:
6668:
6301:
6287:
6090:
6088:
5767:lifts limits uniquely and reflects limits.
4696:Interchange of limits and colimits of sets
4688:as a representation of the functor Cocone(
4584:, there is a unique canonical isomorphism
3648:(which is just a morphism in the category
2917:. Indeed, there may not even be a cone to
2335:be a diagram that picks out three objects
2263:. The limit of this diagram is called the
2237:, for instance, the products are given by
1821:{\displaystyle u\circ \phi _{X}=\psi _{X}}
1049:{\displaystyle \phi _{X}\circ u=\psi _{X}}
5287:then lim is a functor and the morphisms Ο
4956:
3672:for all X implies that the components of
80:Learn how and when to remove this message
6193:Categories for the Working Mathematician
6186:
6101:Categories for the Working Mathematician
6094:
5962:
43:This article includes a list of general
6226:
6085:
6068: β Construction in category theory
5871:) is left adjoint to forgetful functor
3105:. In this case, the limit of a diagram
14:
6940:
4891:
3717:) is the same thing as a co-cone from
2896:
1298:. For this reason one often speaks of
6667:
6320:
6282:
5945:for morphisms. The forgetful functor
3842:the unique morphism lim Ξ· : lim
3548:is the functor that maps each object
3453:
6141:commutativity of limits and colimits
6037:{\displaystyle \operatorname {Hom} }
6017:{\displaystyle \operatorname {Hom} }
5498:if there is a unique preimage cone (
5386:
4077:, and one has a natural isomorphism
3078:is one that has all small colimits.
2760:{\displaystyle I_{x,A}:F_{x,A}\to F}
1406:together with a family of morphisms
1275:Limits may also be characterized as
1224:, since they are characterized by a
684:of morphisms indexed by the objects
186:are defined by means of diagrams in
29:
6308:
6227:Borceux, Francis (1994). "Limits".
5887:is the free group generated by the
3652:) is the same thing as a cone from
2391:. It can nicely be visualized as a
2293:is a pair of parallel morphisms in
1450:{\displaystyle \psi _{X}:F(X)\to N}
677:{\displaystyle \psi _{X}:N\to F(X)}
24:
6156:
5370:a canonical natural isomorphism).
4047:
4036:
4029:
3995:Dually, if every diagram of shape
3976:is not connected. For example, if
3948:which is natural in the variables
3812:
3801:
3794:
3736:is a universal morphism from Ξ to
3529:
3522:
3511:
3503:
2966:(it need not have large products),
2398:
2208:consists of a family of morphisms
1870:
1098:
456:category. A diagram is said to be
124:generalizes constructions such as
49:it lacks sufficient corresponding
25:
6959:
6250:
5632:Creation and reflection of limits
4664:which is natural in the variable
4327:which is natural in the variable
4069:to the diagonal functor Ξ :
3854:to the diagonal functor Ξ :
3660:. To see this, first note that Ξ(
2487:is just a morphism with codomain
1879:Colimits are also referred to as
6921:
6911:
6902:
6901:
6654:
6321:
6166:Abstract and Concrete Categories
5799:, this is true of the covariant
5081:. For example, one can say that
4750:
4747:
4744:
4526:with the limiting cone given by
2144:: the empty one (similar to the
1698:such that for any other co-cone
34:
6231:Handbook of categorical algebra
5987:are types of colimits, whereas
4884:In words, filtered colimits in
4185:Limit and colimit of presheaves
3473:be a small index category. The
2837:are colimits of empty diagrams.
2301:of such a diagram is called an
2032:, there is a universal diagram
1885:. They can be characterized as
1750:there exists a unique morphism
1500:, such that for every morphism
1218:Limits are also referred to as
724:, such that for every morphism
6200:. Vol. 5 (2nd ed.).
6134:
6108:. Vol. 5 (2nd ed.).
5354:preserves all limits of shape
5318:
5242:
4940:. A natural question is then:
4865:
4853:
4824:
4821:
4809:
4740:
4648:
4633:
4618:
4600:
4432:
4420:
4408:
4393:
4311:
4296:
4281:
4266:
4179:As representations of functors
4135:
4123:
4111:
4093:
4042:
3932:
3920:
3908:
3893:
3870:and the set of all cones from
3807:
3755:
3697:), which all share the domain
3516:
3444:existence theorem for colimits
3417:
3411:
3385:
3379:
3342:
3336:
3310:
3304:
3287:
3281:
3231:
3228:
3222:
3213:
3205:
3199:
3179:
3176:
3170:
3162:
3156:
3020:
3014:
2982:
2921:, let alone a universal cone.
2751:
2681:the set of filters finer than
2653:
2647:
2475:is essentially just an object
2048:
1766:
1717:
1705:
1665:
1653:
1630:
1581:
1575:
1516:
1441:
1438:
1432:
1350:
1171:with the unique factorization
1158:
1146:
1126:
1114:
994:
941:
929:
889:
877:
854:
792:
786:
740:
671:
665:
659:
514:
315:
13:
1:
6198:Graduate Texts in Mathematics
6106:Graduate Texts in Mathematics
6078:
5774:
5597:also has all limits of shape
5064:preserve all limits of shape
5040:. (Note that if the limit of
4549:of the functor Cone(–,
3747:is a universal morphism from
3247:given (in component form) by
2891:
2823:
2180:is essentially nothing but a
1939:
1916:As with limits, if a diagram
158:
5859:(which assigns to every set
5683:there exists a unique cone (
4495:. If one is given an object
3629:(thought of as an object in
3083:existence theorem for limits
2526:, then any diagram of shape
7:
6596:Constructions on categories
6047:
5612:lifts all small limits and
5553:lifts limits for a diagram
4514:, –) β Cone(–,
2943:. Specifically, a category
2448:be a diagram. The limit of
2204:of these objects. The cone
2102:
2097:
1309:
100:, the abstract notion of a
10:
6964:
6703:Higher-dimensional algebra
5581:lifts limits of all shape
5136:is one that preserves all
5100:preserves the colimits of
5089:is one that preserves all
4182:
3556:to the constant functor Ξ(
3085:states that if a category
3064:if every diagram of shape
3004:if it has limits of shape
2972:if it has limits of shape
2954:if it has limits of shape
2935:if every diagram of shape
2785:is a categorical limit of
2777:is a topological limit of
2503:is a limit of the diagram
1313:
491:
6897:
6830:
6794:
6742:
6735:
6686:
6676:
6663:
6652:
6595:
6537:
6488:
6443:
6434:
6331:
6327:
6316:
6172:. John Wiley & Sons.
6072:Product (category theory)
6054:Cartesian closed category
5293:form the components of a
5283:have all limits of shape
5170:An important property of
4704:be a finite category and
3050:for every small category
1723:{\displaystyle (N,\psi )}
1671:{\displaystyle (L,\phi )}
1164:{\displaystyle (L,\phi )}
1139:factors through the cone
1132:{\displaystyle (N,\psi )}
947:{\displaystyle (N,\psi )}
922:such that for every cone
895:{\displaystyle (L,\phi )}
487:
327:{\displaystyle F:J\to C.}
163:Limits and colimits in a
6948:Limits (category theory)
5648:be a diagram. A functor
5589:has all limits of shape
5569:preserves the limits of
5271:preserves the limits of
5051:preserves the limits of
4973:induces a map from Cone(
3964:. If the index category
3101:has all limits of shape
2603:a particular filter and
2519:is any category with an
2057:{\displaystyle F:J\to C}
1775:{\displaystyle u:L\to N}
1639:{\displaystyle F:J\to C}
1525:{\displaystyle f:X\to Y}
1359:{\displaystyle F:J\to C}
1211:is sometimes called the
1003:{\displaystyle u:N\to L}
863:{\displaystyle F:J\to C}
749:{\displaystyle f:X\to Y}
523:{\displaystyle F:J\to C}
6513:Cokernels and quotients
6436:Universal constructions
6060:Equaliser (mathematics)
5605:preserves these limits.
