7048:
2298:
5710:
5372:
4991:
5641:
5585:
5001:
5631:
5423:
9787:
7812:
1932:
1844:
2293:{\displaystyle {\begin{aligned}&\theta \circ {\Big (}\theta _{1}\circ (\theta _{1,1},\ldots ,\theta _{1,k_{1}}),\ldots ,\theta _{n}\circ (\theta _{n,1},\ldots ,\theta _{n,k_{n}}){\Big )}\\={}&{\Big (}\theta \circ (\theta _{1},\ldots ,\theta _{n}){\Big )}\circ (\theta _{1,1},\ldots ,\theta _{1,k_{1}},\ldots ,\theta _{n,1},\ldots ,\theta _{n,k_{n}})\end{aligned}}}
7843:-dimensional semidisks, centered at the base of the unit semidisk and sitting inside of it. The operadic composition comes from gluing configurations of "little" disks inside the unit disk into the "little" disks in another unit semidisk and configurations of "little" disks and semidisks inside the unit semidisk into the other unit semidisk.
6663:
1615:
3191:
7882:
Another class of examples of operads are those capturing the structures of algebraic structures, such as associative algebras, commutative algebras and Lie algebras. Each of these can be exhibited as a finitely presented operad, in each of these three generated by binary operations.
2634:
3737:
5578:
10087:
6526:
8968:
This point of view formalizes the notion that linear combinations are the most general sort of operation on a vector space – saying that a vector space is an algebra over the operad of linear combinations is precisely the statement that
4032:
9424:
1839:{\displaystyle {\begin{aligned}\circ :P(n)\times P(k_{1})\times \cdots \times P(k_{n})&\to P(k_{1}+\cdots +k_{n})\\(\theta ,\theta _{1},\ldots ,\theta _{n})&\mapsto \theta \circ (\theta _{1},\ldots ,\theta _{n}),\end{aligned}}}
9336:
7271:
7133:
centered at the origin. The symmetric group acts on such configurations by permuting the list of little disks. The operadic composition for little disks is illustrated in the accompanying figure to the right, where an element
1186:
9170:
and operadic composition given by substituting polynomials (with renumbered variables) for variables. A similar operad can be defined whose algebras are the associative, commutative algebras over some fixed base field. The
210:"The name operad and the formal definition appear first in the early 1970's in J. Peter May's "The Geometry of Iterated Loop Spaces", but a year or more earlier, Boardman and Vogt described the same concept under the name
7351:
9958:
8895:
3024:
8705:
695:
8801:
1916:
6245:
9021:
sum to 1, the terms are all non-negative, or both, respectively. Graphically, these are the infinite affine hyperplane, the infinite hyper-octant, and the infinite simplex. This formalizes what is meant by
5203:
5129:
4476:
593:
8565:
7972:
8243:
5289:
4644:
can be written involving operations without ambiguity from the omitted compositions, just as associativity for operations allows products to be written without ambiguity from the omitted parentheses.
3479:
6956:
5970:
9672:
6521:
6455:
4623:
2490:
1937:
1620:
2784:
9620:
3604:
3593:
8963:
7664:
7030:
6389:
4343:
4298:
4203:
1454:
9558:
5434:
4828:
1309:
6815:
6739:
6712:
6298:
9227:
8508:
3298:
2714:
401:
3245:
6658:{\displaystyle V^{\otimes k_{1}}\otimes \cdots \otimes V^{\otimes k_{n}}\ {\overset {g_{1}\otimes \cdots \otimes g_{n}}{\longrightarrow }}\ V^{\otimes n}\ {\overset {f}{\to }}\ V}
5817:
9715:
744:
9965:
7871:
6334:
4123:
3017:
9141:
8593:
6003:
5365:
5327:
4256:
2419:
338:
9766:
are the special case of operads that are also closed under identifying arguments together ("reusing" some data). Clones can be equivalently defined as operads that are also a
9049:
7480:
7131:
3840:
1602:
7167:
4425:
9586:
9519:
9495:
6984:
6901:
6873:
6131:
6067:
5903:
5879:
5624:
9753:
7757:
6845:
3799:
4700:
4390:
4366:
8084:
7378:
9879:”finiteness" refers to the fact that only a finite number of inputs are allowed in the definition of an operad. For example, the condition is satisfied if one can write
9461:
9019:
8034:
7035:
Depending on applications, variations of the above are possible: for example, in algebraic topology, instead of vector spaces and tensor products between them, one uses
5851:
5745:
5695:
5416:
5051:
2479:
1257:
9825:
Operads are particularly important and useful in categories with a good notion of "homotopy", where they play a key role in organizing hierarchies of higher homotopies.
8290:
1224:
503:
4056:
3398:
8467:
8388:
8263:
7440:
7402:
4941:
4921:
4866:
4790:
4749:
4665:
4519:
3547:
2661:
1395:
4901:
4561:
3912:
3521:
1332:
9254:
9168:
8620:
8447:
8352:
8185:
8158:
7999:
7904:
7719:
7606:
6769:
3352:
3325:
2905:
2858:
2831:
2362:
1008:
892:
865:
838:
811:
214:, inspired by PROPs and PACTs of Adams and Mac Lane. In fact, there is an abundance of prehistory. Weibel points out that the concept first arose a century ago in
8417:
8128:
7571:
7090:
4729:
3917:
3888:
2391:
1529:
1037:
9344:
4964:
7801:
7781:
7688:
6107:
6087:
6043:
6023:
5923:
5778:
4984:
2970:
2945:
2925:
2878:
2804:
2439:
2332:
1556:
1500:
1475:
1352:
978:
952:
932:
912:
784:
764:
465:
445:
425:
310:
161:
137:
117:
94:
72:
49:
10586:
31:, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad
9262:
7172:
9767:
4125:. The structure maps of the operad (the composition and the actions of the symmetric groups) are then assumed to be continuous. The result is called a
1045:
10939:
192:
3186:{\displaystyle \theta \circ (\theta _{1}*s_{1},\ldots ,\theta _{n}*s_{n})=(\theta \circ (\theta _{1},\ldots ,\theta _{n}))*(s_{1},\ldots ,s_{n})}
7276:
9884:
8977:
for the operad of all linear combinations, while the linear combinations operad canonically encodes all possible operations on a vector space.
8806:
8625:
7506:
10253:"Operads in Algebra, Topology and Physics": Martin Markl, Steve Shnider, Jim Stasheff, Mathematical Surveys and Monographs, Volume: 96; 2002
601:
10974:
8710:
7863:
8973:
algebraic operations in a vector space are linear combinations. The basic operations of vector addition and scalar multiplication are a
5715:
meaning that the three operations obtained are equal: pre- or post- composing with the identity makes no difference. As for categories,
1859:
6175:
5134:
5060:
4434:
4058:
denotes the tensor product of the monoidal category), and the actions of the symmetric group elements are given by isomorphisms in
7859:
508:
8517:
7912:
4129:. Similarly, in the definition of a morphism of operads, it would be necessary to assume that the maps involved are continuous.
8190:
5208:
3406:
3360:
The permutation actions in this definition are vital to most applications, including the original application to loop spaces.
11125:
11110:
11077:
11016:
10923:
10894:
10652:
10569:
10230:
10176:
6914:
5928:
2629:{\displaystyle (\theta *t)\circ (\theta _{1},\ldots ,\theta _{n})=(\theta \circ (\theta _{t(1)},\ldots ,\theta _{t(n)}))*t'}
10439:
Jones, J. D. S.; Getzler, Ezra (8 March 1994). "Operads, homotopy algebra and iterated integrals for double loop spaces".
9628:
6460:
6394:
4566:
4403:. For example, Nikolai Durov defines his generalized rings as monoid objects in the monoidal category of endofunctors on
3732:{\displaystyle f(\theta \circ (\theta _{1},\ldots ,\theta _{n}))=f(\theta )\circ (f(\theta _{1}),\ldots ,f(\theta _{n}))}
8469:
operad has the structure of a braided operad, which generalizes the notion of an operad from symmetric to braid groups.
10418:
Kontsevich, Maxim; Soibelman, Yan (26 January 2000). "Deformations of algebras over operads and
Deligne's conjecture".
2719:
9591:
11055:
10838:
10540:
10504:
9469:
Like a group or a ring, the free construction allows to express an operad in terms of generators and relations. By a
4259:
3552:
9055:
is the image of a simplex. Here suboperads correspond to more restricted operations and thus more general theories.
11150:
10560:. Proceedings of the NATO Advanced Study Institute on Axiomatic, Enriched and Motivic Homotopy Theory. Cambridge,
8900:
7611:
6997:
6342:
10116:
5709:
5573:{\displaystyle \theta _{(ab)c\cdot d}\circ ((\theta _{ab\cdot c},1_{d})\circ ((\theta _{a\cdot b},1_{c}),1_{d}))}
4310:
4265:
4170:
1409:
9524:
4795:
1262:
11069:
6774:
6717:
6671:
6257:
10378:
9190:
8484:
3250:
2666:
346:
76:
to be a set together with concrete operations on this set which behave just like the abstract operations of
10913:
10082:{\displaystyle \gamma (V):T_{n}\otimes T_{i_{1}}\otimes \cdots \otimes T_{i_{n}}\to T_{i_{1}+\dots +i_{n}}}
7036:
3198:
1314:
The definition of a symmetric operad given below captures the essential properties of these two operations
270:
5785:
163:
abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a
9681:
3855:
703:
11028:(2012), "Encyclopedia of types of algebras 2010", in Bai, Chengming; Guo, Li; Loday, Jean-Louis (eds.),
6303:
4080:
2975:
9074:
8570:
5975:
5332:
5294:
4223:
2396:
315:
9025:
7456:
7107:
6986:. (Notice the analogy between operads&operad algebras and rings&modules: a module over a ring
3812:
11165:
11032:, Nankai Series in Pure, Applied Mathematics and Theoretical Physics, vol. 9, pp. 217–298,
8965:
for instance represents the operation of forming a linear combination with coefficients 2,3,-5,0,...
7137:
5371:
4990:
4406:
1561:
10213:
9567:
9500:
9476:
6965:
6882:
6854:
6112:
6048:
5884:
5860:
5640:
5593:
5584:
5000:
11170:
9720:
7724:
6820:
4641:
3748:
258:
250:
9187:
Typical algebraic constructions (e.g., free algebra construction) can be extended to operads. Let
5205:, if the outer compositions are performed first (operations are read from right to left). Writing
4670:
4373:
4349:
8039:
7356:
4427:
that commute with filtered colimits. This is a generalization of a ring since each ordinary ring
4345:
is an operad. Similarly, a symmetric operad can be defined as a monoid object in the category of
4301:
227:
28:
9437:
8995:
8004:
5823:
5718:
5653:
5380:
5009:
2451:
1229:
10999:
10208:
9840:
8268:
7521:
5630:
5422:
3806:
1194:
473:
273:
233:
Interest in operads was considerably renewed in the early 90s when, based on early insights of
215:
10544:
9051:
being or the standard simplex being model spaces, and such observations as that every bounded
4041:
3371:
9835:
9798:
8452:
8373:
8248:
7498:
7407:
7387:
4926:
4906:
4833:
4757:
4734:
4650:
4491:
4393:
4027:{\displaystyle P(n)\otimes P(k_{1})\otimes \cdots \otimes P(k_{n})\to P(k_{1}+\cdots +k_{n})}
3532:
1380:
246:
10669:
9419:{\displaystyle \Gamma :\prod _{n\in \mathbb {N} }\mathbf {Set} ^{S_{n}}\to {\mathsf {Oper}}}
5760:
The most basic operads are the ones given in the section on "Intuition", above. For any set
5590:
If the bottom two rows of operations are composed first (puts a downward parenthesis at the
4871:
4546:
3897:
3491:
1317:
11087:
11043:
10882:
10691:
10386:
10332:
9232:
9146:
8598:
8425:
8330:
8163:
8136:
7984:
7889:
7867:
7847:
7697:
7584:
6747:
5428:
which then evaluates unambiguously to yield a 4-ary operation. As an annotated expression:
4133:
3330:
3303:
2883:
2836:
2809:
2340:
986:
870:
843:
816:
789:
168:
11095:
10394:
10348:
8393:
8104:
7886:
For example, the associative operad is a symmetric operad generated by a binary operation
7547:
7066:
4705:
3864:
2367:
1505:
1013:
8:
9845:
8985:
8304:
7047:
6146:
164:
11047:
10886:
10695:
5377:
If the top two rows of operations are composed first (puts an upward parenthesis at the
4946:
2641:
11033:
10948:
10933:
10872:
10844:
10799:
10781:
10738:
10707:
10681:
10619:
10510:
10482:
10440:
10419:
10336:
8989:
8981:
8511:
7855:
7832:
7786:
7766:
7673:
7097:
6092:
6072:
6028:
6008:
5908:
5763:
4969:
4066:
2955:
2930:
2910:
2863:
2789:
2424:
2317:
1541:
1485:
1460:
1337:
963:
937:
917:
897:
769:
749:
450:
430:
410:
295:
266:
180:
146:
122:
102:
79:
57:
34:
10830:
10496:
7532:, and "little disks" is a case of "folklore" derived from the "little convex bodies".
11121:
11106:
11073:
11051:
11012:
10919:
10890:
10834:
10803:
10648:
10565:
10536:
10500:
10340:
10320:
10267:
10236:
10226:
10182:
10172:
10138:
9257:
8354:
acting trivially. The algebras over this operad are the commutative semigroups; the
4070:
281:
10711:
10603:
10370:
10133:
9331:{\displaystyle {\mathsf {Oper}}\to \prod _{n\in \mathbb {N} }\mathbf {Set} ^{S_{n}}}
11091:
11025:
10995:
10848:
10826:
10791:
10748:
10699:
10598:
10514:
10492:
10459:
N. Durov, New approach to
Arakelov geometry, University of Bonn, PhD thesis, 2007;
10390:
10366:
10344:
10312:
10218:
10164:
10156:
10128:
10112:
9339:
7851:
7827:
is a two-colored topological operad defined in terms of configurations of disjoint
7502:
7266:{\displaystyle (\theta _{1},\theta _{2},\theta _{3})\in P(2)\times P(3)\times P(4)}
4161:
4137:
277:
262:
242:
234:
188:
11007:, Grundlehren der Mathematischen Wissenschaften, vol. 346, Berlin, New York:
10618:
Stasheff, Jim (1998). "Grafting
Boardman's Cherry Trees to Quantum Field Theory".