5514:′) is a limit of
5011:preserve the limits of
4944:βHow are the limits of
4331:. Here the functor Hom(
3460:universal constructions
2811:{\displaystyle I_{x,A}}
2233:of the product. In the
1107:One says that the cone
636:together with a family
64:more precise citations.
6670:Higher category theory
6416:Natural transformation
6038:
6018:
5917:The forgetful functor
5902:The inclusion functor
5741:is already a limit of
5468:lifts limits of shape
5431:there exists a limit (
5381:contravariant functors
5341:
5295:natural transformation
5258:
4957:Preservation of limits
4875:
4761:
4676:) with the set Cocone(
4655:
4442:
4318:
4145:
4056:
4007:small) there exists a
3939:
3832:natural transformation
3821:
3772:small) there exists a
3635:natural transformation
3539:
3469:be a category and let
3433:
3238:
3059:has colimits of shape
3030:
2992:
2812:
2761:
2675:
2563:the set of filters on
2483:. A cone to an object
2403:
2078:
2058:
2026:
2002:
1982:
1959:
1930:
1907:
1876:
1862:
1842:
1822:
1776:
1744:
1724:
1692:
1672:
1640:
1601:
1546:
1526:
1494:
1474:
1451:
1400:
1380:
1360:
1262:
1242:
1205:
1185:
1165:
1133:
1104:
1090:
1070:
1050:
1004:
968:
948:
916:
896:
864:
825:
770:
750:
718:
698:
678:
630:
610:
590:
564:
544:
530:be a diagram of shape
524:
478:
442:
419:
399:
375:
351:
328:
290:
270:
246:
226:
200:
180:
6039:
6019:
5963:A note on terminology
5891:of the generators of
5782:representable functor
5620:is also complete and
5565:also has a limit and
5538:). One can show that
5492:lifts limits uniquely
5342:
5259:
5044:does not exist, then
4876:
4762:
4656:
4564:Dually, if a diagram
4443:
4319:
4249:canonical isomorphism
4146:
4057:
3940:
3822:
3540:
3434:
3239:
3031:
2993:
2930:have limits of shape
2813:
2762:
2685:and that converge to
2676:
2515:. More generally, if
2402:
2079:
2064:whose image contains
2059:
2027:
2003:
1983:
1960:
1931:
1908:
1874:
1863:
1843:
1823:
1777:
1745:
1725:
1693:
1673:
1641:
1602:
1547:
1527:
1495:
1475:
1452:
1401:
1381:
1361:
1263:
1243:
1206:
1186:
1166:
1134:
1102:
1091:
1071:
1051:
1005:
969:
949:
917:
897:
865:
826:
771:
751:
719:
699:
679:
631:
611:
591:
565:
545:
525:
479:
443:
420:
400:
376:
352:
329:
291:
271:
247:
227:
201:
181:
6539:Algebraic categories
6257:Interactive Web page
6028:
6008:
5943:continuous functions
5506:′) such that (
5303:
5217:
5178:For a given diagram
5134:cocontinuous functor
4948:related to those of
4780:
4722:
4591:
4381:
4257:
4218:preserves all limits
4084:
4017:
3884:
3782:
3500:
3254:
3124:
3008:
2976:
2789:
2713:
2607:
2305:of those morphisms.
2068:
2036:
2016:
1992:
1972:
1949:
1920:
1897:
1891:category of co-cones
1852:
1832:
1786:
1754:
1734:
1702:
1682:
1650:
1618:
1556:
1536:
1504:
1484:
1464:
1413:
1390:
1370:
1338:
1252:
1232:
1195:
1175:
1143:
1111:
1080:
1060:
1014:
982:
958:
926:
906:
874:
842:
780:
760:
728:
708:
688:
640:
620:
600:
580:
554:
534:
502:
468:
432:
409:
389:
365:
357:is thought of as an
341:
303:
280:
260:
236:
216:
190:
170:
149:universal properties
27:Mathematical concept
6708:Homotopy hypothesis
6386:Commutative diagram
5937:be the category of
5879:of two free groups
5573:. It follows that:
5364:commute with limits
4892:Functors and limits
4772:natural isomorphism
4522:will be a limit of
4505:natural isomorphism
3075:cocomplete category
2897:Existence of limits
2589:neighborhood filter
1875:A universal co-cone
6421:Universal property
6188:Mac Lane, Saunders
6096:Mac Lane, Saunders
6034:
6014:
5733:whose image under
5616:is complete, then
5561:has a limit, then
5337:
5254:
5128:) is a colimit of
5116:) is a colimit of
5087:continuous functor
4871:
4836:
4805:
4757:
4651:
4438:
4314:
4209:, –) :
4141:
4052:
3982:diagonal morphisms
3935:
3817:
3606:for each morphism
3535:
3454:Universal property
3429:
3427:
3234:
3209:
3166:
3026:
2988:
2962:discrete category
2808:
2757:
2671:
2546:Topological limits
2404:
2393:commutative square
2239:Cartesian products
2074:
2054:
2022:
1998:
1978:
1955:
1926:
1903:
1882:universal co-cones
1877:
1858:
1838:
1818:
1772:
1740:
1720:
1688:
1668:
1636:
1597:
1542:
1522:
1490:
1470:
1447:
1396:
1376:
1356:
1258:
1238:
1226:universal property
1213:mediating morphism
1201:
1181:
1161:
1129:
1105:
1086:
1066:
1046:
1000:
964:
944:
912:
892:
860:
821:
766:
746:
714:
694:
674:
626:
606:
586:
560:
540:
520:
474:
438:
415:
395:
371:
361:, and the diagram
347:
324:
286:
266:
242:
222:
196:
176:
6935:
6934:
6893:
6892:
6889:
6888:
6871:monoidal category
6826:
6825:
6698:Enriched category
6650:
6649:
6646:
6645:
6623:Quotient category
6618:Opposite category
6533:
6532:
5980:coequalizers, and
5835:filtered colimits
5820:forgetful functor
5807:,–) :
5387:Lifting of limits
5149:complete category
4827:
4796:
4710:filtered category
4576:has a colimit in
4243:, denoted by lim
3999:has a colimit in
3182:
3139:
3068:has a colimit in
3043:complete category
2554:topological space
2550:limits of filters
2174:discrete category
2077:{\displaystyle G}
2025:{\displaystyle G}
2001:{\displaystyle J}
1981:{\displaystyle G}
1958:{\displaystyle J}
1929:{\displaystyle F}
1906:{\displaystyle F}
1861:{\displaystyle J}
1841:{\displaystyle X}
1743:{\displaystyle F}
1691:{\displaystyle F}
1545:{\displaystyle J}
1493:{\displaystyle J}
1473:{\displaystyle X}
1460:for every object
1399:{\displaystyle C}
1379:{\displaystyle N}
1281:category of cones