10316:
10300:
11083:
11008:
10966:
10769:
10382:
10328:
10296:
9763:
9052:
8992:
can be considered to correspond to the sub-operads where the terms of the vector
8131:
3851:
2335:
1181:{\displaystyle (f*s)(x_{1},\ldots ,x_{n})=f(x_{s^{-1}(1)},\ldots ,x_{s^{-1}(n)})}
981:
238:
11151:
https://golem.ph.utexas.edu/category/2011/05/an_operadic_introduction_to_en.html
10561:
10352:
9338:, which simply forgets the operadic composition. It is possible to construct a
9068:
8974:
8478:
7525:
6158:
5854:
254:
52:
10753:
10726:
10703:
11159:
10905:
10644:
10529:
10324:
10240:
10186:
10142:
9860:
9855:
9850:
7978:
7783:
and forming a larger tree, whose root is taken to be the same as the root of
4307:
For example, a monoid object in the category of "polynomial endofunctors" on
4218:
4141:
5705:
The identity axiom (for a binary operation) can be visualized in a tree as:
10962:
10909:
10263:
10200:
9427:
7510:
6165:
We can then define endomorphism operads in this category, as follows. Let
6150:
6136:
958:
203:
196:
7346:{\displaystyle \theta \circ (\theta _{1},\theta _{2},\theta _{3})\in P(9)}
10903:
10795:
9953:{\displaystyle T(V)=\bigoplus _{n=1}^{\infty }T_{n}\otimes V^{\otimes n}}
9172:
8420:
7541:
4483:
223:
140:
20:
9786:
8890:{\displaystyle {\vec {x}}=(x^{(1)},\ldots ,x^{(n)})\in \mathbb {R} ^{n}}
7608:
operates on this set by permuting the leaf labels. Operadic composition
11138:
10445:
10222:
10168:
9176:
8700:{\displaystyle {\vec {x}}\circ ({\vec {y_{1}}},\ldots ,{\vec {y_{n}}})}
7858:
on
Hochschild cohomology. Kontsevich's conjecture was proven partly by
184:
97:
8481:, real vector spaces can be considered to be algebras over the operad
7811:
690:{\displaystyle f\circ (f_{1},\ldots ,f_{n})\in P(k_{1}+\cdots +k_{n})}
10953:
10877:
10743:
10686:
10624:
10487:
10460:
10424:
8296:
7517:
7101:
4149:
8796:{\displaystyle x^{(1)}{\vec {y_{1}}},\ldots ,x^{(n)}{\vec {y_{n}}}}
4346:
4145:
257:
of operads. Operads have since found many applications, such as in
11038:
10786:
5972:. Intuitively, such a morphism turns each "abstract" operation of
5755:
5131:, if the inner compositions are performed first, or it could mean
10534:
Geometric and
Algebraic Topological Methods in Quantum Mechanics,
3854:
of sets. More generally, it is possible to define operads in any
1911:{\displaystyle \theta \circ (1,\ldots ,1)=\theta =1\circ \theta }
8299:: sets together with a single binary associative operation. The
6958:; this amounts to specifying concrete multilinear operations on
6240:{\displaystyle {\mathcal {End}}_{V}=\{{\mathcal {End}}_{V}(n)\}}
5418:
line; does the inner composition first), the following results:
4563:
is associative), analogous to the axiom in category theory that
8319:
The terminal symmetric operad is the operad which has a single
8303:-linear algebras over the associative operad are precisely the
5636:
which then evaluates unambiguously to yield a 4-ary operation:
5198:{\displaystyle (\theta \circ (\theta ,1))\circ ((\theta ,1),1)}
5124:{\displaystyle \theta \circ ((\theta ,1)\circ ((\theta ,1),1))}
4923:
on the first two, and the identity on the third), and then the
4073:. In this case, a topological operad is given by a sequence of
9229:
denote the category whose objects are sets on which the group
6937:
6934:
6787:
6784:
6684:
6681:
6486:
6483:
6420:
6417:
6361:
6358:
6270:
6267:
6214:
6211:
6188:
6185:
5951:
5948:
5798:
5795:
4471:{\displaystyle \Sigma _{R}:{\textbf {Set}}\to {\textbf {Set}}}
4132:
Other common settings to define operads include, for example,
10161:
Homotopy
Invariant Algebraic Structures on Topological Spaces
5626:
line; does the outer composition first), following results:
4392:
means a symmetric group. A monoid object in the category of
4069:
and continuous maps, with the monoidal product given by the
11142:
10564:: Springer Science & Business Media. pp. 154–156.
9426:
to this forgetful functor (this is the usual definition of
8295:
The algebras over the associative operad are precisely the
5853:. These operads are important because they serve to define
588:{\displaystyle f_{1}\in P(k_{1}),\ldots ,f_{n}\in P(k_{n})}
8560:{\displaystyle \mathbb {R} ^{\infty }(n)=\mathbb {R} ^{n}}
7967:{\displaystyle \psi \circ (\psi ,1)=\psi \circ (1,\psi ).}
6109:
that follow the rules abstractely specified by the operad
4399:
An operad in the above sense is sometimes thought of as a
9058:
8238:{\displaystyle \sigma \circ (\tau _{1},\dots ,\tau _{n})}
7509:
in a similar way, in terms of configurations of disjoint
5284:{\displaystyle x=\theta ,y=(\theta ,1),z=((\theta ,1),1)}
6137:
Endomorphism operad in vector spaces and operad algebras
3474:{\displaystyle (f_{n}:P(n)\to Q(n))_{n\in \mathbb {N} }}
11066:
Homotopy of
Operads and Grothendieck-Teichmüller Groups
9179:(whose algebras are the Lie algebras), and vice versa.
7042:
5367:. That is, the tree is missing "vertical parentheses":
407:
the set of all functions from the cartesian product of
6951:{\displaystyle {\mathcal {O}}\to {\mathcal {End}}_{V}}
6157:; this becomes a monoidal category using the ordinary
5965:{\displaystyle {\mathcal {O}}\to {\mathcal {End}}_{X}}
1564:
218:"A Treatise on Universal Algebra", published in 1898."
9968:
9887:
9723:
9684:
9631:
9594:
9570:
9527:
9503:
9479:
9440:
9347:
9265:
9235:
9193:
9149:
9077:
9028:
8998:
8903:
8809:
8713:
8628:
8601:
8573:
8520:
8487:
8455:
8428:
8396:
8376:
8333:
8271:
8251:
8193:
8166:
8139:
8107:
8042:
8007:
7987:
7915:
7892:
7789:
7769:
7727:
7700:
7676:
7614:
7587:
7550:
7459:
7410:
7390:
7359:
7279:
7175:
7140:
7110:
7069:
7000:
6968:
6917:
6885:
6857:
6823:
6777:
6750:
6720:
6674:
6529:
6463:
6397:
6345:
6306:
6260:
6178:
6149:, we can consider the category of finite-dimensional
6115:
6095:
6075:
6051:
6031:
6011:
5978:
5931:
5911:
5887:
5863:
5826:
5788:
5766:
5721:
5656:
5596:
5437:
5383:
5335:
5297:
5211:
5137:
5063:
5012:
4972:
4949:
4929:
4909:
4874:
4836:
4798:
4760:
4737:
4708:
4673:
4653:
4569:
4549:
4494:
4437:
4409:
4376:
4352:
4313:
4268:
4226:
4173:
4083:
4044:
3920:
3900:
3867:
3815:
3751:
3607:
3555:
3535:
3494:
3409:
3374:
3333:
3306:
3253:
3201:
3027:
2978:
2958:
2933:
2913:
2886:
2866:
2839:
2812:
2792:
2722:
2669:
2644:
2493:
2454:
2427:
2399:
2370:
2343:
2320:
1935:
1862:
1618:
1544:
1508:
1488:
1463:
1412:
1383:
1340:
1320:
1265:
1232:
1197:
1048:
1016:
989:
966:
940:
920:
900:
873:
846:
819:
792:
772:
752:
706:
604:
511:
476:
453:
433:
413:
349:
318:
298:
230:" (and also because his mother was an opera singer).
149:
125:
105:
82:
60:
37:
9667:{\displaystyle {\mathcal {O}}=\{{\mathcal {O}}(n)\}}
6516:{\displaystyle g_{n}\in {\mathcal {End}}_{V}(k_{n})}
6450:{\displaystyle g_{1}\in {\mathcal {End}}_{V}(k_{1})}
10417:
4618:{\displaystyle f\circ (g\circ h)=(f\circ g)\circ h}
10727:"On Kontsevich's Hochschild cohomology conjecture"
10725:Hu, Po; Kriz, Igor; Voronov, Alexander A. (2006).
10247:
10081:
9952:
9747:
9717:is the generator and the relation is contained in
9709:
9666:
9614:
9580:
9552:
9513:
9489:
9455:
9418:
9330:
9248:
9221:
9162:
9135:
9043:
9013:
8957:
8889:
8795:
8699:
8614:
8587:
8559:
8502:
8461:
8441:
8411:
8382:
8346:
8284:
8257:
8237:
8179:
8152:
8122:
8078:
8028:
7993:
7966:
7898:
7795:
7775:
7751:
7713:
7682:
7658:
7600:
7565:
7474:
7434:
7396:
7372:
7345:
7265:
7161:
7125:
7084:
7024:
6978:
6950:
6895:
6867:
6839:
6809:
6763:
6733:
6706:
6657:
6515:
6449:
6383:
6328:
6292:
6239:
6125:
6101:
6081:
6061:
6037:
6017:
5997:
5964:
5917:
5897:
5873:
5845:
5811:
5772:
5739:
5689:
5618:
5572:
5410:
5359:
5321:
5283:
5197:
5123:
5045:
4978:
4958:
4935:
4915:
4895:
4860:
4822:
4784:
4743:
4723:
4694:
4659:
4617:
4555:
4513:
4470:
4419:
4384:
4360:
4337:
4292:
4250:
4197:
4117:
4050:
4026:
3906:
3882:
3834:
3793:
3731:
3587:
3541:
3515:
3473:
3392:
3346:
3319:
3292:
3239:
3185:
3011:
2964:
2939:
2919:
2899:
2872:
2852:
2825:
2798:
2778:
2708:
2655:
2628:
2473:
2433:
2413:
2385:
2356:
2326:
2292:
1910:
1838:
1596:
1550:
1523:
1494:
1469:
1448:
1389:
1346:
1326:
1303:
1251:
1218:
1180:
1031:
1002:
972:
946:
926:
906:
886:
859:
832:
805:
778:
758:
738:
689:
587:
497:
459:
439:
419:
395:
332:
304:
155:
131:
111:
88:
66:
43:
11103:Set Operads in Combinatorics and Computer Science
10670:"Operads and Motives in Deformation Quantization"
10643:. Contemporary Mathematics. Baltimore, Maryland,
8358:-linear algebras are the commutative associative
8265:, and within blocks according to the appropriate
2779:{\displaystyle \{1,2,\dots ,k_{1}+\dots +k_{n}\}}
2167:
2119:
2099:
1949:
11157:
10947:Markl, Martin (June 2006). "Operads and PROPs".
10295:
9615:{\displaystyle {\mathcal {F}}\to {\mathcal {O}}}
8365:
7577:leaves, where the leaves are numbered from 1 to
4633:are associative as operations. Compare with the
3850:So far operads have only been considered in the
2663:on the right hand side refers to the element of
183:; they were introduced to characterize iterated
10558:Axiomatic, Enriched and Motivic Homotopy Theory
10207:. Lecture Notes in Mathematics. Vol. 271.
10163:. Lecture Notes in Mathematics. Vol. 347.
5756:Endomorphism operad in sets and operad algebras
3588:{\displaystyle \theta _{1},\ldots ,\theta _{n}}
6903:is given by a finite-dimensional vector space
6817:by permuting the components of the tensors in
3354:, etc., and keeps their overall order intact).
2334:as above, together with a right action of the
957:We can also permute arguments, i.e. we have a
10994:
10724:
10121:Bulletin of the American Mathematical Society
9777:
8090:should not be confused with associativity of
4148:(or even the category of categories itself),
10975:Notices of the American Mathematical Society
10938:: CS1 maint: multiple names: authors list (
10866:
10438:
10155:
10111:
9661:
9642:
9071:are the commutative rings. It is defined by
8958:{\displaystyle {\vec {x}}=(2,3,-5,0,\dots )}
8314:
8187:acts by right multiplication. The composite
7659:{\displaystyle T\circ (S_{1},\ldots ,S_{n})}
7025:{\displaystyle R\to \operatorname {End} (M)}
6384:{\displaystyle f\in {\mathcal {End}}_{V}(n)}
6234:
6202:
4100:
4084:
2773:
2723:
390:
365:
10817:Markl, Martin (2006). "Operads and PROPs".
10473:Markl, Martin (2006). "Operads and PROPs".
8707:given by the concatentation of the vectors
8245:permutes its inputs in blocks according to
8095:
7353:obtained by shrinking the configuration of
4986:. This is clearer when depicted as a tree:
4667:is a binary operation, which is written as
4543:of operations is associative (the function
4529:
3300:that permutes the first of these blocks by
1849:satisfying the following coherence axioms:
10667:
7449:by considering configurations of disjoint
6169:be a finite-dimensional vector space The
5646:The operad axiom of associativity is that
4640:Associativity in operad theory means that
4338:{\displaystyle R{\text{-}}{\mathsf {Mod}}}
4293:{\displaystyle R{\text{-}}{\mathsf {Mod}}}
4198:{\displaystyle R{\text{-}}{\mathsf {Mod}}}
1449:{\displaystyle (P(n))_{n\in \mathbb {N} }}
222:The word "operad" was created by May as a
16:Generalization of associativity properties
11037:
10952:
10876:
10785:
10752:
10742:
10685:
10623:
10602:
10555:
10486:
10444:
10423:
10413:
10411:
10212:
10132:
9553:{\displaystyle {\mathcal {F}}=\Gamma (E)}
9366:
9297:
9094:
9031:
8877:
8581:
8547:
8523:
8490:
8036:multiplicatively, the above condition is
7528:. Later it was generalized by May to the
7462:
7113:
6523:, their composition is given by the map
4792:is unambiguously written operadically as
4378:
4354:
3465:
2407:
1440:
326:
27:is a structure that consists of abstract
10961:
10915:Operads in Algebra, Topology and Physics
10617:
9818:
9678:if it has a free presentation such that
7810:
7803:and whose leaves are numbered in order.
7046:
4823:{\displaystyle \theta \circ (\theta ,1)}
4304:) satisfying some finiteness condition.
4155:
1304:{\displaystyle x_{1},\ldots ,x_{n}\in X}
894:to the second block, etc. We then apply
212:categories of operators in standard form
11068:, Mathematical Surveys and Monographs,
11024:
10638:
10528:Giovanni Giachetta, Luigi Mangiarotti,
7854:to formulate a Swiss-cheese version of
7846:The Swiss-cheese operad was defined by
6810:{\displaystyle {\mathcal {End}}_{V}(n)}
6734:{\displaystyle \operatorname {id} _{V}}
6707:{\displaystyle {\mathcal {End}}_{V}(1)}
6293:{\displaystyle {\mathcal {End}}_{V}(n)}
4634:
1362:
11158:
11063:
10770:"Kontsevich's Swiss cheese conjecture"
10767:
10408:
9411:
9408:
9405:
9402:
9277:
9274:
9271:
9268:
9222:{\displaystyle \mathbf {Set} ^{S_{n}}}
9059:Commutative-ring operad and Lie operad
8622:permuting components, and composition
8503:{\displaystyle \mathbb {R} ^{\infty }}
7806:
5747:is a corollary of the identity axiom.
4534:
4330:
4327:
4324:
4285:
4282:
4279:
4190:
4187:
4184:
3845:
3827:
3824:
3821:
3818:
3293:{\displaystyle S_{k_{1}+\dots +k_{n}}}
2709:{\displaystyle S_{k_{1}+\dots +k_{n}}}
2310:A symmetric operad (often just called
470:We can compose these functions: given
396:{\displaystyle P(n):=\{f:X^{n}\to X\}}
11120:. Springer International Publishing.
11118:Nonsymmetric Operads in Combinatorics
10946:
10816:
10472:
10365:
7906:, subject only to the condition that
7877:
7039:and cartesian products between them.
6089:together with concrete operations on
5881:is an operad, an operad algebra over
3240:{\displaystyle (s_{1},\ldots ,s_{n})}
10205:The Geometry of Iterated Loop Spaces
9781:
9430:). Given a collection of operations
7573:is the set of all rooted trees with
5812:{\displaystyle {\mathcal {End}}_{X}}
4065:A common example is the category of
10584:
10199:
9710:{\displaystyle E={\mathcal {O}}(2)}
8094:which holds in any operad; see the
6962:that behave like the operations of
6668:(identity) The identity element in
4463:
4453:
4412:
3745:preserves the permutation actions:
2305:
1456:of sets, whose elements are called
1402:operad) consists of the following:
739:{\displaystyle k_{1}+\cdots +k_{n}}
13:
10117:"Homotopy-everything $ H$ -spaces"
9919:
9724:
9693:
9647:
9634:
9607:
9597:
9588:and the kernel of the epimorphism
9573:
9538:
9530:
9506:
9482:
9441:
9348:
8528:
8495:
8456:
8377:
6994:together with a ring homomorphism
6971:
6931:
6920:
6888:
6860:
6781:
6678:
6480:
6414:
6355:
6329:{\displaystyle V^{\otimes n}\to V}
6264:
6208:
6182:
6118:
6054:
5981:
5945:
5934:
5890:
5866:
5792:
4439:
4118:{\displaystyle \{P(n)\}_{n\geq 0}}
3012:{\displaystyle s_{i}\in S_{k_{i}}}
1384:
14:
11182:
11132:
11105:. SpringerBriefs in Mathematics.
10918:. American Mathematical Society.
10869:Higher Operads, Higher Categories
10115:; Vogt, R. M. (1 November 1968).
9136:{\displaystyle P(n)=\mathbb {Z} }
8588:{\displaystyle n\in \mathbb {N} }
8472:
5998:{\displaystyle {\mathcal {O}}(n)}
5700:
5360:{\displaystyle (x\circ y)\circ z}
5322:{\displaystyle x\circ (y\circ z)}
4260:monoidal category of endofunctors
4251:{\displaystyle (T,\gamma ,\eta )}
3525:preserves composition: for every
2414:{\displaystyle n\in \mathbb {N} }
333:{\displaystyle n\in \mathbb {N} }
202:Martin Markl, Steve Shnider, and
10268:"Operads, Algebras, and Modules"
9785:
9380:
9377:
9374:
9311:
9308:
9305:
9202:
9199:
9196:
9044:{\displaystyle \mathbb {R} ^{n}}
8101:In the associative operad, each
7475:{\displaystyle \mathbb {R} ^{n}}
7445:Analogously, one can define the
7126:{\displaystyle \mathbb {R} ^{2}}
5708:
5639:
5629:
5583:
5421:
5370:
4999:
4996:which yields a 3-ary operation:
4989:
3835:{\displaystyle {\mathsf {Oper}}}
1597:{\textstyle k_{1},\ldots ,k_{n}}
840:arguments, etc., and then apply
206:write in their book on operads:
10810:
10761:
10718:
10661:
10632:
10611:
10604:10.1090/s0002-9904-1977-14318-8
10578:
10549:
10522:
10466:
10453:
10432:
10262:
10134:10.1090/S0002-9904-1968-12070-1
9521:as a quotient of a free operad
9182:
7535:
7453:-balls inside the unit ball of
7162:{\displaystyle \theta \in P(3)}
7037:(reasonable) topological spaces
5650:, and thus that the expression
5603:
4751:may or may not be associative.
4420:{\displaystyle {\textbf {Set}}}
11064:Fresse, Benoit (17 May 2017),
10871:. Cambridge University Press.
10639:Voronov, Alexander A. (1999).
10359:
10289:
10256:
10193:
10149:
10105:
10040:
9978:
9972:
9897:
9891:
9873:
9742:
9736:
9733:
9727:
9704:
9698:
9658:
9652:
9602:
9581:{\displaystyle {\mathcal {O}}}
9547:
9541:
9514:{\displaystyle {\mathcal {O}}}
9490:{\displaystyle {\mathcal {O}}}
9450:
9444:
9397:
9282:
9130:
9098:
9087:
9081:
9005:
8952:
8919:
8910:
8869:
8864:
8858:
8839:
8833:
8825:
8816:
8787:
8769:
8763:
8743:
8725:
8719:
8694:
8688:
8660:
8644:
8635:
8539:
8533:
8406:
8400:
8232:
8200:
8117:
8111:
8073:
8064:
8052:
8043:
8023:
8011:
7977:This condition corresponds to
7958:
7946:
7934:
7922:
7653:
7621:
7560:
7554:
7340:
7334:
7325:
7286:
7260:
7254:
7245:
7239:
7230:
7224:
7215:
7176:
7156:
7150:
7079:
7073:
7063:is a topological operad where
7019:
7013:
7004:
6979:{\displaystyle {\mathcal {O}}}
6925:
6896:{\displaystyle {\mathcal {O}}}
6868:{\displaystyle {\mathcal {O}}}
6804:
6798:
6701:
6695:
6641:
6584:
6510:
6497:
6444:
6431:
6378:
6372:
6320:
6287:
6281:
6231:
6225:
6126:{\displaystyle {\mathcal {O}}}
6062:{\displaystyle {\mathcal {O}}}
5992:
5986:
5939:
5898:{\displaystyle {\mathcal {O}}}
5874:{\displaystyle {\mathcal {O}}}
5837:
5684:
5678:
5672:
5663:
5660:
5657:
5619:{\displaystyle ab\quad c\ \ d}
5567:
5564:
5548:
5516:
5513:
5507:
5472:
5469:
5452:
5443:
5393:
5384:
5348:
5336:
5316:
5304:
5278:
5269:
5257:
5254:
5242:
5230:
5192:
5183:
5171:
5168:
5162:
5159:
5147:
5138:
5118:
5115:
5106:
5094:
5091:
5085:
5073:
5070:
5040:
5034:
5028:
5019:
5016:
5013:
4890:
4875:
4855:
4837:
4817:
4805:
4779:
4773:
4764:
4761:
4754:Then what is commonly written
4718:
4709:
4689:
4677:
4606:
4594:
4588:
4576:
4506:
4500:
4458:
4245:
4227:
4096:
4090:
4021:
3989:
3983:
3980:
3967:
3952:
3939:
3930:
3924:
3877:
3871:
3782:
3776:
3767:
3755:
3726:
3723:
3710:
3695:
3682:
3676:
3670:
3664:
3655:
3652:
3620:
3611:
3504:
3498:
3454:
3450:
3444:
3438:
3435:
3429:
3410:
3384:
3234:
3202:
3180:
3148:
3142:
3139:
3107:
3098:
3092:
3034:
2612:
2609:
2604:
2598:
2576:
2570:
2559:
2550:
2544:
2512:
2506:
2494:
2380:
2374:
2283:
2175:
2162:
2130:
2094:
2043:
2018:
1967:
1887:
1869:
1826:
1794:
1785:
1778:
1740:
1733:
1701:
1695:
1688:
1675:
1660:
1647:
1638:
1632:
1518:
1512:
1429:
1425:
1419:
1413:
1213:
1207:
1175:
1170:
1164:
1132:
1126:
1105:
1096:
1064:
1061:
1049:
1026:
1020:
684:
652:
643:
611:
582:
569:
541:
528:
492:
486:
384:
359:
353:
1:
11070:American Mathematical Society
11030:Operads and universal algebra
10860:
10831:10.1016/S1570-7954(07)05002-4
10497:10.1016/S1570-7954(07)05002-4
10371:"La renaissance des opérades"
10317:10.1215/S0012-7094-94-07608-4
9748:{\displaystyle \Gamma (E)(3)}
9143:, with the obvious action of
8595:, with the obvious action of
8366:Operads from the braid groups
7752:{\displaystyle i=1,\ldots ,n}
7544:form a natural operad. Here,
7092:consists of ordered lists of
6990:is given by an abelian group
6840:{\displaystyle V^{\otimes n}}
4482:to the underlying set of the
4396:is an operad in finite sets.
3794:{\displaystyle f(x*s)=f(x)*s}
1357:
786:blocks, the first one having
700:is defined as follows: given
10587:"Infinite loop space theory"
10556:Greenlees, J. P. C. (2002).
10301:"Koszul duality for operads"
10299:; Kapranov, Mikhail (1994).
10098:
8514:. This operad is defined by
8086:. This associativity of the
7169:is composed with an element
7051:Operadic composition in the
6879:-linear operad algebra over
5820:consisting of all functions
4695:{\displaystyle \theta (a,b)}
4385:{\displaystyle \mathbb {S} }
4361:{\displaystyle \mathbb {S} }
3363:
2947:, keeping each block intact)
2314:) is a non-symmetric operad
287:
119:such that the algebras over
7:
9829:
8079:{\displaystyle (ab)c=a(bc)}
7530:little convex bodies operad
7373:{\displaystyle \theta _{i}}
6300:= the space of linear maps
5750:
5648:these yield the same result
4539:"Associativity" means that
3856:symmetric monoidal category
1373:operad without permutations
96:. For instance, there is a
10:
11187:
10998:; Vallette, Bruno (2012),
10668:Kontsevich, Maxim (1999).
10379:Séminaire Nicolas Bourbaki
9778:Operads in homotopy theory
9456:{\displaystyle \Gamma (E)}
9014:{\displaystyle {\vec {x}}}
8370:Similarly, there is a non-
8029:{\displaystyle \psi (a,b)}
7666:is given by replacing the
7380:and inserting it into the
7043:"Little something" operads
6025:-ary operation on the set
5846:{\displaystyle X^{n}\to X}
5740:{\displaystyle 1\circ 1=1}
5690:{\displaystyle (((ab)c)d)}
5411:{\displaystyle (ab)c\ \ d}
5046:{\displaystyle (((ab)c)d)}
2907:, and then permutes these
2806:blocks, the first of size
2474:{\displaystyle t\in S_{n}}
1538:for all positive integers
1252:{\displaystyle s\in S_{n}}
813:arguments, the second one
174:
11101:Miguel A. Mendéz (2015).
10754:10.1112/S0010437X05001521
10647:: AMS. pp. 365–373.
10305:Duke Mathematical Journal
9758:
9622:describes the relations.
8315:Terminal symmetric operad
8285:{\displaystyle \tau _{i}}
7493:(initially called little
6744:(symmetric group action)
6045:. An operad algebra over
5057:ambiguous: it could mean
4943:on the left "multiplies"
4629:mean that the operations
4167:we consider the category
3805:Operads therefore form a
1219:{\displaystyle f\in P(n)}
498:{\displaystyle f\in P(n)}
253:could be explained using
11116:Samuele Giraudo (2018).
9866:
9564:describes generators of
8323:-ary operation for each
7981:of the binary operation
5006:However, the expression
4530:Understanding the axioms
4051:{\displaystyle \otimes }
3488:preserves the identity:
3393:{\displaystyle f:P\to Q}
259:deformation quantization
251:rational homotopy theory
10768:Thomas, Justin (2016).
10704:10.1023/A:1007555725247
10641:The Swiss-cheese operad
9065:commutative-ring operad
8462:{\displaystyle \Sigma }
8383:{\displaystyle \Sigma }
8258:{\displaystyle \sigma }
7491:little intervals operad
7435:{\displaystyle i=1,2,3}
7397:{\displaystyle \theta }
6911:and an operad morphism
6069:thus consists of a set
5925:and an operad morphism
4936:{\displaystyle \theta }
4916:{\displaystyle \theta }
4861:{\displaystyle (a,b,c)}
4785:{\displaystyle ((ab)c)}
4744:{\displaystyle \theta }
4660:{\displaystyle \theta }
4514:{\displaystyle R^{(X)}}
3542:{\displaystyle \theta }
3400:consists of a sequence
3247:denotes the element of
1390:{\displaystyle \Sigma }
10967:"What Is...an Operad?"
10731:Compositio Mathematica
10159:; Vogt, R. M. (1973).
10083:
9954:
9923:
9841:Algebra over an operad
9749:
9711:
9668:
9616:
9582:
9554:
9515:
9491:
9463:is the free operad on
9457:
9420:
9332:
9256:acts. Then there is a
9250:
9223:
9175:of this operad is the
9164:
9137:
9045:
9015:
8959:
8891:
8797:
8701:
8616:
8589:
8561:
8504:
8463:
8443:
8419:is given by the Artin
8413:
8390:operad for which each
8384:
8348:
8286:
8259:
8239:
8181:
8154:
8124:
8096:axiom of associativity
8080:
8030:
7995:
7968:
7900:
7820:
7797:
7777:
7753:
7715:
7684:
7660:
7602:
7567:
7476:
7436:
7398:
7374:
7347:
7267:
7163:
7127:
7086:
7056:
7055:explained in the text.