1261:{\displaystyle L}
1241:{\displaystyle L}
1204:{\displaystyle u}
1184:{\displaystyle u}
1089:{\displaystyle J}
1069:{\displaystyle X}
967:{\displaystyle F}
915:{\displaystyle F}
769:{\displaystyle J}
717:{\displaystyle J}
697:{\displaystyle X}
629:{\displaystyle C}
609:{\displaystyle N}
589:{\displaystyle F}
563:{\displaystyle C}
543:{\displaystyle J}
477:{\displaystyle J}
441:{\displaystyle J}
418:{\displaystyle J}
398:{\displaystyle C}
374:{\displaystyle F}
350:{\displaystyle J}
289:{\displaystyle C}
269:{\displaystyle J}
245:{\displaystyle C}
225:{\displaystyle J}
199:{\displaystyle C}
179:{\displaystyle C}
90:
89:
82:
16:(Redirected from
6955:
6925:
6924:
6915:
6914:
6905:
6904:
6740:
6739:
6718:Simplex category
6693:Categorification
6684:
6683:
6665:
6664:
6658:
6628:Product category
6613:Kleisli category
6608:Functor category
6453:Terminal objects
6441:
6440:
6376:Adjoint functors
6329:
6328:
6318:
6317:
6303:
6296:
6289:
6280:
6279:
6246:
6234:
6223:
6183:
6171:
6150:
6138:
6132:
6131:
6092:
6043:
6041:
6040:
6035:
6023:
6021:
6020:
6015:
5729:if each cone to
5679:) is a limit of
5427:) is a limit of
5346:
5344:
5343:
5338:
5333:
5332:
5263:
5261:
5260:
5255:
5229:
5228:
5172:adjoint functors
5036:) is a limit of
5024:) is a limit of
4908:is a diagram in
4880:
4878:
4877:
4872:
4846:
4845:
4835:
4804:
4792:
4791:
4766:
4764:
4763:
4758:
4753:
4660:
4658:
4657:
4652:
4580:, denoted colim
4503:together with a
4476:
4447:
4445:
4444:
4439:
4343:, –) with
4323:
4321:
4320:
4315:
4203:category of sets
4150:
4148:
4147:
4142:
4061:
4059:
4058:
4053:
4051:
4050:
4041:
4040:
4039:
4033:
4032:
3944:
3942:
3941:
3936:
3826:
3824:
3823:
3818:
3816:
3815:
3806:
3805:
3804:
3798:
3797:
3617:Given a diagram
3584:for each object
3544:
3542:
3541:
3536:
3534:
3533:
3532:
3526:
3525:
3515:
3514:
3491:diagonal functor
3475:functor category
3442:There is a dual
3438:
3436:
3435:
3430:
3428:
3421:
3420:
3397:
3396:
3389:
3388:
3367:
3366:
3346:
3345:
3322:
3321:
3314:
3313:
3277:
3276:
3243:
3241:
3240:
3235:
3208:
3165:
3035:
3033:
3032:
3027:
2997:
2995:
2994:
2989:
2901:A given diagram
2817:
2815:
2814:
2809:
2807:
2806:
2766:
2764:
2763:
2758:
2750:
2749:
2731:
2730:
2709:. The injection
2680:
2678:
2677:
2672:
2625:
2624:
2458:projective limit
2253:constant functor
2235:category of sets
2133:Terminal objects
2083:
2081:
2080:
2075:
2063:
2061:
2060:
2055:
2031:
2029:
2028:
2023:
2007:
2005:
2004:
1999:
1987:
1985:
1984:
1979:
1964:
1962:
1961:
1956:
1935:
1933:
1932:
1927:
1912:
1910:
1909:
1904:
1867:
1865:
1864:
1859:
1847:
1845:
1844:
1839:
1827:
1825:
1824:
1819:
1817:
1816:
1804:
1803:
1781:
1779:
1778:
1773:
1749:
1747:
1746:
1741:
1729:
1727:
1726:
1721:
1697:
1695:
1694:
1689:
1677:
1675:
1674:
1669:
1645:
1643:
1642:
1637:
1606:
1604:
1603:
1598:
1596:
1595:
1568:
1567:
1551:
1549:
1548:
1543:
1531:
1529:
1528:
1523:
1499:
1497:
1496:
1491:
1479:
1477:
1476:
1471:
1456:
1454:
1453:
1448:
1425:
1424:
1405:
1403:
1402:
1397:
1385:
1383:
1382:
1377:
1365:
1363:
1362:
1357:
1277:terminal objects
1267:
1265:
1264:
1259:
1247:
1245:
1244:
1239:
1210:
1208:
1207:
1202:
1190:
1188:
1187:
1182:
1170:
1168:
1167:
1162:
1138:
1136:
1135:
1130:
1103:A universal cone
1095:
1093:
1092:
1087:
1075:
1073:
1072:
1067:
1055:
1053:
1052:
1047:
1045:
1044:
1026:
1025:
1009:
1007:
1006:
1001:
973:
971:
970:
965:
953:
951:
950:
945:
921:
919:
918:
913:
901:
899:
898:
893:
869:
867:
866:
861:
830:
828:
827:
822:
820:
819:
807:
806:
775:
773:
772:
767:
755:
753:
752:
747:
723:
721:
720:
715:
703:
701:
700:
695:
683:
681:
680:
675:
652:
651:
635:
633:
632:
627:
615:
613:
612:
607:
595:
593:
592:
587:
569:
567:
566:
561:
549:
547:
546:
541:
529:
527:
526:
521:
483:
481:
480:
475:
447:
445:
444:
439:
424:
422:
421:
416:
404:
402:
401:
396:
380:
378:
377:
372:
356:
354:
353:
348:
333:
331:
330:
325:
295:
293:
292:
287:
275:
273:
272:
267:
251:
249:
248:
243:
231:
229:
228:
223:
205:
203:
202:
197:
185:
183:
182:
177:
153:adjoint functors
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
6963:
6962:
6958:
6957:
6956:
6954:
6953:
6952:
6938:
6937:
6936:
6931:
6885:
6855:
6822:
6799:
6790:
6747:
6731:
6682:
6672:
6659:
6642:
6591:
6529:
6498:Initial objects
6484:
6430:
6323:
6312:
6310:Category theory
6307:
6253:
6243:
6212:
6202:Springer-Verlag
6180:
6169:
6159:
6157:Further reading
6154:
6153:
6139:
6135:
6120:
6110:Springer-Verlag
6093:
6086:
6081:
6050:
6029:
6026:
6025:
6009:
6006:
6005:
5997:equalizers, and
5965:
5953:
5936:
5777:
5760:creates limits.
5634:
5389:
5362:can be said to
5328:
5324:
5304:
5301:
5300:
5292:
5224:
5220:
5218:
5215:
5214:
4997:is a cone from
4985:is a cone from
4959:
4894:
4841:
4837:
4831:
4800:
4787:
4783:
4781:
4778:
4777:
4743:
4723:
4720:
4719:
4698:
4592:
4589:
4588:
4540:
4534:
4494:
4481:
4474:
4456:
4382:
4379:
4378:
4258:
4255:
4254:
4239:has a limit in
4187:
4181:
4085:
4082:
4081:
4046:
4045:
4035:
4034:
4028:
4027:
4026:
4018:
4015:
4014:
4009:colimit functor
3885:
3882:
3881:
3811:
3810:
3800:
3799:
3793:
3792:
3791:
3783:
3780:
3779:
3764:has a limit in
3758:
3684:
3605:
3528:
3527:
3521:
3520:
3519:
3510:
3509:
3501:
3498:
3497:
3456:
3426:
3425:
3398:
3392:
3391:
3390:
3372:
3368:
3362:
3361:
3354:
3348:
3347:
3323:
3317:
3316:
3315:
3297:
3293:
3272:
3271:
3264:
3257:
3255:
3252:
3251:
3186:
3143:
3125:
3122:
3121:
3009:
3006:
3005:
2977:
2974:
2973:
2970:have equalizers
2939:has a limit in
2899:
2894:
2834:Initial objects
2826:
2796:
2792:
2790:
2787:
2786:
2781:if and only if
2739:
2735:
2720:
2716:
2714:
2711:
2710:
2701:if and only if
2614:
2610:
2608:
2605:
2604:
2507:if and only if
2428:if and only if
2216:
2176:then a diagram
2158:terminal object
2152:. The limit of
2115:) of a diagram
2105:
2100:
2069:
2066:
2065:
2037:
2034:
2033:
2017:
2014:
2013:
1993:
1990:
1989:
1973:
1970:
1969:
1950:
1947:
1946:
1942:
1921:
1918:
1917:
1898:
1895:
1894:
1887:initial objects
1853:
1850:
1849:
1833:
1830:
1829:
1812:
1808:
1799:
1795:
1787:
1784:
1783:
1755:
1752:
1751:
1735:
1732:
1731:
1703:
1700:
1699:
1683:
1680:
1679:
1651:
1648:
1647:
1619:
1616:
1615:
1591:
1587:
1563:
1559:
1557:
1554:
1553:
1537:
1534:
1533:
1505:
1502:
1501:
1485:
1482:
1481:
1465:
1462:
1461:
1420:
1416:
1414:
1411:
1410:
1391:
1388:
1387:
1371:
1368:
1367:
1339:
1336:
1335:
1318:
1312:
1253:
1250:
1249:
1233:
1230:
1229:
1221:universal cones
1196:
1193:
1192:
1191:. The morphism
1176:
1173:
1172:
1144:
1141:
1140:
1112:
1109:
1108:
1081:
1078:
1077:
1061:
1058:
1057:
1040:
1036:
1021:
1017:
1015:
1012:
1011:
983:
980:
979:
974:there exists a
959:
956:
955:
927:
924:
923:
907:
904:
903:
875:
872:
871:
843:
840:
839:
838:of the diagram
815:
811:
802:
798:
781:
778:
777:
761:
758:
757:
729:
726:
725:
709:
706:
705:
689:
686:
685:
647:
643:
641:
638:
637:
621:
618:
617:
601:
598:
597:
581:
578:
577:
555:
552:
551:
535:
532:
531:
503:
500:
499:
496:
490:
469:
466:
465:
433:
430:
429:
410:
407:
406:
390:
387:
386:
366:
363:
362:
342:
339:
338:
304:
301:
300:
281:
278:
277:
261:
258:
257:
237:
234:
233:
217:
214:
213:
191:
188:
187:
171:
168:
167:
161:
126:disjoint unions
94:category theory
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
6961:
6951:
6950:
6933:
6932:
6930:
6929:
6919:
6909:
6898:
6895:
6894:
6891:
6890:
6887:
6886:
6884:
6883:
6878:
6873:
6859:
6853:
6848:
6843:
6837:
6835:
6828:
6827:
6824:
6823:
6821:
6820:
6815:
6804:
6802:
6797:
6792:
6791:
6789:
6788:
6783:
6778:
6773:
6768:
6763:
6752:
6750:
6745:
6737:
6733:
6732:
6730:
6725:
6723:String diagram
6720:
6715:
6713:Model category
6710:
6705:
6700:
6695:
6690:
6688:
6681:
6680:
6677:
6674:
6673:
6661:
6660:
6653:
6651:
6648:
6647:
6644:
6643:
6641:
6640:
6635:
6633:Comma category
6630:
6625:
6620:
6615:
6610:
6605:
6599:
6597:
6593:
6592:
6590:
6589:
6579:
6569:
6567:Abelian groups
6564:
6559:
6554:
6549:
6543:
6541:
6535:
6534:
6531:
6530:
6528:
6527:
6522:
6517:
6516:
6515:
6505:
6500:
6494:
6492:
6486:
6485:
6483:
6482:
6477:
6472:
6471:
6470:
6460:
6455:
6449:
6447:
6438:
6432:
6431:
6429:
6428:
6423:
6418:
6413:
6408:
6403:
6398:
6393:
6388:
6383:
6378:
6373:
6372:
6371:
6366:
6361:
6356:
6351:
6346:
6335:
6333:
6325:
6324:
6314:
6313:
6306:
6305:
6298:
6291:
6283:
6277:
6276:
6264:
6252:
6251:External links
6249:
6248:
6247:
6241:
6224:
6210:
6184:
6178:
6158:
6155:
6152:
6151:
6133:
6118:
6083:
6082:
6080:
6077:
6076:
6075:
6069:
6063:
6057:
6049:
6046:
6033:
6013:
6002:
6001:
5998:
5995:
5992:
5985:
5984:
5981:
5978:
5975:
5964:
5961:
5960:
5959:
5949:
5932:
5926:
5915:
5900:
5889:disjoint union
5842:
5816:
5776:
5773:
5769:
5768:
5761:
5747:
5746:
5737:is a limit of
5723:reflect limits
5720:
5633:
5630:
5626:
5625:
5624:is continuous.
5606:
5494:for a diagram
5407:for a diagram
5388:
5385:
5379:functors. For
5348:
5347:
5336:
5331:
5327:
5323:
5320:
5317:
5314:
5311:
5308:
5288:
5265:
5264:
5253:
5250:
5247:
5244:
5241:
5238:
5235:
5232:
5227:
5223:
5005:. The functor
4958:
4955:
4954:
4953:
4893:
4890:
4882:
4881:
4870:
4867:
4864:
4861:
4858:
4855:
4852:
4849:
4844:
4840:
4834:
4830:
4826:
4823:
4820:
4817:
4814:
4811:
4808:
4803:
4799:
4795:
4790:
4786:
4768:
4767:
4756:
4752:
4749:
4746:
4742:
4739:
4736:
4733:
4730:
4727:
4697:
4694:
4662:
4661:
4650:
4647:
4644:
4641:
4638:
4635:
4632:
4629:
4626:
4623:
4620:
4617:
4614:
4611:
4608:
4605:
4602:
4599:
4596:
4547:representation
4536:
4530:
4518:), the object
4490:
4477:
4452:
4449:
4448:
4437:
4434:
4431:
4428:
4425:
4422:
4419:
4416:
4413:
4410:
4407:
4404:
4401:
4398:
4395:
4392:
4389:
4386:
4325:
4324:
4313:
4310:
4307:
4304:
4301:
4298:
4295:
4292:
4289:
4286:
4283:
4280:
4277:
4274:
4271:
4268:
4265:
4262:
4180:
4177:
4152:
4151:
4140:
4137:
4134:
4131:
4128:
4125:
4122:
4119:
4116:
4113:
4110:
4107:
4104:
4101:
4098:
4095:
4092:
4089:
4063:
4062:
4049:
4044:
4038:
4031:
4025:
4022:
3946:
3945:
3934:
3931:
3928:
3925:
3922:
3919:
3916:
3913:
3910:
3907:
3904:
3901:
3898:
3895:
3892:
3889:
3828:
3827:
3814:
3809:
3803:
3796:
3790:
3787:
3757:
3754:
3753:
3752:
3741:
3680:
3676:are morphisms
3601:
3546:
3545:
3531:
3524:
3518:
3513:
3508:
3505:
3455:
3452:
3440:
3439:
3424:
3419:
3416:
3413:
3410:
3407:
3404:
3401:
3395:
3387:
3384:
3381:
3378:
3375:
3371:
3365:
3360:
3357:
3355:
3353:
3350:
3349:
3344:
3341:
3338:
3335:
3332:
3329:
3326:
3320:
3312:
3309:
3306:
3303:
3300:
3296:
3292:
3289:
3286:
3283:
3280:
3275:
3270:
3267:
3265:
3263:
3260:
3259:
3245:
3244:
3233:
3230:
3227:
3224:
3221:
3218:
3215:
3212:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3185:
3181:
3178:
3175:
3172:
3169:
3164:
3161:
3158:
3155:
3152:
3149:
3146:
3142:
3138:
3135:
3132:
3129:
3038:
3037:
3025:
3022:
3019:
3016:
3013:
3002:have pullbacks
2999:
2987:
2984:
2981:
2967:
2898:
2895:
2893:
2890:
2889:
2888:
2880:
2872:
2871:
2870:
2854:
2853:
2852:
2838:
2825:
2822:
2821:
2820:
2819:
2818:
2805:
2802:
2799:
2795:
2769:
2768:
2756:
2753:
2748:
2745:
2742:
2738:
2734:
2729:
2726:
2723:
2719:
2689:. The filters
2670:
2667:
2664:
2661:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2631:
2628:
2623:
2620:
2617:
2613:
2543:
2521:initial object
2461:
2409:Inverse limits
2397:
2396:
2324:
2323:
2322:
2278:
2277:
2276:
2212:
2200:is called the
2184:of objects of
2161:
2146:empty function
2104:
2101:
2099:
2096:
2073:
2053:
2050:
2047:
2044:
2041:
2021:
1997:
1977:
1967:directed graph
1954:
1941:
1938:
1925:
1902:
1857:
1837:
1815:
1811:
1807:
1802:
1798:
1794:
1791:
1771:
1768:
1765:
1762:
1759:
1739:
1719:
1716:
1713:
1710:
1707:
1687:
1667:
1664:
1661:
1658:
1655:
1635:
1632:
1629:
1626:
1623:
1594:
1590:
1586:
1583:
1580:
1577:
1574:
1571:
1566:
1562:
1541:
1521:
1518:
1515:
1512:
1509:
1489:
1469:
1458:
1457:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1423:
1419:
1395:
1375:
1355:
1352:
1349:
1346:
1343:
1311:
1308:
1257:
1237:
1200:
1180:
1160:
1157:
1154:
1151:
1148:
1128:
1125:
1122:
1119:
1116:
1085:
1065:
1043:
1039:
1035:
1032:
1029:
1024:
1020:
999:
996:
993:
990:
987:
963:
943:
940:
937:
934:
931:
911:
891:
888:
885:
882:
879:
859:
856:
853:
850:
847:
818:
814:
810:
805:
801:
797:
794:
791:
788:
785:
765:
745:
742:
739:
736:
733:
713:
693:
673:
670:
667:
664:
661:
658:
655:
650:
646:
625:
605:
585:
559:
550:in a category
539:
519:
516:
513:
510:
507:
489:
486:
473:
437:
414:
394:
370:
359:index category
346:
335:
334:
323:
320:
317:
314:
311:
308:
285:
265:
241:
221:
206:. Formally, a
195:
175:
160:
157:
114:inverse limits
96:, a branch of
88:
87:
42:
40:
33:
26:
9:
6:
4:
3:
2:
6960:
6949:
6946:
6945:
6943:
6928:
6920:
6918:
6910:
6908:
6900:
6899:
6896:
6882:
6879:
6877:
6874:
6872:
6868:
6864:
6860:
6858:
6856:
6849:
6847:
6844:
6842:
6839:
6838:
6836:
6833:
6829:
6819:
6816:
6813:
6809:
6806:
6805:
6803:
6801:
6793:
6787:
6784:
6782:
6779:
6777:
6774:
6772:
6771:Tetracategory
6769:
6767:
6764:
6761:
6760:pseudofunctor
6757:
6754:
6753:
6751:
6749:
6741:
6738:
6734:
6729:
6726:
6724:
6721:
6719:
6716:
6714:
6711:
6709:
6706:
6704:
6701:
6699:
6696:
6694:
6691:
6689:
6685:
6679:
6678:
6675:
6671:
6666:
6662:
6657:
6639:
6636:
6634:
6631:
6629:
6626:
6624:
6621:
6619:
6616:
6614:
6611:
6609:
6606:
6604:
6603:Free category
6601:
6600:
6598:
6594:
6587:
6586:Vector spaces
6583:
6580:
6577:
6573:
6570:
6568:
6565:
6563:
6560:
6558:
6555:
6553:
6550:
6548:
6545:
6544:
6542:
6540:
6536:
6526:
6523:
6521:
6518:
6514:
6511:
6510:
6509:
6506:
6504:
6501:
6499:
6496:
6495:
6493:
6491:
6487:
6481:
6480:Inverse limit
6478:
6476:
6473:
6469:
6466:
6465:
6464:
6461:
6459:
6456:
6454:
6451:
6450:
6448:
6446:
6442:
6439:
6437:
6433:
6427:
6424:
6422:
6419:
6417:
6414:
6412:
6409:
6407:
6406:Kan extension
6404:
6402:
6399:
6397:
6394:
6392:
6389:
6387:
6384:
6382:
6379:
6377:
6374:
6370:
6367:
6365:
6362:
6360:
6357:
6355:
6352:
6350:
6347:
6345:
6342:
6341:
6340:
6337:
6336:
6334:
6330:
6326:
6319:
6315:
6311:
6304:
6299:
6297:
6292:
6290:
6285:
6284:
6281:
6275:
6273:
6268:
6265:
6262:
6261:Jocelyn Paine
6258:
6255:
6254:
6244:
6242:0-521-44178-1
6238:
6233:
6232:
6225:
6221:
6217:
6213:
6211:0-387-98403-8
6207:
6203:
6199:
6195:
6194:
6189:
6185:
6181:
6179:0-471-60922-6
6175:
6168:
6167:
6161:
6160:
6149:
6147:
6142:
6137:
6129:
6125:
6121:
6119:0-387-98403-8
6115:
6111:
6107:
6103:
6102:
6097:
6091:
6089:
6084:
6073:
6070:
6067:
6066:Inverse limit
6064:
6061:
6058:
6055:
6052:
6051:
6045:
6031:
6011:
5999:
5996:
5993:
5990:
5989:
5988:
5982:
5979:
5976:
5973:
5972:
5971:
5968:
5957:
5952:
5948:
5944:
5940:
5939:metric spaces
5935:
5931:
5927:
5924:
5920:
5916:
5913:
5909:
5905:
5901:
5898:
5894:
5890:
5886:
5882:
5878:
5874:
5870:
5866:
5862:
5858:
5854:
5850:
5847:
5843:
5840:
5836:
5832:
5828:
5824:
5821:
5817:
5814:
5810:
5806:
5802:
5798:
5794:
5790:
5786:
5783:
5779:
5778:
5772:
5766:
5762:
5759:
5755:
5754:
5753:
5750:
5744:
5740:
5736:
5732:
5728:
5724:
5721:
5718:
5714:
5710:
5706:
5702:
5698:
5694:
5690:
5686:
5682:
5678:
5674:
5671:if whenever (
5670:
5666:
5665:create limits
5663:
5662:
5661:
5659:
5655:
5651:
5647:
5643:
5639:
5629:
5623:
5619:
5615:
5611:
5607:
5604:
5600:
5596:
5592:
5588:
5584:
5580:
5576:
5575:
5574:
5572:
5568:
5564:
5560:
5556:
5552:
5547:
5545:
5541:
5537:
5533:
5529:
5525:
5521:
5517:
5513:
5509:
5505:
5501:
5497:
5493:
5490:
5485:
5483:
5480:
5476:
5472:
5471:
5466:
5463:). A functor
5462:
5458:
5454:
5450:
5446:
5442:
5438:
5434:
5430:
5426:
5422:
5419:if whenever (
5418:
5414:
5410:
5406:
5402:
5398:
5394:
5384:
5382:
5378:
5377:
5371:
5369:
5365:
5361:
5357:
5353:
5334:
5329:
5325:
5312:
5309:
5306:
5299:
5298:
5297:
5296:
5291:
5286:
5282:
5278:
5274:
5270:
5251:
5248:
5239:
5233:
5230:
5225:
5221:
5213:
5212:
5211:
5209:
5205:
5201:
5197:
5193:
5189:
5185:
5181:
5176:
5173:
5168:
5166:
5162:
5158:
5154:
5150:
5146:
5141:
5139:
5135:
5131:
5127:
5123:
5119:
5115:
5111:
5107:
5103:
5099:
5094:
5092:
5088:
5084:
5080:
5076:
5072:
5068:
5067:
5061:
5056:
5054:
5050:
5047:
5043:
5039:
5035:
5031:
5027:
5023:
5019:
5015:
5014:
5008:
5004:
5000:
4996:
4992:
4988:
4984:
4980:
4976:
4972:
4968:
4964:
4951:
4947:
4943:
4942:
4941:
4939:
4935:
4931:
4927:
4923:
4919:
4915:
4911:
4907:
4903:
4899:
4889:
4887:
4868:
4862:
4859:
4856:
4850:
4847:
4842:
4838:
4832:
4818:
4815:
4812:
4806:
4801:
4793:
4788:
4784:
4776:
4775:
4774:
4773:
4754:
4737:
4734:
4731:
4728:
4725:
4718:
4717:
4716:
4715:
4711:
4707:
4703:
4693:
4691:
4687:
4683:
4679:
4675:
4671:
4667:
4645:
4642:
4639:
4636:
4630:
4627:
4621:
4615:
4612:
4609:
4606:
4603:
4597:
4594:
4587:
4586:
4585:
4583:
4579:
4575:
4571:
4567:
4562:
4560:
4556:
4552:
4548:
4544:
4539:
4533:
4529:
4525:
4521:
4517:
4513:
4509:
4506:
4502:
4498:
4493:
4489:
4485:
4480:
4472:
4468:
4464:
4460:
4457: : Cone(
4455:
4435:
4429:
4426:
4423:
4417:
4414:
4411:
4405:
4402:
4399:
4396:
4390:
4387:
4377:
4376:
4375:
4373:
4369:
4365:
4361:
4357:
4353:
4348:
4346:
4342:
4338:
4334:
4330:
4308:
4305:
4302:
4299:
4293:
4290:
4284:
4278:
4272:
4269:
4263:
4260:
4253:
4252:
4251:
4250:
4247:, there is a
4246:
4242:
4238:
4234:
4230:
4227:If a diagram
4225:
4223:
4219:
4216:
4212:
4208:
4204:
4200:
4197:to limits in
4196:
4192:
4186:
4176:
4174:
4173:
4167:
4165:
4161:
4157:
4138:
4132:
4129:
4126:
4120:
4117:
4114:
4108:
4105:
4102:
4099:
4096:
4090:
4087:
4080:
4079:
4078:
4076:
4072:
4068:
4023:
4020:
4013:
4012:
4011:
4010:
4006:
4002:
3998:
3993:
3991:
3987:
3983:
3979:
3975:
3971:
3967:
3963:
3959:
3955:
3951:
3929:
3926:
3923:
3917:
3914:
3911:
3905:
3899:
3896:
3890:
3887:
3880:
3879:
3878:
3877:
3873:
3869:
3865:
3861:
3857:
3853:
3852:right adjoint
3849:
3845:
3841:
3837:
3833:
3788:
3778:
3777:
3776:
3775:
3774:limit functor
3771:
3767:
3763:
3750:
3746:
3743:A colimit of
3742:
3739:
3735:
3731:
3730:
3729:
3726:
3724:
3720:
3716:
3712:
3708:
3704:
3700:
3696:
3692:
3688:
3683:
3679:
3675:
3671:
3667:
3663:
3659:
3655:
3651:
3647:
3643:
3639:
3636:
3632:
3628:
3624:
3620:
3615:
3613:
3609:
3604:
3599:
3595:
3591:
3587:
3583:
3579:
3575:
3572:. That is, Ξ(
3571:
3567:
3563:
3559:
3555:
3551:
3506:
3496:
3495:
3494:
3493:
3492:
3487:
3483:
3479:
3476:
3472:
3468:
3463:
3461:
3451:
3449:
3445:
3422:
3414:
3408:
3405:
3402:
3399:
3382:
3376:
3373:
3369:
3358:
3356:
3351:
3339:
3333:
3330:
3327:
3324:
3307:
3301:
3298:
3294:
3290:
3284:
3278:
3268:
3266:
3261:
3250:
3249:
3248:
3225:
3219:
3216:
3210:
3202:
3196:
3193:
3190:
3187:
3183:
3173:
3167:
3159:
3153:
3150:
3147:
3144:
3140:
3136:
3133:
3130:
3127:
3120:
3119:
3118:
3116:
3112:
3108:
3104:
3100:
3096:
3092:
3088:
3084:
3079:
3077:
3076:
3071:
3067:
3063:
3062:
3055:
3053:
3049:
3045:
3044:
3023:
3017:
3011:
3003:
3000:
2985:
2979:
2971:
2968:
2965:
2961:
2957:
2953:
2952:have products
2950:
2949:
2948:
2946:
2942:
2938:
2934:
2933:
2927:
2922:
2920:
2916:
2912:
2908:
2904:
2886:
2885:
2884:Direct limits
2881:
2878:
2877:
2873:
2868:
2867:
2863:
2862:
2860:
2859:
2855:
2850:
2847:
2846:
2844:
2843:
2839:
2836:
2835:
2831:
2830:
2829:
2803:
2800:
2797:
2793:
2784:
2780:
2776:
2773:
2772:
2771:
2770:
2754:
2746:
2743:
2740:
2736:
2732:
2727:
2724:
2721:
2717:
2708:
2704:
2700:
2696:
2692:
2688:
2684:
2665:
2662:
2659:
2656:
2650:
2644:
2641:
2638:
2635:
2632:
2626:
2621:
2618:
2615:
2611:
2602:
2598:
2594:
2590:
2586:
2582:
2578:
2574:
2570:
2566:
2562:
2558:
2555:
2551:
2547:
2544:
2541:
2537:
2533:
2529:
2525:
2522:
2518:
2514:
2510:
2506:
2502:
2498:
2494:
2491:. A morphism
2490:
2486:
2482:
2478:
2474:
2470:
2466:
2462:
2459:
2455:
2454:inverse limit
2452:is called an
2451:
2447:
2443:
2439:
2435:
2431:
2427:
2423:
2419:
2415:
2411:
2410:
2406:
2405:
2401:
2394:
2390:
2389:fiber product
2386:
2382:
2378:
2374:
2370:
2366:
2362:
2358:
2354:
2350:
2346:
2342:
2338:
2334:
2330:
2329:
2325:
2320:
2319:zero morphism
2316:
2312:
2311:
2307:
2306:
2304:
2300:
2296:
2292:
2288:
2284:
2283:
2279:
2274:
2270:
2266:
2262:
2258:
2255:to an object
2254:
2250:
2246:
2243:
2242:
2240:
2236:
2232:
2229:) called the
2228:
2224:
2220:
2215:
2211:
2207:
2203:
2199:
2195:
2191:
2188:, indexed by
2187:
2183:
2179:
2175:
2171:
2167:
2166:
2162:
2159:
2155:
2151:
2147:
2143:
2139:
2135:
2134:
2130:
2129:
2128:
2126:
2122:
2118:
2114:
2110:
2095:
2093:
2092:weak colimits
2089:
2085:
2071:
2051:
2045:
2042:
2039:
2019:
2012:generated by
2011:
2010:free category
1995:
1975:
1968:
1952:
1937:
1923:
1914:
1900:
1892:
1888:
1884:
1883:
1873:
1869:
1855:
1835:
1813:
1809:
1805:
1800:
1796:
1792:
1789:
1769:
1763:
1760:
1757:
1737:
1714:
1711:
1708:
1685:
1662:
1659:
1656:
1646:is a co-cone
1633:
1627:
1624:
1621:
1614:of a diagram
1613:
1608:
1592:
1588:
1584:
1578:
1572:
1569:
1564:
1560:
1539:
1519:
1513:
1510:
1507:
1487:
1467:
1444:
1435:
1429:
1426:
1421:
1417:
1409:
1408:
1407:
1393:
1373:
1366:is an object
1353:
1347:
1344:
1341:
1334:of a diagram
1333:
1332:
1326:
1323:
1317:
1307:
1305:
1301:
1297:
1293:
1288:
1286:
1282:
1278:
1273:
1271:
1255:
1235:
1227:
1223:
1222:
1216:
1214:
1198:
1178:
1155:
1152:
1149:
1123:
1120:
1117:
1101:
1097:
1083:
1063:
1041:
1037:
1033:
1030:
1027:
1022:
1018:
997:
991:
988:
985:
977:
961:
938:
935:
932:
909:
886:
883:
880:
857:
851:
848:
845:
837:
832:
816:
812:
808:
803:
799:
795:
789:
783:
763:
743:
737:
734:
731:
711:
691:
668:
662:
656:
653:
648:
644:
623:
603:
596:is an object
583:
575:
574:
557:
537:
517:
511:
508:
505:
495:
494:Inverse limit
485:
471:
463:
459:
455:
451:
435:
426:
412:
405:patterned on
392:
384:
368:
360:
344:
337:The category
321:
318:
312:
309:
306:
299:
298:
297:
283:
263:
255:
239:
219:
211:
210:
193:
173:
166:
156:
154:
150:
145:
143:
142:direct limits
139:
135:
131:
127:
123:
119:
115:
111:
107:
103:
99:
95:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
19:
6851:
6832:Categorified
6736:n-categories
6687:Key concepts
6525:Direct limit
6508:Coequalizers
6444:
6426:Yoneda lemma
6332:Key concepts
6322:Key concepts
6271:
6230:
6191:
6165:
6145:
6136:
6099:
6003:
5986:
5969:
5966:
5955:
5950:
5946:
5933:
5929:
5922:
5918:
5907:
5903:
5896:
5892:
5884:
5880:
5877:free product
5872:
5868:
5860:
5856:
5852:
5848:
5846:free functor
5838:
5830:
5826:
5822:
5812:
5808:
5804:
5796:
5792:
5788:
5784:
5770:
5764:
5763:The functor
5757:
5756:The functor
5751:
5748:
5742:
5738:
5734:
5730:
5726:
5722:
5716:
5712:
5708:
5707:′) = (
5704:
5700:
5696:
5692:
5691:′) to
5688:
5684:
5680:
5676:
5672:
5668:
5664:
5657:
5653:
5649:
5645:
5641:
5637:
5635:
5627:
5621:
5617:
5613:
5609:
5602:
5598:
5594:
5590:
5586:
5582:
5578:
5570:
5566:
5562:
5558:
5554:
5550:
5548:
5539:
5535:
5531:
5530:′) = (
5527:
5523:
5519:
5515:
5511:
5507:
5503:
5499:
5495:
5491:
5488:
5486:
5482:lifts limits
5481:
5478:
5474:
5469:
5467:
5464:
5460:
5456:
5455:′) = (
5452:
5448:
5444:
5440:
5439:′) of
5436:
5432:
5428:
5424:
5420:
5416:
5412:
5408:
5404:
5400:
5396:
5392:
5390:
5374:
5372:
5363:
5359:
5355:
5351:
5350:The functor
5349:
5289:
5284:
5280:
5276:
5272:
5268:
5266:
5207:
5203:
5199:
5195:
5191:
5190:and functor
5187:
5183:
5179:
5177:
5169:
5164:
5160:
5156:
5152:
5144:
5142:
5137:
5133:
5129:
5125:
5121:
5117:
5113:
5109:
5105:
5101:
5097:
5095:
5090:
5086:
5082:
5078:
5074:
5070:
5065:
5063:
5059:
5057:
5052:
5045:
5041:
5037:
5033:
5029:
5025:
5021:
5017:
5012:
5010:
5006:
5002:
4998:
4994:
4990:
4986:
4982:
4978:
4974:
4970:
4966:
4962:
4960:
4949:
4945:
4937:
4933:
4929:
4921:
4917:
4913:
4909:
4905:
4901:
4897:
4895:
4885:
4883:
4769:
4705:
4701:
4699:
4692:, –).
4689:
4685:
4681:
4677:
4673:
4669:
4665:
4663:
4581:
4577:
4573:
4569:
4565:
4563:
4558:
4554:
4550:
4542:
4537:
4531:
4527:
4523:
4519:
4515:
4511:
4510: : Hom(
4507:
4500:
4496:
4491:
4487:
4483:
4478:
4470:
4466:
4462:
4458:
4453:
4450:
4371:
4367:
4363:
4359:
4355:
4351:
4349:
4344:
4340:
4336:
4332:
4328:
4326:
4244:
4240:
4236:
4232:
4228:
4226:
4221:
4214:
4210:
4206:
4198:
4194:
4191:Hom functors
4189:One can use
4188:
4171:
4168:
4163:
4159:
4155:
4153:
4074:
4070:
4067:left adjoint
4064:
4008:
4004:
4000:
3996:
3994:
3989:
3985:
3977:
3973:
3965:
3961:
3957:
3953:
3949:
3947:
3875:
3871:
3867:
3863:
3859:
3855:
3847:
3843:
3839:
3835:
3829:
3773:
3769:
3765:
3761:
3759:
3748:
3744:
3737:
3733:
3727:
3722:
3718:
3714:
3710:
3706:
3702:
3698:
3694:
3690:
3686:
3681:
3677:
3673:
3669:
3665:
3661:
3657:
3653:
3649:
3645:
3641:
3637:
3630:
3626:
3622:
3618:
3616:
3611:
3607:
3602:
3597:
3593:
3589:
3585:
3581:
3577:
3573:
3569:
3565:
3561:
3557:
3553:
3549:
3547:
3489:
3485:
3481:
3477:
3470:
3466:
3464:
3457:
3447:
3443:
3441:
3246:
3114:
3110:
3106:
3102:
3098:
3094:
3090:
3086:
3082:
3080:
3073:
3069:
3065:
3060:
3058:
3056:
3051:
3047:
3041:
3039:
3001:
2969:
2963:
2959:
2955:
2951:
2944:
2940:
2936:
2931:
2929:
2925:
2923:
2918:
2914:
2910:
2906:
2902:
2900:
2882:
2874:
2864:
2858:Coequalizers
2856:
2848:
2840:
2832:
2827:
2782:
2778:
2774:
2706:
2702:
2698:
2694:
2690:
2686:
2682:
2600:
2596:
2592:
2584:
2580:
2576:
2572:
2568:
2564:
2560:
2559:, denote by
2556:
2545:
2539:
2535:
2531:
2527:
2523:
2516:
2508:
2504:
2500:
2496:
2492:
2488:
2484:
2480:
2476:
2472:
2468:
2464:
2457:
2453:
2449:
2445:
2441:
2437:
2433:
2429:
2425:
2421:
2418:directed set
2413:
2407:
2388:
2384:
2383:is called a
2380:
2376:
2375:. The limit
2372:
2368:
2364:
2360:
2356:
2352:
2348:
2344:
2340:
2336:
2332:
2326:
2314:
2308:
2302:
2298:
2297:. The limit
2294:
2290:
2286:
2280:
2272:
2271:and denoted
2268:
2264:
2260:
2256:
2248:
2244:
2230:
2226:
2222:
2218:
2213:
2209:
2205:
2201:
2197:
2193:
2192:. The limit
2189:
2185:
2177:
2169:
2163:
2157:
2153:
2149:
2141:
2137:
2131:
2124:
2120:
2116:
2112:
2108:
2106:
2091:
2087:
2086:
1988:. If we let
1943:
1915:
1880:
1878:
1611:
1609:
1459:
1329:
1327:
1322:dual notions
1319:
1316:Direct limit
1303:
1299:
1289:
1284:
1274:
1269:
1219:
1217:
1212:
1106:
975:
835:
833:
571:
497:
461:
457:
427:
336:
207:
162:
146:
121:
101:
91:
76:
67:
48:
6800:-categories
6776:Kan complex
6766:Tricategory
6748:-categories
6638:Subcategory
6396:Exponential
6364:Preadditive
6359:Pre-abelian
6044:bifunctor.
5977:coproducts,
5837:; however,
5801:Hom functor
5660:is said to
5405:lift limits
5403:is said to
5062:is said to
5009:is said to
4770:there is a
4708:be a small
3756:Adjunctions
3732:A limit of
2947:is said to
2928:is said to
2924:A category
2513:isomorphism
2231:projections
1296:isomorphism
130:direct sums
118:dual notion
98:mathematics
62:introducing
6818:3-category
6808:2-category
6781:β-groupoid
6756:Bicategory
6503:Coproducts
6463:Equalizers
6369:Bicategory
6220:0906.18001
6128:0906.18001
6079:References
5974:cokernels,
5912:direct sum
5865:free group
5695:such that
5487:A functor
5443:such that
5391:A functor
5202:, if both
5140:colimits.
5120:whenever (
5058:A functor
5028:whenever (
4977:) to Cone(
4961:A functor
4712:. For any
4183:See also:
4175:functors.