7053:little 2-disks operad,
7026:
6980:
6952:
6897:
6869:
6841:
6811:
6765:
6735:
6708:
6659:
6517:
6451:
6385:
6330:
6294:
6241:
6127:
6103:
6083:
6063:
6039:
6019:
5999:
5966:
5919:
5899:
5875:
5847:
5813:
5774:
5741:
5691:
5620:
5574:
5412:
5361:
5323:
5285:
5199:
5125:
5047:
4980:
4960:
4937:
4917:
4897:
4896:{\displaystyle (ab,c)}
4862:
4824:
4786:
4745:
4725:
4696:
4661:
4619:
4557:
4556:{\displaystyle \circ }
4515:
4472:
4421:
4386:
4362:
4339:
4294:
4252:
4199:
4119:
4052:
4028:
3908:
3907:{\displaystyle \circ }
3884:
3836:
3795:
3733:
3589:
3543:
3517:
3516:{\displaystyle f(1)=1}
3475:
3394:
3368:A morphism of operads
3348:
3321:
3294:
3241:
3187:
3013:
2966:
2941:
2921:
2901:
2874:
2854:
2827:
2800:
2780:
2710:
2657:
2630:
2475:
2448:: given a permutation
2435:
2415:
2387:
2358:
2328:
2294:
1912:
1840:
1598:
1552:
1525:
1496:
1471:
1450:
1391:
1348:
1328:
1327:{\displaystyle \circ }
1305:
1253:
1220:
1182:
1033:
1004:
974:
948:
928:
908:
888:
861:
834:
807:
780:
766:, we divide them into
760:
740:
691:
589:
499:
461:
441:
421:
397:
334:
306:
157:
133:
113:
90:
68:
45:
11026:Zinbiel, Guillaume W.
10867:Tom Leinster (2004).
10591:Bull. Amer. Math. Soc
10530:Gennadi Sardanashvily
10084:
9955:
9903:
9836:PRO (category theory)
9750:
9712:
9669:
9625:A (symmetric) operad
9617:
9583:
9555:
9516:
9492:
9458:
9421:
9333:
9251:
9249:{\displaystyle S_{n}}
9224:
9165:
9163:{\displaystyle S_{n}}
9138:
9046:
9016:
8960:
8892:
8798:
8702:
8617:
8615:{\displaystyle S_{n}}
8590:
8562:
8505:
8464:
8449:. Moreover, this non-
8444:
8442:{\displaystyle B_{n}}
8414:
8385:
8349:
8347:{\displaystyle S_{n}}
8287:
8260:
8240:
8182:
8180:{\displaystyle S_{n}}
8155:
8153:{\displaystyle S_{n}}
8125:
8081:
8031:
7996:
7994:{\displaystyle \psi }
7969:
7901:
7899:{\displaystyle \psi }
7814:
7798:
7778:
7759:, thus attaching the
7754:
7716:
7714:{\displaystyle S_{i}}
7685:
7661:
7603:
7601:{\displaystyle S_{n}}
7568:
7487:little n-cubes operad
7477:
7447:little n-disks operad
7437:
7399:
7375:
7348:
7273:to yield the element
7268:
7164:
7128:
7087:
7061:little 2-disks operad
7050:
7027:
6981:
6953:
6898:
6870:
6842:
6812:
6766:
6764:{\displaystyle S_{n}}
6736:
6709:
6660:
6518:
6452:
6386:
6331:
6295:
6242:
6128:
6104:
6084:
6064:
6040:
6020:
6000:
5967:
5920:
5900:
5876:
5848:
5814:
5775:
5742:
5692:
5621:
5575:
5413:
5362:
5324:
5286:
5200:
5126:
5048:
4981:
4961:
4938:
4918:
4898:
4863:
4825:
4787:
4746:
4726:
4697:
4662:
4620:
4558:
4516:
4473:
4422:
4394:combinatorial species
4387:
4363:
4340:
4295:
4253:
4200:
4156:Algebraist definition
4120:
4053:
4029:
3909:
3885:
3861:. In that case, each
3837:
3796:
3734:
3590:
3544:
3518:
3476:
3395:
3349:
3347:{\displaystyle s_{2}}
3322:
3320:{\displaystyle s_{1}}
3295:
3242:
3188:
3014:
2967:
2942:
2922:
2902:
2900:{\displaystyle k_{n}}
2875:
2855:
2853:{\displaystyle k_{2}}
2833:, the second of size
2828:
2826:{\displaystyle k_{1}}
2801:
2781:
2716:that acts on the set
2711:
2658:
2631:
2476:
2436:
2416:
2388:
2359:
2357:{\displaystyle S_{n}}
2329:
2295:
1913:
1841:
1599:
1553:
1526:
1497:
1472:
1451:
1392:
1371:(sometimes called an
1349:
1329:
1306:
1254:
1221:
1183:
1034:
1005:
1003:{\displaystyle S_{n}}
975:
949:
934:values obtained from
929:
909:
889:
887:{\displaystyle f_{2}}
862:
860:{\displaystyle f_{1}}
835:
833:{\displaystyle k_{2}}
808:
806:{\displaystyle k_{1}}
781:
761:
741:
692:
590:
500:
462:
442:
422:
398:
335:
307:
245:discovered that some
226:of "operations" and "
179:Operads originate in
169:group representations
158:
134:
114:
91:
69:
46:
10796:10.2140/gt.2016.20.1
9966:
9885:
9721:
9682:
9629:
9592:
9568:
9525:
9501:
9477:
9438:
9345:
9263:
9233:
9191:
9147:
9075:
9026:
8996:
8986:conical combinations
8901:
8807:
8711:
8626:
8599:
8571:
8518:
8485:
8453:
8426:
8412:{\displaystyle P(n)}
8394:
8374:
8331:
8269:
8249:
8191:
8164:
8137:
8123:{\displaystyle P(n)}
8105:
8040:
8005:
7985:
7913:
7890:
7868:Alexander A. Voronov
7856:Deligne's conjecture
7848:Alexander A. Voronov
7787:
7767:
7725:
7698:
7674:
7612:
7585:
7566:{\displaystyle P(n)}
7548:
7457:
7408:
7388:
7357:
7277:
7173:
7138:
7108:
7085:{\displaystyle P(n)}
7067:
6998:
6966:
6915:
6883:
6855:
6821:
6775:
6748:
6718:
6714:is the identity map
6672:
6527:
6461:
6395:
6343:
6339:(composition) given
6304:
6258:
6176:
6113:
6093:
6073:
6049:
6029:
6009:
5976:
5929:
5909:
5885:
5861:
5824:
5786:
5782:endomorphism operad
5764:
5719:
5654:
5594:
5435:
5381:
5333:
5295:
5209:
5135:
5061:
5010:
4970:
4947:
4927:
4907:
4872:
4834:
4796:
4758:
4735:
4724:{\displaystyle (ab)}
4706:
4671:
4651:
4567:
4547:
4492:
4435:
4407:
4374:
4350:
4311:
4266:
4224:
4217:can be defined as a
4171:
4081:
4042:
3918:
3898:
3883:{\displaystyle P(n)}
3865:
3813:
3749:
3605:
3553:
3533:
3492:
3407:
3372:
3331:
3304:
3251:
3199:
3025:
2976:
2956:
2931:
2911:
2884:
2864:
2837:
2810:
2790:
2786:by breaking it into
2720:
2667:
2642:
2491:
2452:
2425:
2397:
2386:{\displaystyle P(n)}
2368:
2341:
2318:
1933:
1860:
1616:
1562:
1542:
1524:{\displaystyle P(1)}
1506:
1486:
1461:
1410:
1381:
1369:non-symmetric operad
1363:Non-symmetric operad
1338:
1318:
1263:
1230:
1195:
1046:
1032:{\displaystyle P(n)}
1014:
987:
964:
938:
918:
898:
871:
867:to the first block,
844:
817:
790:
770:
750:
704:
602:
509:
474:
451:
431:
411:
347:
316:
296:
147:
123:
103:
80:
58:
35:
11048:2011arXiv1101.0267Z
10887:2004hohc.book.....L
10819:Handbook of Algebra
10696:1999math......4055K
10585:May, J. P. (1977).
10475:Handbook of Algebra
9846:Higher-order operad
9821:, Stasheff writes:
9471:free representation
8990:convex combinations
8982:affine combinations
8512:linear combinations
7825:Swiss-cheese operad
7817:Swiss-cheese operad
7807:Swiss-cheese operad
7690:by the root of the
6171:endomorphism operad
4535:Associativity axiom
3846:In other categories
189:J. Michael Boardman
10965:(June–July 2004).
10223:10.1007/bfb0067491
10169:10.1007/bfb0068547
10079:
9950:
9797:. You can help by
9745:
9707:
9664:
9612:
9578:
9550:
9511:
9497:, we mean writing
9487:
9453:
9416:
9371:
9328:
9302:
9246:
9219:
9160:
9133:
9041:
9011:
8955:
8887:
8793:
8697:
8612:
8585:
8557:
8500:
8459:
8439:
8409:
8380:
8344:
8282:
8255:
8235:
8177:
8150:
8120:
8076:
8026:
7991:
7964:
7896:
7878:Associative operad
7870:and then fully by
7821:
7793:
7773:
7749:
7711:
7680:
7656:
7598:
7563:
7472:
7432:
7394:
7370:
7343:
7263:
7159:
7123:
7082:
7057:
7022:
6976:
6948:
6893:
6865:
6837:
6807:
6761:
6731:
6704:
6655:
6513:
6447:
6381:
6326:
6290:
6237:
6123:
6099:
6079:
6059:
6035:
6015:
6005:into a "concrete"
5995:
5962:
5915:
5905:is given by a set
5895:
5871:
5843:
5809:
5770:
5737:
5687:
5616:
5570:
5408:
5357:
5319:
5281:
5195:
5121:
5043:
4976:
4959:{\displaystyle ab}
4956:
4933:
4913:
4893:
4858:
4820:
4782:
4741:
4721:
4692:
4657:
4635:associative operad
4615:
4553:
4511:
4468:
4417:
4382:
4358:
4335:
4290:
4248:
4195:
4127:topological operad
4115:
4077:(instead of sets)
4067:topological spaces
4048:
4024:
3904:
3894:, the composition
3880:
3832:
3791:
3729:
3585:
3539:
3513:
3471:
3390:
3344:
3317:
3290:
3237:
3183:
3009:
2962:
2937:
2917:
2897:
2870:
2850:
2823:
2796:
2776:
2706:
2656:{\displaystyle t'}
2653:
2626:
2471:
2431:
2411:
2383:
2354:
2324:
2290:
2288:
1908:
1836:
1834:
1594:
1548:
1521:
1492:
1467:
1446:
1387:
1344:
1324:
1301:
1249:
1216:
1178:
1029:
1000:
970:
944:
924:
904:
884:
857:
830:
803:
776:
756:
736:
687:
585:
495:
457:
437:
417:
393:
330:
302:
267:Deligne conjecture
181:algebraic topology
153:
139:are precisely the
129:
109:
86:
64:
41:
11126:978-3-030-02073-6
11111:978-3-319-11712-6
11079:978-1-4704-3480-9
11018:978-3-642-30361-6
11001:Algebraic Operads
10996:Loday, Jean-Louis
10925:978-0-8218-4362-8
10896:978-0-521-53215-0
10654:978-0-8218-7829-3
10571:978-1-4020-1834-3
10367:Loday, Jean-Louis
10275:math.uchicago.edu
10232:978-3-540-05904-2
10178:978-3-540-06479-4
9815:
9814:
9354:
9285:
9258:forgetful functor
9008:
8913:
8819:
8790:
8746:
8691:
8663:
8638:
7850:. It was used by
7796:{\displaystyle T}
7776:{\displaystyle T}
7683:{\displaystyle T}
7540:In graph theory,
7501:) was defined by
6651:
6647:
6638:
6622:
6618:
6581:
6102:{\displaystyle X}
6082:{\displaystyle X}
6038:{\displaystyle X}
6018:{\displaystyle n}
5918:{\displaystyle X}
5773:{\displaystyle X}
5612:
5609:
5404:
5401:
4979:{\displaystyle c}
4647:For instance, if
4478:that sends a set
4465:
4455:
4414:
4320:
4275:
4180:
4071:cartesian product
2965:{\displaystyle n}
2940:{\displaystyle t}
2920:{\displaystyle n}
2880:th block of size
2873:{\displaystyle n}
2799:{\displaystyle n}
2434:{\displaystyle *}
2327:{\displaystyle P}
1551:{\displaystyle n}
1495:{\displaystyle 1}
1470:{\displaystyle n}
1347:{\displaystyle *}
973:{\displaystyle *}
947:{\displaystyle X}
927:{\displaystyle n}
907:{\displaystyle f}
779:{\displaystyle n}
759:{\displaystyle X}
460:{\displaystyle X}
440:{\displaystyle X}
420:{\displaystyle n}
312:is a set and for
305:{\displaystyle X}
282:Thomas Willwacher
263:Poisson manifolds
156:{\displaystyle L}
132:{\displaystyle L}
112:{\displaystyle L}
89:{\displaystyle O}
67:{\displaystyle O}
51:, one defines an
44:{\displaystyle O}
11178:
11166:Abstract algebra
11098:
11060:
11041:
11021:
11006:
10991:
10989:
10987:
10971:
10958:
10956:
10943:
10937:
10929:
10900:
10880:
10854:
10852:
10814:
10808:
10807:
10789:
10765:
10759:
10758:
10756:
10746:
10722:
10716:
10715:
10689:
10674:Lett. Math. Phys
10665:
10659:
10658:
10636:
10630:
10629:
10627:
10615:
10609:
10608:
10606:
10582:
10576:
10575:
10553:
10547:
10526:
10520:
10518:
10490:
10470:
10464:
10457:
10451:
10450:
10448:
10436:
10430:
10429:
10427:
10415:
10406:
10405:
10403:
10401:
10363:
10357:
10356:
10297:Ginzburg, Victor
10293:
10287:
10286:
10284:
10282:
10272:
10260:
10254:
10251:
10245:
10244:
10216:
10197:
10191:
10190:
10153:
10147:
10146:
10136:
10127:(6): 1117–1123.
10109:
10092:
10088:
10086:
10085:
10080:
10078:
10077:
10076:
10075:
10057:
10056:
10039:
10038:
10037:
10036:
10013:
10012:
10011:
10010:
9993:
9992:
9959:
9957:
9956:
9951:
9949:
9948:
9933:
9932:
9922:
9917:
9877:
9810:
9807:
9789:
9782:
9754:
9752:
9751:
9746:
9716:
9714:
9713:
9708:
9697:
9696:
9673:
9671:
9670:
9665:
9651:
9650:
9638:
9637:
9621:
9619:
9618:
9613:
9611:
9610:
9601:
9600:
9587:
9585:
9584:
9579:
9577:
9576:
9559:
9557:
9556:
9551:
9534:
9533:
9520:
9518:
9517:
9512:
9510:
9509:
9496:
9494:
9493:
9488:
9486:
9485:
9462:
9460:
9459:
9454:
9425:
9423:
9422:
9417:
9415:
9414:
9396:
9395:
9394:
9393:
9383:
9370:
9369:
9337:
9335:
9334:
9329:
9327:
9326:
9325:
9324:
9314:
9301:
9300:
9281:
9280:
9255:
9253:
9252:
9247:
9245:
9244:
9228:
9226:
9225:
9220:
9218:
9217:
9216:
9215:
9205:
9169:
9167:
9166:
9161:
9159:
9158:
9142:
9140:
9139:
9134:
9129:
9128:
9110:
9109:
9097:
9050:
9048:
9047:
9042:
9040:
9039:
9034:
9020:
9018:
9017:
9012:
9010:
9009:
9001:
8964:
8962:
8961:
8956:
8915:
8914:
8906:
8896:
8894:
8893:
8888:
8886:
8885:
8880:
8868:
8867:
8843:
8842:
8821:
8820:
8812:
8802:
8800:
8799:
8794:
8792:
8791:
8786:
8785:
8776:
8773:
8772:
8748:
8747:
8742:
8741:
8732:
8729:
8728:
8706:
8704:
8703:
8698:
8693:
8692:
8687:
8686:
8677:
8665:
8664:
8659:
8658:
8649:
8640:
8639:
8631:
8621:
8619:
8618:
8613:
8611:
8610:
8594:
8592:
8591:
8586:
8584:
8566:
8564:
8563:
8558:
8556:
8555:
8550:
8532:
8531:
8526:
8509:
8507:
8506:
8501:
8499:
8498:
8493:
8468:
8466:
8465:
8460:
8448:
8446:
8445:
8440:
8438:
8437:
8418:
8416:
8415:
8410:
8389:
8387:
8386:
8381:
8353:
8351:
8350:
8345:
8343:
8342:
8291:
8289:
8288:
8283:
8281:
8280:
8264:
8262:
8261:
8256:
8244:
8242:
8241:
8236:
8231:
8230:
8212:
8211:
8186:
8184:
8183:
8178:
8176:
8175:
8159:
8157:
8156:
8151:
8149:
8148:
8130:is given by the
8129:
8127:
8126:
8121:
8085:
8083:
8082:
8077:
8035:
8033:
8032:
8027:
8000:
7998:
7997:
7992:
7973:
7971:
7970:
7965:
7905:
7903:
7902:
7897:
7852:Maxim Kontsevich
7802:
7800:
7799:
7794:
7782:
7780:
7779:
7774:
7758:
7756:
7755:
7750:
7720:
7718:
7717:
7712:
7710:
7709:
7689:
7687:
7686:
7681:
7665:
7663:
7662:
7657:
7652:
7651:
7633:
7632:
7607:
7605:
7604:
7599:
7597:
7596:
7572:
7570:
7569:
7564:
7503:Michael Boardman
7481:
7479:
7478:
7473:
7471:
7470:
7465:
7441:
7439:
7438:
7433:
7403:
7401:
7400:
7395:
7379:
7377:
7376:
7371:
7369:
7368:
7352:
7350:
7349:
7344:
7324:
7323:
7311:
7310:
7298:
7297:
7272:
7270:
7269:
7264:
7214:
7213:
7201:
7200:
7188:
7187:
7168:
7166:
7165:
7160:
7132:
7130:
7129:
7124:
7122:
7121:
7116:
7091:
7089:
7088:
7083:
7031:
7029:
7028:
7023:
6985:
6983:
6982:
6977:
6975:
6974:
6957:
6955:
6954:
6949:
6947:
6946:
6941:
6940:
6924:
6923:
6902:
6900:
6899:
6894:
6892:
6891:
6875:is an operad, a
6874:
6872:
6871:
6866:
6864:
6863:
6846:
6844:
6843:
6838:
6836:
6835:
6816:
6814:
6813:
6808:
6797:
6796:
6791:
6790:
6770:
6768:
6767:
6762:
6760:
6759:
6740:
6738:
6737:
6732:
6730:
6729:
6713:
6711:
6710:
6705:
6694:
6693:
6688:
6687:
6664:
6662:
6661:
6656:
6649:
6648:
6640:
6636:
6635:
6634:
6620:
6619:
6617:
6616:
6615:
6597:
6596:
6583:
6579:
6578:
6577:
6576:
6575:
6549:
6548:
6547:
6546:
6522:
6520:
6519:
6514:
6509:
6508:
6496:
6495:
6490:
6489:
6473:
6472:
6456:
6454:
6453:
6448:
6443:
6442:
6430:
6429:
6424:
6423:
6407:
6406:
6390:
6388:
6387:
6382:
6371:
6370:
6365:
6364:
6335:
6333:
6332:
6327:
6319:
6318:
6299:
6297:
6296:
6291:
6280:
6279:
6274:
6273:
6246:
6244:
6243:
6238:
6224:
6223:
6218:
6217:
6198:
6197:
6192:
6191:
6132:
6130:
6129:
6124:
6122:
6121:
6108:
6106:
6105:
6100:
6088:
6086:
6085:
6080:
6068:
6066:
6065:
6060:
6058:
6057:
6044:
6042:
6041:
6036:
6024:
6022:
6021:
6016:
6004:
6002:
6001:
5996:
5985:
5984:
5971:
5969:
5968:
5963:
5961:
5960:
5955:
5954:
5938:
5937:
5924:
5922:
5921:
5916:
5904:
5902:
5901:
5896:
5894:
5893:
5880:
5878:
5877:
5872:
5870:
5869:
5852:
5850:
5849:
5844:
5836:
5835:
5818:
5816:
5815:
5810:
5808:
5807:
5802:
5801:
5780:, we obtain the
5779:
5777:
5776:
5771:
5746:
5744:
5743:
5738:
5712:
5697:is unambiguous.
5696:
5694:
5693:
5688:
5643:
5633:
5625:
5623:
5622:
5617:
5610:
5607:
5587:
5579:
5577:
5576:
5571:
5563:
5562:
5547:
5546:
5534:
5533:
5506:
5505:
5493:
5492:
5465:
5464:
5425:
5417:
5415:
5414:
5409:
5402:
5399:
5374:
5366:
5364:
5363:
5358:
5328:
5326:
5325:
5320:
5290:
5288:
5287:
5282:
5204:
5202:
5201:
5196:
5130:
5128:
5127:
5122:
5052:
5050:
5049:
5044:
5003:
4993:
4985:
4983:
4982:
4977:
4965:
4963:
4962:
4957:
4942:
4940:
4939:
4934:
4922:
4920:
4919:
4914:
4902:
4900:
4899:
4894:
4867:
4865:
4864:
4859:
4829:
4827:
4826:
4821:
4791:
4789:
4788:
4783:
4750:
4748:
4747:
4742:
4730:
4728:
4727:
4722:
4701:
4699:
4698:
4693:
4666:
4664:
4663:
4658:
4624:
4622:
4621:
4616:
4562:
4560:
4559:
4554:
4520:
4518:
4517:
4512:
4510:
4509:
4477:
4475:
4474:
4469:
4467:
4466:
4457:
4456:
4447:
4446:
4431:defines a monad
4426:
4424:
4423:
4418:
4416:
4415:
4401:generalized ring
4391:
4389:
4388:
4383:
4381:
4367:
4365:
4364:
4359:
4357:
4344:
4342:
4341:
4336:
4334:
4333:
4321:
4318:
4299:
4297:
4296:
4291:
4289:
4288:
4276:
4273:
4257:
4255:
4254:
4249:
4205:of modules over
4204:
4202:
4201:
4196:
4194:
4193:
4181:
4178:
4162:commutative ring
4138:commutative ring
4124:
4122:
4121:
4116:
4114:
4113:
4057:
4055:
4054:
4049:
4033:
4031:
4030:
4025:
4020:
4019:
4001:
4000:
3979:
3978:
3951:
3950:
3913:
3911:
3910:
3905:
3890:is an object of
3889:
3887:
3886:
3881:
3841:
3839:
3838:
3833:
3831:
3830:
3800:
3798:
3797:
3792:
3738:
3736:
3735:
3730:
3722:
3721:
3694:
3693:
3651:
3650:
3632:
3631:
3594:
3592:
3591:
3586:
3584:
3583:
3565:
3564:
3548:
3546:
3545:
3540:
3522:
3520:
3519:
3514:
3480:
3478:
3477:
3472:
3470:
3469:
3468:
3422:
3421:
3399:
3397:
3396:
3391:
3353:
3351:
3350:
3345:
3343:
3342:
3327:, the second by
3326:
3324:
3323:
3318:
3316:
3315:
3299:
3297:
3296:
3291:
3289:
3288:
3287:
3286:
3268:
3267:
3246:
3244:
3243:
3238:
3233:
3232:
3214:
3213:
3192:
3190:
3189:
3184:
3179:
3178:
3160:
3159:
3138:
3137:
3119:
3118:
3091:
3090:
3078:
3077:
3059:
3058:
3046:
3045:
3018:
3016:
3015:
3010:
3008:
3007:
3006:
3005:
2988:
2987:
2971:
2969:
2968:
2963:
2946:
2944:
2943:
2938:
2926:
2924:
2923:
2918:
2906:
2904:
2903:
2898:
2896:
2895:
2879:
2877:
2876:
2871:
2859:
2857:
2856:
2851:
2849:
2848:
2832:
2830:
2829:
2824:
2822:
2821:
2805:
2803:
2802:
2797:
2785:
2783:
2782:
2777:
2772:
2771:
2753:
2752:
2715:
2713:
2712:
2707:
2705:
2704:
2703:
2702:
2684:
2683:
2662:
2660:
2659:
2654:
2652:
2635:
2633:
2632:
2627:
2625:
2608:
2607:
2580:
2579:
2543:
2542:
2524:
2523:
2480:
2478:
2477:
2472:
2470:
2469:
2440:
2438:
2437:
2432:
2420:
2418:
2417:
2412:
2410:
2392:
2390:
2389:
2384:
2363:
2361:
2360:
2355:
2353:
2352:
2333:
2331:
2330:
2325:
2306:Symmetric operad
2299:
2297:
2296:
2291:
2289:
2282:
2281:
2280:
2279:
2250:
2249:
2225:
2224:
2223:
2222:
2193:
2192:
2171:
2170:
2161:
2160:
2142:
2141:
2123:
2122:
2112:
2103:
2102:
2093:
2092:
2091:
2090:
2061:
2060:
2039:
2038:
2017:
2016:
2015:
2014:
1985:
1984:
1963:
1962:
1953:
1952:
1939:
1917:
1915:
1914:
1909:
1845:
1843:
1842:
1837:
1835:
1825:
1824:
1806:
1805:
1777:
1776:
1758:
1757:
1732:
1731:
1713:
1712:
1687:
1686:
1659:
1658:
1603:
1601:
1600:
1595:
1593:
1592:
1574:
1573:
1557:
1555:
1554:
1549:
1530:
1528:
1527:
1522:
1501:
1499:
1498:
1493:
1476:
1474:
1473:
1468:
1455:
1453:
1452:
1447:
1445:
1444:
1443:
1396:
1394:
1393:
1388:
1353:
1351:
1350:
1345:
1333:
1331:
1330:
1325:
1310:
1308:
1307:
1302:
1294:
1293:
1275:
1274:
1258:
1256:
1255:
1250:
1248:
1247:
1225:
1223:
1222:
1217:
1187:
1185:
1184:
1179:
1174:
1173:
1163:
1162:
1136:
1135:
1125:
1124:
1095:
1094:
1076:
1075:
1038:
1036:
1035:
1030:
1009:
1007:
1006:
1001:
999:
998:
979:
977:
976:
971:
953:
951:
950:
945:
933:
931:
930:
925:
913:
911:
910:
905:
893:
891:
890:
885:
883:
882:
866:
864:
863:
858:
856:
855:
839:
837:
836:
831:
829:
828:
812:
810:
809:
804:
802:
801:
785:
783:
782:
777:
765:
763:
762:
757:
745:
743:
742:
737:
735:
734:
716:
715:
696:
694:
693:
688:
683:
682:
664:
663:
642:
641:
623:
622:
594:
592:
591:
586:
581:
580:
562:
561:
540:
539:
521:
520:
504:
502:
501:
496:
466:
464:
463:
458:
446:
444:
443:
438:
426:
424:
423:
418:
402:
400:
399:
394:
383:
382:
339:
337:
336:
331:
329:
311:
309:
308:
303:
278:Maxim Kontsevich
243:Mikhail Kapranov
235:Maxim Kontsevich
216:A.N. Whitehead's
162:
160:
159:
154:
138:
136:
135:
130:
118:
116:
115:
110:
95:
93:
92:
87:
73:
71:
70:
65:
50:
48:
47:
42:
11186:
11185:
11181:
11180:
11179:
11177:
11176:
11175:
11171:Category theory
11156:
11155:
11135:
11080:
11058:
11019:
11009:Springer-Verlag
11004:
10985:
10983:
10969:
10931:
10930:
10926:
10897:
10863:
10858:
10857:
10841:
10815:
10811:
10766:
10762:
10723:
10719:
10666:
10662:
10655:
10637:
10633:
10616:
10612:
10583:
10579:
10572:
10554:
10550:
10527:
10523:
10507:
10471:
10467:
10461:arXiv:0704.2030
10458:
10454:
10437:
10433:
10416:
10409:
10399:
10397:
10364:
10360:
10294:
10290:
10280:
10278:
10270:
10261:
10257:
10252:
10248:
10233:
10214:10.1.1.146.3172
10198:
10194:
10179:
10157:Boardman, J. M.
10154:
10150:
10113:Boardman, J. M.