3640: : Ξ(
3093:) and Hom(
2958:for every
2892:Properties
2842:Coproducts
2436:) and let
2282:Equalizers
2088:Weak limit
1940:Variations
1782:such that
1552:, we have
1314:See also:
1010:such that
870:is a cone
776:, we have
492:See also:
159:Definition
134:coproducts
70:March 2013
45:references
6867:Symmetric
6812:2-functor
6552:Relations
6475:Pullbacks
5983:codomains
5703:′,
5687:′,
5526:′,
5510:′,
5502:′,
5451:′,
5435:′,
5376:covariant
5319:→
5307:τ
5243:→
5222:τ
5049:vacuously
4848:
4825:→
4794:
4741:→
4735:×
4714:bifunctor
4672:–,
4640:−
4631:
4622:≅
4607:
4598:
4553:) :
4418:
4406:−
4391:
4309:−
4294:
4285:≅
4264:
4172:covariant
4158:to colim
4121:
4115:≅
4100:
4091:
4043:→
3984:Ξ΄ :
3970:connected
3918:
3912:≅
3891:
3834:Ξ· :
3808:→
3560:) :
3517:→
3504:Δ
3409:
3403:∈
3377:
3370:π
3334:
3328:∈
3302:
3295:π
3291:∘
3220:
3197:
3191:∈
3184:∏
3180:⇉
3154:
3148:∈
3141:∏
3024:∙
3021:←
3018:∙
3015:→
3012:∙
2986:∙
2983:⇉
2980:∙
2866:Cokernels
2752:→
2663:⊂
2657:∪
2642:∣
2636:∈
2575:a point,
2328:Pullbacks
2303:equalizer
2049:→
1810:ψ
1797:ϕ
1793:∘
1767:→
1715:ψ
1663:ϕ
1631:→
1589:ψ
1570:∘
1561:ψ
1517:→
1442:→
1418:ψ
1351:→
1302:limit of
1294:a unique
1156:ϕ
1124:ψ
1038:ψ
1028:∘
1019:ϕ
995:→
978:morphism
939:ψ
887:ϕ
855:→
813:ψ
800:ψ
796:∘
741:→
660:→
645:ψ
515:→
464:whenever
383:morphisms
316:→
212:of shape
110:pullbacks
6942:Category
6927:Glossary
6907:Category
6881:n-monoid
6834:concepts
6490:Colimits
6458:Products
6411:Morphism
6354:Concrete
6349:Additive
6339:Category
6190:(1998).
6098:(1998).
6048:See also
5994:products
5991:kernels,
5851: :
5825: :
5775:Examples
5652: :
5640: :
5544:amnestic
5411: :
5395: :
5194: :
5182: :
5155: :
5093:limits.
5073: :
4965: :
4932: :
4916: :
4900: :
4568: :
4473:) where
4465:) β Hom(
4231: :
3709: :
3685: :
3109: :
3097:), then
2905: :
2876:Pushouts
2849:Copowers
2824:Colimits
2495: :
2440: :
2385:pullback
2367: :
2355: :
2217: :
2165:Products
2119: :
2098:Examples
1828:for all
1310:Colimits
1056:for all
452:or even
165:category
138:pushouts
106:products
6917:Outline
6876:n-group
6841:2-group
6796:Strict
6786:β-topos
6582:Modules
6520:Pushout
6468:Kernels
6401:Functor
6344:Abelian
6269:at the
6143:at the
6000:domains
5593:, then
4926:functor
3866:to lim
2310:Kernels
2265:J power
2202:product
2008:be the
1889:in the
1612:colimit
1331:co-cone
1279:in the
254:functor
209:diagram
122:colimit
116:. The
58:improve
18:Colimit
6863:Traced
6846:2-ring
6576:Fields
6562:Groups
6557:Magmas
6445:Limits
6239:
6218:
6208:
6176:
6126:
6116:
5780:Every
4981:): if
4201:, the
4118:Cocone
3846:β lim
3600:) = id
3592:and Ξ(
3488:. The
2511:is an
2412:. Let
2343:, and
2331:. Let
2315:kernel
2245:Powers
2182:family
2103:Limits
976:unique
488:Limits
462:finite
454:finite
47:, but
6857:-ring
6744:Weak
6728:Topos
6572:Rings
6267:Limit
6170:(PDF)
5941:with
5867:over
5368:up to
5147:is a
5138:small
5091:small
4993:then
4924:is a
4839:colim
4785:colim
4604:colim
4545:is a
4162:. If
4097:colim
4021:colim
4003:(for
3768:(for
3751:to Ξ.
3633:), a
2960:small
2416:be a
2387:or a
2285:. If
2251:is a
2172:is a
2168:. If
2136:. If
1893:from
1292:up to
836:limit
458:small
450:small
448:is a
256:from
252:is a
120:of a
102:limit
6547:Sets
6237:ISBN
6206:ISBN
6174:ISBN
6114:ISBN
5928:Let
5895:and
5883:and
5863:the
5844:The
5818:The
5803:Hom(
5725:for
5667:for
5636:Let
5601:and
5585:and
5557:and
5518:and
5279:and
5206:and
5132:. A
5016:if (
4912:and
4700:Let
4486:) =
4415:Cone
3952:and
3915:Cone
3713:β Ξ(
3668:) =
3644:) β
3580:) =
3465:Let
3081:The
3072:. A
2587:the
2583:) β
2363:and
2313:. A
2090:and
1320:The
573:cone
570:. A
498:Let
484:is.
151:and
140:and
112:and
6391:End
6381:CCC
6274:Lab
6216:Zbl
6148:Lab
6124:Zbl
6032:Hom
6012:Hom
5956:Set
5947:Met
5930:Met
5923:Set
5919:Top
5908:Grp
5857:Grp
5853:Set
5831:Set
5827:Grp
5813:Set
5795:of
5789:Set
5608:If
5577:If
5322:lim
5316:lim
5246:lim
5237:lim
5143:If
5104:if
5055:.)
5001:to
4989:to
4896:If
4886:Set
4829:lim
4798:lim
4628:Hom
4625:lim
4595:Hom
4559:Set
4535:(id
4499:of
4388:Hom
4385:lim
4370:to
4354:in
4291:Hom
4288:lim
4276:lim
4261:Hom
4220:in
4215:Set
4199:Set
4088:Hom
3968:is
3960:to
3903:lim
3888:Hom
3874:to
3786:lim
3725:.)
3721:to
3656:to
3610:in
3588:in
3568:to
3552:in
3484:in
3406:Hom
3374:cod
3331:Hom
3299:dom
3217:cod
3194:Hom
3054:).
2591:of
2479:of
2463:If
2456:or
2379:of
2347:in
2267:of
2259:of
2196:of
1848:in
1730:of
1678:of
1532:in
1480:of
1386:of
1300:the
1283:to
1270:one
1076:in
954:to
902:to
756:in
704:of
616:of
576:to
460:or
385:in
276:to
232:in
92:In
6944::
6869:)
6865:)(
6214:.
6204:.
6196:.
6122:.
6112:.
6104:.
6087:^
5954:β
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5904:Ab
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5811:β
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5739:GF
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5675:,
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5559:GF
5546:.
5534:,
5459:,
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5719:.
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5699:(
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4869:.
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4860:,
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4854:(
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4634:(
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4508:Ξ¦
4501:C
4497:L
4492:X
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4484:Ο
4482:(
4479:X
4475:Ο
4467:N
4463:F
4459:N
4454:X
4436:.
4433:)
4430:F
4427:,
4424:N
4421:(
4412:=
4409:)
4403:F
4400:,
4397:N
4394:(
4372:F
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4364:F
4360:N
4356:C
4352:F
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4312:)
4306:F
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4300:N
4297:(
4282:)
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4270:N
4267:(
4245:F
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4233:J
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4139:.
4136:)
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4112:)
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4094:(
4075:C
4071:C
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4024::
4005:J
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3876:F
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3802:J
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3789::
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3766:C
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3749:F
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3738:F
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3699:N
3695:X
3693:(
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3646:F
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3507::
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