10110:
10106:
10101:
10096:
10095:
10071:
10067:
10052:
10048:
10047:
10043:
10032:
10028:
10027:
10023:
10006:
10002:
10001:
9997:
9988:
9984:
9967:
9964:
9963:
9941:
9937:
9928:
9924:
9918:
9907:
9886:
9883:
9882:
9878:
9874:
9869:
9832:
9819:Stasheff (2004)
9811:
9805:
9802:
9795:needs expansion
9780:
9761:
9722:
9719:
9718:
9692:
9691:
9683:
9680:
9679:
9646:
9645:
9633:
9632:
9630:
9627:
9626:
9606:
9605:
9596:
9595:
9593:
9590:
9589:
9572:
9571:
9569:
9566:
9565:
9529:
9528:
9526:
9523:
9522:
9505:
9504:
9502:
9499:
9498:
9481:
9480:
9478:
9475:
9474:
9439:
9436:
9435:
9401:
9400:
9389:
9385:
9384:
9373:
9372:
9365:
9358:
9346:
9343:
9342:
9320:
9316:
9315:
9304:
9303:
9296:
9289:
9267:
9266:
9264:
9261:
9260:
9240:
9236:
9234:
9231:
9230:
9211:
9207:
9206:
9195:
9194:
9192:
9189:
9188:
9185:
9154:
9150:
9148:
9145:
9144:
9124:
9120:
9105:
9101:
9093:
9076:
9073:
9072:
9061:
9053:convex polytope
9035:
9030:
9029:
9027:
9024:
9023:
9000:
8999:
8997:
8994:
8993:
8905:
8904:
8902:
8899:
8898:
8881:
8876:
8875:
8857:
8853:
8832:
8828:
8811:
8810:
8808:
8805:
8804:
8781:
8777:
8775:
8774:
8762:
8758:
8737:
8733:
8731:
8730:
8718:
8714:
8712:
8709:
8708:
8682:
8678:
8676:
8675:
8654:
8650:
8648:
8647:
8630:
8629:
8627:
8624:
8623:
8606:
8602:
8600:
8597:
8596:
8580:
8572:
8569:
8568:
8551:
8546:
8545:
8527:
8522:
8521:
8519:
8516:
8515:
8494:
8489:
8488:
8486:
8483:
8482:
8475:
8454:
8451:
8450:
8433:
8429:
8427:
8424:
8423:
8395:
8392:
8391:
8375:
8372:
8371:
8368:
8338:
8334:
8332:
8329:
8328:
8317:
8276:
8272:
8270:
8267:
8266:
8250:
8247:
8246:
8226:
8222:
8207:
8203:
8192:
8189:
8188:
8171:
8167:
8165:
8162:
8161:
8144:
8140:
8138:
8135:
8134:
8132:symmetric group
8106:
8103:
8102:
8041:
8038:
8037:
8006:
8003:
8002:
7986:
7983:
7982:
7914:
7911:
7910:
7891:
7888:
7887:
7880:
7809:
7788:
7785:
7784:
7768:
7765:
7764:
7726:
7723:
7722:
7705:
7701:
7699:
7696:
7695:
7675:
7672:
7671:
7647:
7643:
7628:
7624:
7613:
7610:
7609:
7592:
7588:
7586:
7583:
7582:
7549:
7546:
7545:
7538:
7520:(n-dimensional
7485:Originally the
7466:
7461:
7460:
7458:
7455:
7454:
7409:
7406:
7405:
7389:
7386:
7385:
7364:
7360:
7358:
7355:
7354:
7319:
7315:
7306:
7302:
7293:
7289:
7278:
7275:
7274:
7209:
7205:
7196:
7192:
7183:
7179:
7174:
7171:
7170:
7139:
7136:
7135:
7117:
7112:
7111:
7109:
7106:
7105:
7068:
7065:
7064:
7045:
6999:
6996:
6995:
6970:
6969:
6967:
6964:
6963:
6942:
6930:
6929:
6928:
6919:
6918:
6916:
6913:
6912:
6887:
6886:
6884:
6881:
6880:
6859:
6858:
6856:
6853:
6852:
6828:
6824:
6822:
6819:
6818:
6792:
6780:
6779:
6778:
6776:
6773:
6772:
6755:
6751:
6749:
6746:
6745:
6725:
6721:
6719:
6716:
6715:
6689:
6677:
6676:
6675:
6673:
6670:
6669:
6639:
6627:
6623:
6611:
6607:
6592:
6588:
6587:
6582:
6571:
6567:
6563:
6559:
6542:
6538:
6534:
6530:
6528:
6525:
6524:
6504:
6500:
6491:
6479:
6478:
6477:
6468:
6464:
6462:
6459:
6458:
6438:
6434:
6425:
6413:
6412:
6411:
6402:
6398:
6396:
6393:
6392:
6366:
6354:
6353:
6352:
6344:
6341:
6340:
6311:
6307:
6305:
6302:
6301:
6275:
6263:
6262:
6261:
6259:
6256:
6255:
6219:
6207:
6206:
6205:
6193:
6181:
6180:
6179:
6177:
6174:
6173:
6139:
6117:
6116:
6114:
6111:
6110:
6094:
6091:
6090:
6074:
6071:
6070:
6053:
6052:
6050:
6047:
6046:
6030:
6027:
6026:
6010:
6007:
6006:
5980:
5979:
5977:
5974:
5973:
5956:
5944:
5943:
5942:
5933:
5932:
5930:
5927:
5926:
5910:
5907:
5906:
5889:
5888:
5886:
5883:
5882:
5865:
5864:
5862:
5859:
5858:
5855:operad algebras
5831:
5827:
5825:
5822:
5821:
5803:
5791:
5790:
5789:
5787:
5784:
5783:
5765:
5762:
5761:
5758:
5753:
5720:
5717:
5716:
5703:
5655:
5652:
5651:
5595:
5592:
5591:
5558:
5554:
5542:
5538:
5523:
5519:
5501:
5497:
5479:
5475:
5442:
5438:
5436:
5433:
5432:
5382:
5379:
5378:
5334:
5331:
5330:
5296:
5293:
5292:
5210:
5207:
5206:
5136:
5133:
5132:
5062:
5059:
5058:
5011:
5008:
5007:
4971:
4968:
4967:
4948:
4945:
4944:
4928:
4925:
4924:
4908:
4905:
4904:
4873:
4870:
4869:
4835:
4832:
4831:
4797:
4794:
4793:
4759:
4756:
4755:
4736:
4733:
4732:
4707:
4704:
4703:
4672:
4669:
4668:
4652:
4649:
4648:
4568:
4565:
4564:
4548:
4545:
4544:
4537:
4532:
4499:
4495:
4493:
4490:
4489:
4462:
4461:
4452:
4451:
4442:
4438:
4436:
4433:
4432:
4411:
4410:
4408:
4405:
4404:
4377:
4375:
4372:
4371:
4353:
4351:
4348:
4347:
4323:
4322:
4317:
4312:
4309:
4308:
4278:
4277:
4272:
4267:
4264:
4263:
4225:
4222:
4221:
4183:
4182:
4177:
4172:
4169:
4168:
4158:
4142:chain complexes
4103:
4099:
4082:
4079:
4078:
4043:
4040:
4039:
4015:
4011:
3996:
3992:
3974:
3970:
3946:
3942:
3919:
3916:
3915:
3899:
3896:
3895:
3866:
3863:
3862:
3848:
3817:
3816:
3814:
3811:
3810:
3750:
3747:
3746:
3717:
3713:
3689:
3685:
3646:
3642:
3627:
3623:
3606:
3603:
3602:
3579:
3575:
3560:
3556:
3554:
3551:
3550:
3549:and operations
3534:
3531:
3530:
3529:-ary operation
3493:
3490:
3489:
3464:
3457:
3453:
3417:
3413:
3408:
3405:
3404:
3373:
3370:
3369:
3366:
3338:
3334:
3332:
3329:
3328:
3311:
3307:
3305:
3302:
3301:
3282:
3278:
3263:
3259:
3258:
3254:
3252:
3249:
3248:
3228:
3224:
3209:
3205:
3200:
3197:
3196:
3174:
3170:
3155:
3151:
3133:
3129:
3114:
3110:
3086:
3082:
3073:
3069:
3054:
3050:
3041:
3037:
3026:
3023:
3022:
3001:
2997:
2996:
2992:
2983:
2979:
2977:
2974:
2973:
2957:
2954:
2953:
2932:
2929:
2928:
2912:
2909:
2908:
2891:
2887:
2885:
2882:
2881:
2865:
2862:
2861:
2844:
2840:
2838:
2835:
2834:
2817:
2813:
2811:
2808:
2807:
2791:
2788:
2787:
2767:
2763:
2748:
2744:
2721:
2718:
2717:
2698:
2694:
2679:
2675:
2674:
2670:
2668:
2665:
2664:
2645:
2643:
2640:
2639:
2618:
2594:
2590:
2566:
2562:
2538:
2534:
2519:
2515:
2492:
2489:
2488:
2465:
2461:
2453:
2450:
2449:
2441:and satisfying
2426:
2423:
2422:
2406:
2398:
2395:
2394:
2369:
2366:
2365:
2348:
2344:
2342:
2339:
2338:
2336:symmetric group
2319:
2316:
2315:
2308:
2287:
2286:
2275:
2271:
2264:
2260:
2239:
2235:
2218:
2214:
2207:
2203:
2182:
2178:
2166:
2165:
2156:
2152:
2137:
2133:
2118:
2117:
2113:
2111:
2105:
2104:
2098:
2097:
2086:
2082:
2075:
2071:
2050:
2046:
2034:
2030:
2010:
2006:
1999:
1995:
1974:
1970:
1958:
1954:
1948:
1947:
1936:
1934:
1931:
1930:
1861:
1858:
1857:
1833:
1832:
1820:
1816:
1801:
1797:
1781:
1772:
1768:
1753:
1749:
1737:
1736:
1727:
1723:
1708:
1704:
1691:
1682:
1678:
1654:
1650:
1619:
1617:
1614:
1613:
1588:
1584:
1569:
1565:
1563:
1560:
1559:
1543:
1540:
1539:
1507:
1504:
1503:
1487:
1484:
1483:
1477:-ary operations
1462:
1459:
1458:
1439:
1432:
1428:
1411:
1408:
1407:
1382:
1379:
1378:
1365:
1360:
1339:
1336:
1335:
1319:
1316:
1315:
1289:
1285:
1270:
1266:
1264:
1261:
1260:
1243:
1239:
1231:
1228:
1227:
1196:
1193:
1192:
1155:
1151:
1150:
1146:
1117:
1113:
1112:
1108:
1090:
1086:
1071:
1067:
1047:
1044:
1043:
1015:
1012:
1011:
994:
990:
988:
985:
984:
982:symmetric group
965:
962:
961:
954:in such a way.
939:
936:
935:
919:
916:
915:
914:to the list of
899:
896:
895:
878:
874:
872:
869:
868:
851:
847:
845:
842:
841:
824:
820:
818:
815:
814:
797:
793:
791:
788:
787:
771:
768:
767:
751:
748:
747:
746:arguments from
730:
726:
711:
707:
705:
702:
701:
678:
674:
659:
655:
637:
633:
618:
614:
603:
600:
599:
595:, the function
576:
572:
557:
553:
535:
531:
516:
512:
510:
507:
506:
475:
472:
471:
452:
449:
448:
432:
429:
428:
412:
409:
408:
378:
374:
348:
345:
344:
325:
317:
314:
313:
297:
294:
293:
290:
276:in the work of
239:Victor Ginzburg
195:in 1968 and by
177:
148:
145:
144:
124:
121:
120:
104:
101:
100:
81:
78:
77:
59:
56:
55:
36:
33:
32:
17:
12:
11:
5:
11184:
11174:
11173:
11168:
11154:
11153:
11148:
11134:
11133:External links
11131:
11130:
11129:
11114:
11099:
11078:
11061:
11056:
11022:
11017:
10992:
10959:
10944:
10924:
10904:Martin Markl,
10901:
10895:
10862:
10859:
10856:
10855:
10839:
10809:
10760:
10737:(1): 143–168.
10717:
10660:
10653:
10631:
10610:
10597:(4): 456–494.
10577:
10570:
10562:United Kingdom
10548:
10521:
10505:
10465:
10452:
10446:hep-th/9403055
10431:
10407:
10375:www.numdam.org
10358:
10353:Project Euclid
10311:(1): 203–272.
10288:
10255:
10246:
10231:
10192:
10177:
10148:
10103:
10102:
10100:
10097:
10094:
10093:
10091:
10090:
10074:
10070:
10066:
10063:
10060:
10055:
10051:
10046:
10042:
10035:
10031:
10026:
10022:
10019:
10016:
10009:
10005:
10000:
9996:
9991:
9987:
9983:
9980:
9977:
9974:
9971:
9961:
9947:
9944:
9940:
9936:
9931:
9927:
9921:
9916:
9913:
9910:
9906:
9902:
9899:
9896:
9893:
9890:
9871:
9870:
9868:
9865:
9864:
9863:
9858:
9853:
9848:
9843:
9838:
9831:
9828:
9827:
9826:
9813:
9812:
9792:
9790:
9779:
9776:
9760:
9757:
9744:
9741:
9738:
9735:
9732:
9729:
9726:
9706:
9703:
9700:
9695:
9690:
9687:
9663:
9660:
9657:
9654:
9649:
9644:
9641:
9636:
9609:
9604:
9599:
9575:
9549:
9546:
9543:
9540:
9537:
9532:
9508:
9484:
9452:
9449:
9446:
9443:
9413:
9410:
9407:
9404:
9399:
9392:
9388:
9382:
9379:
9376:
9368:
9364:
9361:
9357:
9353:
9350:
9323:
9319:
9313:
9310:
9307:
9299:
9295:
9292:
9288:
9284:
9279:
9276:
9273:
9270:
9243:
9239:
9214:
9210:
9204:
9201:
9198:
9184:
9181:
9157:
9153:
9132:
9127:
9123:
9119:
9116:
9113:
9108:
9104:
9100:
9096:
9092:
9089:
9086:
9083:
9080:
9069:whose algebras
9060:
9057:
9038:
9033:
9007:
9004:
8975:generating set
8954:
8951:
8948:
8945:
8942:
8939:
8936:
8933:
8930:
8927:
8924:
8921:
8918:
8912:
8909:
8884:
8879:
8874:
8871:
8866:
8863:
8860:
8856:
8852:
8849:
8846:
8841:
8838:
8835:
8831:
8827:
8824:
8818:
8815:
8789:
8784:
8780:
8771:
8768:
8765:
8761:
8757:
8754:
8751:
8745:
8740:
8736:
8727:
8724:
8721:
8717:
8696:
8690:
8685:
8681:
8674:
8671:
8668:
8662:
8657:
8653:
8646:
8643:
8637:
8634:
8609:
8605:
8583:
8579:
8576:
8554:
8549:
8544:
8541:
8538:
8535:
8530:
8525:
8497:
8492:
8479:linear algebra
8474:
8473:Linear algebra
8471:
8458:
8436:
8432:
8408:
8405:
8402:
8399:
8379:
8367:
8364:
8341:
8337:
8316:
8313:
8279:
8275:
8254:
8234:
8229:
8225:
8221:
8218:
8215:
8210:
8206:
8202:
8199:
8196:
8174:
8170:
8147:
8143:
8119:
8116:
8113:
8110:
8075:
8072:
8069:
8066:
8063:
8060:
8057:
8054:
8051:
8048:
8045:
8025:
8022:
8019:
8016:
8013:
8010:
7990:
7975:
7974:
7963:
7960:
7957:
7954:
7951:
7948:
7945:
7942:
7939:
7936:
7933:
7930:
7927:
7924:
7921:
7918:
7895:
7879:
7876:
7839:-semidisk and
7835:inside a unit
7808:
7805:
7792:
7772:
7748:
7745:
7742:
7739:
7736:
7733:
7730:
7708:
7704:
7679:
7655:
7650:
7646:
7642:
7639:
7636:
7631:
7627:
7623:
7620:
7617:
7595:
7591:
7562:
7559:
7556:
7553:
7537:
7534:
7526:unit hypercube
7469:
7464:
7431:
7428:
7425:
7422:
7419:
7416:
7413:
7393:
7367:
7363:
7342:
7339:
7336:
7333:
7330:
7327:
7322:
7318:
7314:
7309:
7305:
7301:
7296:
7292:
7288:
7285:
7282:
7262:
7259:
7256:
7253:
7250:
7247:
7244:
7241:
7238:
7235:
7232:
7229:
7226:
7223:
7220:
7217:
7212:
7208:
7204:
7199:
7195:
7191:
7186:
7182:
7178:
7158:
7155:
7152:
7149:
7146:
7143:
7120:
7115:
7081:
7078:
7075:
7072:
7044:
7041:
7021:
7018:
7015:
7012:
7009:
7006:
7003:
6973:
6945:
6939:
6936:
6933:
6927:
6922:
6890:
6862:
6849:
6848:
6834:
6831:
6827:
6806:
6803:
6800:
6795:
6789:
6786:
6783:
6758:
6754:
6742:
6728:
6724:
6703:
6700:
6697:
6692:
6686:
6683:
6680:
6666:
6654:
6646:
6643:
6633:
6630:
6626:
6614:
6610:
6606:
6603:
6600:
6595:
6591:
6586:
6574:
6570:
6566:
6562:
6558:
6555:
6552:
6545:
6541:
6537:
6533:
6512:
6507:
6503:
6499:
6494:
6488:
6485:
6482:
6476:
6471:
6467:
6446:
6441:
6437:
6433:
6428:
6422:
6419:
6416:
6410:
6405:
6401:
6380:
6377:
6374:
6369:
6363:
6360:
6357:
6351:
6348:
6337:
6325:
6322:
6317:
6314:
6310:
6289:
6286:
6283:
6278:
6272:
6269:
6266:
6236:
6233:
6230:
6227:
6222:
6216:
6213:
6210:
6204:
6201:
6196:
6190:
6187:
6184:
6159:tensor product
6138:
6135:
6120:
6098:
6078:
6056:
6034:
6014:
5994:
5991:
5988:
5983:
5959:
5953:
5950:
5947:
5941:
5936:
5914:
5892:
5868:
5842:
5839:
5834:
5830:
5806:
5800:
5797:
5794:
5769:
5757:
5754:
5752:
5749:
5736:
5733:
5730:
5727:
5724:
5702:
5701:Identity axiom
5699:
5686:
5683:
5680:
5677:
5674:
5671:
5668:
5665:
5662:
5659:
5615:
5606:
5602:
5599:
5581:
5580:
5569:
5566:
5561:
5557:
5553:
5550:
5545:
5541:
5537:
5532:
5529:
5526:
5522:
5518:
5515:
5512:
5509:
5504:
5500:
5496:
5491:
5488:
5485:
5482:
5478:
5474:
5471:
5468:
5463:
5460:
5457:
5454:
5451:
5448:
5445:
5441:
5407:
5398:
5395:
5392:
5389:
5386:
5356:
5353:
5350:
5347:
5344:
5341:
5338:
5318:
5315:
5312:
5309:
5306:
5303:
5300:
5280:
5277:
5274:
5271:
5268:
5265:
5262:
5259:
5256:
5253:
5250:
5247:
5244:
5241:
5238:
5235:
5232:
5229:
5226:
5223:
5220:
5217:
5214:
5194:
5191:
5188:
5185:
5182:
5179:
5176:
5173:
5170:
5167:
5164:
5161:
5158:
5155:
5152:
5149:
5146:
5143:
5140:
5120:
5117:
5114:
5111:
5108:
5105:
5102:
5099:
5096:
5093:
5090:
5087:
5084:
5081:
5078:
5075:
5072:
5069:
5066:
5042:
5039:
5036:
5033:
5030:
5027:
5024:
5021:
5018:
5015:
4975:
4955:
4952:
4932:
4912:
4892:
4889:
4886:
4883:
4880:
4877:
4857:
4854:
4851:
4848:
4845:
4842:
4839:
4819:
4816:
4813:
4810:
4807:
4804:
4801:
4781:
4778:
4775:
4772:
4769:
4766:
4763:
4740:
4720:
4717:
4714:
4711:
4691:
4688:
4685:
4682:
4679:
4676:
4656:
4614:
4611:
4608:
4605:
4602:
4599:
4596:
4593:
4590:
4587:
4584:
4581:
4578:
4575:
4572:
4552:
4536:
4533:
4531:
4528:
4508:
4505:
4502:
4498:
4460:
4450:
4445:
4441:
4380:
4356:
4332:
4329:
4326:
4316:
4287:
4284:
4281:
4271:
4247:
4244:
4241:
4238:
4235:
4232:
4229:
4192:
4189:
4186:
4176:
4157:
4154:
4112:
4109:
4106:
4102:
4098:
4095:
4092:
4089:
4086:
4047:
4023:
4018:
4014:
4010:
4007:
4004:
3999:
3995:
3991:
3988:
3985:
3982:
3977:
3973:
3969:
3966:
3963:
3960:
3957:
3954:
3949:
3945:
3941:
3938:
3935:
3932:
3929:
3926:
3923:
3914:is a morphism
3903:
3879:
3876:
3873:
3870:
3847:
3844:
3829:
3826:
3823:
3820:
3803:
3802:
3790:
3787:
3784:
3781:
3778:
3775:
3772:
3769:
3766:
3763:
3760:
3757:
3754:
3742:
3741:
3740:
3739:
3728:
3725:
3720:
3716:
3712:
3709:
3706:
3703:
3700:
3697:
3692:
3688:
3684:
3681:
3678:
3675:
3672:
3669:
3666:
3663:
3660:
3657:
3654:
3649:
3645:
3641:
3638:
3635:
3630:
3626:
3622:
3619:
3616:
3613:
3610:
3597:
3596:
3582:
3578:
3574:
3571:
3568:
3563:
3559:
3538:
3523:
3512:
3509:
3506:
3503:
3500:
3497:
3482:
3481:
3467:
3463:
3460:
3456:
3452:
3449:
3446:
3443:
3440:
3437:
3434:
3431:
3428:
3425:
3420:
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3389:
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2894:
2890:
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2860:, through the
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2037:
2033:
2029:
2026:
2023:
2020:
2013:
2009:
2005:
2002:
1998:
1994:
1991:
1988:
1983:
1980:
1977:
1973:
1969:
1966:
1961:
1957:
1951:
1946:
1943:
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1123:
1120:
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1111:
1107:
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1101:
1098:
1093:
1089:
1085:
1082:
1079:
1074:
1070:
1066:
1063:
1060:
1057:
1054:
1051:
1028:
1025:
1022:
1019:
997:
993:
969:
943:
923:
903:
881:
877:
854:
850:
827:
823:
800:
796:
775:
755:
733:
729:
725:
722:
719:
714:
710:
698:
697:
686:
681:
677:
673:
670:
667:
662:
658:
654:
651:
648:
645:
640:
636:
632:
629:
626:
621:
617:
613:
610:
607:
584:
579:
575:
571:
568:
565:
560:
556:
552:
549:
546:
543:
538:
534:
530:
527:
524:
519:
515:
494:
491:
488:
485:
482:
479:
456:
436:
416:
405:
404:
392:
389:
386:
381:
377:
373:
370:
367:
364:
361:
358:
355:
352:
328:
324:
321:
301:
289:
286:
255:Koszul duality
220:
219:
193:Rainer M. Vogt
176:
173:
152:
128:
108:
85:
63:
40:
15:
9:
6:
4:
3:
2:
11183:
11172:
11169:
11167:
11164:
11163:
11161:
11152:
11149:
11147:
11145:
11140:
11137:
11136:
11127:
11123:
11119:
11115:
11112:
11108:
11104:
11100:
11097:
11093:
11089:
11085:
11081:
11075:
11071:
11067:
11062:
11059:
11057:9789814365116
11053:
11049:
11045:
11040:
11035:
11031:
11027:
11023:
11020:
11014:
11010:
11003:
11002:
10997:
10993:
10981:
10977:
10976:
10968:
10964:
10963:Stasheff, Jim
10960:
10955:
10950:
10945:
10941:
10935:
10927:
10921:
10917:
10916:
10911:
10907:
10906:Steve Shnider
10902:
10898:
10892:
10888:
10884:
10879:
10874:
10870:
10865:
10864:
10853:Definition 37
10850:
10846:
10842:
10840:9780444531018
10836:
10832:
10828:
10824:
10820:
10813:
10805:
10801:
10797:
10793:
10788:
10783:
10779:
10775:
10771:
10764:
10755:
10750:
10745:
10740:
10736:
10732:
10728:
10721:
10713:
10709:
10705:
10701:
10697:
10693:
10688:
10683:
10679:
10675:
10671:
10664:
10656:
10650:
10646:
10645:United States
10642:
10635:
10626:
10621:
10614:
10605:
10600:
10596:
10592:
10588:
10581:
10573:
10567:
10563:
10559:
10552:
10546:
10542:
10541:981-256-129-3
10538:
10535:
10531:
10525:
10516:
10512:
10508:
10506:9780444531018
10502:
10498:
10494:
10489:
10484:
10481:(1): 87–140.
10480:
10476:
10469:
10462:
10456:
10447:
10442:
10435:
10426:
10421:
10414:
10412:
10396:
10392:
10388:
10384:
10380:
10376:
10372:
10368:
10362:
10354:
10350:
10346:
10342:
10338:
10334:
10330:
10326:
10322:
10318:
10314:
10310:
10306:
10302:
10298:
10292:
10276:
10269:
10265:
10264:May, J. Peter
10259:
10250:
10242:
10238:
10234:
10228:
10224:
10220:
10215:
10210:
10206:
10202:
10196:
10188:
10184:
10180:
10174:
10170:
10166:
10162:
10158:
10152:
10144:
10140:
10135:
10130:
10126:
10122:
10118:
10114:
10108:
10104:
10072:
10068:
10064:
10061:
10058:
10053:
10049:
10044:
10033:
10029:
10024:
10020:
10017:
10014:
10007:
10003:
9998:
9994:
9989:
9985:
9981:
9975:
9969:
9962:
9945:
9942:
9938:
9934:
9929:
9925:
9914:
9911:
9908:
9904:
9900:
9894:
9888:
9881:
9880:
9876:
9872:
9862:
9861:Multicategory
9859:
9857:
9856:Pseudoalgebra
9854:
9852:
9849:
9847:
9844:
9842:
9839:
9837:
9834:
9833:
9824:
9823:
9822:
9820:
9809:
9806:December 2018
9800:
9796:
9793:This section
9791:
9788:
9784:
9783:
9775:
9773:
9769:
9765:
9756:
9739:
9730:
9701:
9688:
9685:
9677:
9655:
9639:
9623:
9563:
9544:
9535:
9473:of an operad
9472:
9467:
9466:
9447:
9433:
9429:
9390:
9386:
9362:
9359:
9355:
9351:
9341:
9321:
9317:
9293:
9290:
9286:
9259:
9241:
9237:
9212:
9208:
9180:
9178:
9174:
9155:
9151:
9125:
9121:
9117:
9114:
9111:
9106:
9102:
9090:
9084:
9078:
9070:
9067:is an operad
9066:
9056:
9054:
9036:
9002:
8991:
8987:
8983:
8978:
8976:
8972:
8966:
8949:
8946:
8943:
8940:
8937:
8934:
8931:
8928:
8925:
8922:
8916:
8907:
8897:. The vector
8882:
8872:
8861:
8854:
8850:
8847:
8844:
8836:
8829:
8822:
8813:
8782:
8778:
8766:
8759:
8755:
8752:
8749:
8738:
8734:
8722:
8715:
8683:
8679:
8672:
8669:
8666:
8655:
8651:
8641:
8632:
8607:
8603:
8577:
8574:
8552:
8542:
8536:
8513:
8480:
8470:
8434:
8430:
8422:
8403:
8397:
8363:
8361:
8357:
8339:
8335:
8326:
8322:
8312:
8310:
8308:
8302:
8298:
8293:
8277:
8273:
8252:
8227:
8223:
8219:
8216:
8213:
8208:
8204:
8197:
8194:
8172:
8168:
8145:
8141:
8133:
8114:
8108:
8099:
8097:
8093:
8089:
8070:
8067:
8061:
8058:
8055:
8049:
8046:
8020:
8017:
8014:
8008:
7988:
7980:
7979:associativity
7961:
7955:
7952:
7949:
7943:
7940:
7937:
7931:
7928:
7925:
7919:
7916:
7909:
7908:
7907:
7893:
7884:
7875:
7873:
7872:Justin Thomas
7869:
7865:
7861:
7857:
7853:
7849:
7844:
7842:
7838:
7834:
7831:-dimensional
7830:
7826:
7818:
7813:
7804:
7790:
7770:
7762:
7746:
7743:
7740:
7737:
7734:
7731:
7728:
7706:
7702:
7693:
7677:
7669:
7648:
7644:
7640:
7637:
7634:
7629:
7625:
7618:
7615:
7593:
7589:
7580:
7576:
7557:
7551:
7543:
7533:
7531:
7527:
7524:) inside the
7523:
7519:
7516:-dimensional
7515:
7512:
7508:
7504:
7500:
7496:
7492:
7488:
7483:
7467:
7452:
7448:
7443:
7429:
7426:
7423:
7420:
7417:
7414:
7411:
7391:
7383:
7365:
7361:
7337:
7331:
7328:
7320:
7316:
7312:
7307:
7303:
7299:
7294:
7290:
7283:
7280:
7257:
7251:
7248:
7242:
7236:
7233:
7227:
7221:
7218:
7210:
7206:
7202:
7197:
7193:
7189:
7184:
7180:
7153:
7147:
7144:
7141:
7118:
7103:
7099:
7095:
7076:
7070:
7062:
7054:
7049:
7040:
7038:
7033:
7016:
7010:
7007:
7001:
6993:
6989:
6961:
6943:
6910:
6906:
6878:
6832:
6829:
6825:
6801:
6793:
6756:
6752:
6743:
6726:
6722:
6698:
6690:
6667:
6652:
6644:
6631:
6628:
6624:
6612:
6608:
6604:
6601:
6598:
6593:
6589:
6572:
6568:
6564:
6560:
6556:
6553:
6550:
6543:
6539:
6535:
6531:
6505:
6501:
6492:
6474:
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6439:
6435:
6426:
6408:
6403:
6399:
6375:
6367:
6349:
6346:
6338:
6323:
6315:
6312:
6308:
6284:
6276:
6254:
6253:
6252:
6250:
6228:
6220:
6199:
6194:
6172:
6168:
6164:
6160:
6156:
6152:
6151:vector spaces
6148:
6144:
6134:
6096:
6076:
6032:
6012:
5989:
5957:
5912:
5856:
5840:
5832:
5828:
5819:
5804:
5767:
5748:
5734:
5731:
5728:
5725:
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5713:
5711:
5706:
5698:
5681:
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5555:
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5530:
5527:
5524:
5520:
5510:
5502:
5498:
5494:
5489:
5486:
5483:
5480:
5476:
5466:
5461:
5458:
5455:
5449:
5446:
5439:
5431:
5430:
5429:
5426:
5424:
5419:
5405:
5396:
5390:
5387:
5375:
5373:
5368:
5354:
5351:
5345:
5342:
5339:
5313:
5310:
5307:
5301:
5298:
5275:
5272:
5266:
5263:
5260:
5251:
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5245:
5239:
5236:
5233:
5227:
5224:
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5218:
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5212:
5189:
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5180:
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5112:
5109:
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5022:
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5002:
4997:
4994:
4992:
4987:
4973:
4953:
4950:
4930:
4910:
4887:
4884:
4881:
4878:
4852:
4849:
4846:
4843:
4840:
4830:. This sends
4814:
4811:
4808:
4802:
4799:
4776:
4770:
4767:
4752:
4738:
4715:
4712:
4686:
4683:
4680:
4674:
4654:
4645:
4643:
4638:
4636:
4632:
4628:
4612:
4609:
4603:
4600:
4597:
4591:
4585:
4582:
4579:
4573:
4570:
4550:
4542:
4527:
4525:
4522:generated by
4521:
4503:
4496:
4487:
4481:
4448:
4443:
4430:
4402:
4397:
4395:
4369:
4314:
4305:
4303:
4269:
4261:
4242:
4239:
4236:
4233:
4230:
4220:
4219:monoid object
4216:
4212:
4208:
4174:
4166:
4163:
4153:
4151:
4147:
4143:
4139:
4135:
4130:
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4110:
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4093:
4087:
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3808:
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3175:
3171:
3167:
3164:
3161:
3156:
3152:
3145:
3134:
3130:
3126:
3123:
3120:
3115:
3111:
3104:
3101:
3095:
3087:
3083:
3079:
3074:
3070:
3066:
3063:
3060:
3055:
3051:
3047:
3042:
3038:
3031:
3028:
3021:
3020:
3002:
2998:
2993:
2989:
2984:
2980:
2972:permutations
2959:
2951:
2934:
2914:
2892:
2888:
2867:
2845:
2841:
2818:
2814:
2793:
2768:
2764:
2760:
2757:
2754:
2749:
2745:
2741:
2738:
2735:
2732:
2729:
2726:
2699:
2695:
2691:
2688:
2685:
2680:
2676:
2671:
2649:
2646:
2637:
2622:
2619:
2615:
2601:
2595:
2591:
2587:
2584:
2581:
2573:
2567:
2563:
2556:
2553:
2547:
2539:
2535:
2531:
2528:
2525:
2520:
2516:
2509:
2503:
2500:
2497:
2487:
2486:
2485:
2484:
2466:
2462:
2458:
2455:
2447:
2444:
2443:
2442:
2428:
2421:, denoted by
2403:
2400:
2377:
2371:
2349:
2345:
2337:
2321:
2313:
2276:
2272:
2268:
2265:
2261:
2257:
2254:
2251:
2246:
2243:
2240:
2236:
2232:
2229:
2226:
2219:
2215:
2211:
2208:
2204:
2200:
2197:
2194:
2189:
2186:
2183:
2179:
2172:
2157:
2153:
2149:
2146:
2143:
2138:
2134:
2127:
2124:
2115:
2108:
2087:
2083:
2079:
2076:
2072:
2068:
2065:
2062:
2057:
2054:
2051:
2047:
2040:
2035:
2031:
2027:
2024:
2021:
2011:
2007:
2003:
2000:
1996:
1992:
1989:
1986:
1981:
1978:
1975:
1971:
1964:
1959:
1955:
1944:
1941:
1929:
1928:
1927:
1926:
1922:
1921:associativity
1919:
1905:
1902:
1899:
1896:
1893:
1890:
1884:
1881:
1878:
1875:
1872:
1866:
1863:
1855:
1852:
1851:
1850:
1829:
1821:
1817:
1813:
1810:
1807:
1802:
1798:
1791:
1788:
1783:
1773:
1769:
1765:
1762:
1759:
1754:
1750:
1746:
1743:
1728:
1724:
1720:
1717:
1714:
1709:
1705:
1698:
1693:
1683:
1679:
1672:
1669:
1666:
1663:
1655:
1651:
1644:
1641:
1635:
1629:
1626:
1623:
1612:
1611:
1607:
1589:
1585:
1581:
1578:
1575:
1570:
1566:
1545:
1537:
1534:
1515:
1509:
1489:
1481:
1478:
1464:
1436:
1433:
1422:
1416:
1405:
1404:
1403:
1401:
1397:
1374:
1370:
1355:
1341:
1321:
1312:
1298:
1295:
1290:
1286:
1282:
1279:
1276:
1271:
1267:
1244:
1240:
1236:
1233:
1210:
1204:
1201:
1198:
1167:
1159:
1156:
1152:
1147:
1143:
1140:
1137:
1129:
1121:
1118:
1114:
1109:
1102:
1099:
1091:
1087:
1083:
1080:
1077:
1072:
1068:
1058:
1055:
1052:
1042:
1041:
1040:
1039:, defined by
1023:
1017:
995:
991:
983:
967:
960:
955:
941:
921:
901:
879:
875:
852:
848:
825:
821:
798:
794:
773:
753:
731:
727:
723:
720:
717:
712:
708:
679:
675:
671:
668:
665:
660:
656:
649:
646:
638:
634:
630:
627:
624:
619:
615:
608:
605:
598:
597:
596:
577:
573:
566:
563:
558:
554:
550:
547:
544:
536:
532:
525:
522:
517:
513:
489:
483:
480:
477:
468:
454:
434:
414:
387:
379:
375:
371:
368:
362:
356:
350:
343:
342:
341:
322:
319:
299:
285:
283:
279:
275:
272:
268:
264:
260:
256:
252:
249:phenomena in
248:
244:
240:
236:
231:
229:
225:
217:
213:
209:
208:
207:
205:
200:
198:
194:
190:
186:
182:
172:
170:
166:
150:
143:; in a sense
142:
126:
106:
99:
83:
75:
74:
61:
54:algebra over
38:
30:
26:
22:
11143:
11117:
11102:
11065:
11029:
11000:
10984:. Retrieved
10982:(6): 630–631
10979:
10973:
10954:math/0601129
10914:
10910:Jim Stasheff
10878:math/0305049
10868:
10822:
10818:
10812:
10777:
10773:
10763:
10744:math/0309369
10734:
10730:
10720:
10687:math/9904055
10677:
10673:
10663:
10640:
10634:
10625:math/9803156
10613:
10594:
10590:
10580:
10557:
10551:
10533:
10524:
10488:math/0601129
10478:
10474:
10468:
10455:
10434:
10425:math/0001151
10400:27 September
10398:. Retrieved
10374:
10361:
10351:– via
10308:
10304:
10291:
10281:28 September
10279:. Retrieved
10274:
10258:
10249:
10204:
10195:
10160:
10151:
10124:
10120:
10107:
9875:
9816:
9803:
9799:adding to it
9794:
9771:
9762:
9675:
9624:
9561:
9470:
9468:
9464:
9431:
9428:free functor
9340:left adjoint
9186:
9183:Free Operads
9064:
9062:
8979:
8971:all possible
8970:
8967:
8476:
8369:
8359:
8355:
8327:, with each
8324:
8320:
8318:
8306:
8305:associative
8300:
8294:
8100:
8091:
8087:
7976:
7885:
7881:
7845:
7840:
7836:
7828:
7824:
7822:
7816:
7760:
7691:
7670:-th leaf of
7667:
7578:
7574:
7542:rooted trees
7539:
7536:Rooted trees
7529:
7513:
7511:axis-aligned
7494:
7490:
7486:
7484:
7450:
7446:
7444:
7381:
7093:
7060:
7058:
7052:
7034:
6991:
6987:
6959:
6908:
6904:
6876:
6850:
6771:operates on
6251:consists of
6248:
6170:
6166:
6162:
6154:
6142:
6140:
5781:
5759:
5714:
5707:
5704:
5647:
5645:
5638:
5635:
5628:
5589:
5582:
5427:
5420:
5376:
5369:
5054:
5005:
4998:
4995:
4988:
4753:
4646:
4639:
4630:
4626:
4540:
4538:
4523:
4485:
4479:
4428:
4400:
4398:
4306:
4214:
4210:
4206:
4164:
4159:
4131:
4126:
4074:
4064:
4059:
4035:
3891:
3858:
3849:
3804:
3526:
3483:
3367:
3359:
2446:equivariance
2445:
2311:
2309:
1920:
1853:
1848:
1605:
1532:
1457:
1399:
1376:
1372:
1368:
1366:
1313:
1190:
959:right action
956:
699:
469:
406:
291:
232:
221:
211:
204:Jim Stasheff
201:
197:J. Peter May
178:
141:Lie algebras
53:
24:
18:
10780:(1): 1–48.
10774:Geom. Topol
10545:pp. 474,475
10277:. p. 2
9173:Koszul-dual
8980:Similarly,
8421:braid group
8362:-algebras.
8160:, on which
8092:composition
7507:Rainer Vogt
7384:th disk of
7100:inside the
4642:expressions
4541:composition
3809:denoted by
1606:composition
1531:called the
1482:an element
1406:a sequence
224:portmanteau
185:loop spaces
21:mathematics
11160:Categories
11096:1373.55014
10986:17 January
10861:References
10825:: 87–140.
10395:0866.18007
10349:0855.18006
10201:May, J. P.
9674:is called
9177:Lie operad
8297:semigroups
8001:; writing
7581:The group
7518:hypercubes
5291:, this is
4731:. So that
4631:themselves
4625:; it does
4150:coalgebras
2952:and given
2927:blocks by
1358:Definition
427:copies of
340:we define
167:is to its
98:Lie operad
29:operations
11039:1101.0267
10934:cite book
10804:119320246
10787:1011.1635
10680:: 35–72.
10519:Example 2
10341:115166937
10325:0012-7094
10241:0075-8434
10209:CiteSeerX
10187:0075-8434
10143:0002-9904
10099:Citations
10062:⋯
10041:→
10021:⊗
10018:⋯
10015:⊗
9995:⊗
9970:γ
9943:⊗
9935:⊗
9920:∞
9905:⨁
9851:E∞-operad
9725:Γ
9676:quadratic
9603:→
9539:Γ
9442:Γ
9398:→
9363:∈
9356:∏
9349:Γ
9294:∈
9287:∏
9283:→
9115:…
9006:→
8950:…
8935:−
8911:→
8873:∈
8848:…
8817:→
8788:→
8753:…
8744:→
8689:→
8670:…
8661:→
8642:∘
8636:→
8578:∈
8529:∞
8496:∞
8457:Σ
8378:Σ
8274:τ
8253:σ
8224:τ
8217:…
8205:τ
8198:∘
8195:σ
8098:, above.
8088:operation
8009:ψ
7989:ψ
7956:ψ
7944:∘
7941:ψ
7926:ψ
7920:∘
7917:ψ
7894:ψ
7864:Igor Kriz
7763:trees to
7741:…
7694:-th tree
7638:…
7619:∘
7522:intervals
7392:θ
7362:θ
7329:∈
7317:θ
7304:θ
7291:θ
7284:∘
7281:θ
7249:×
7234:×
7219:∈
7207:θ
7194:θ
7181:θ
7145:∈
7142:θ
7102:unit disk
7096:disjoint
7011:
7005:→
6926:→
6830:⊗
6642:→
6629:⊗
6605:⊗
6602:⋯
6599:⊗
6585:⟶
6565:⊗
6557:⊗
6554:⋯
6551:⊗
6536:⊗
6475:∈
6409:∈
6350:∈
6321:→
6313:⊗
5940:→
5838:→
5726:∘
5528:⋅
5521:θ
5511:∘
5487:⋅
5477:θ
5467:∘
5459:⋅
5440:θ
5352:∘
5343:∘
5311:∘
5302:∘
5261:θ
5234:θ
5219:θ
5175:θ
5166:∘
5151:θ
5145:∘
5142:θ
5098:θ
5089:∘
5077:θ
5068:∘
5065:θ
4931:θ
4911:θ
4809:θ
4803:∘
4800:θ
4739:θ
4675:θ
4655:θ
4637:, below.
4610:∘
4601:∘
4583:∘
4574:∘
4551:∘
4459:→
4440:Σ
4300:(it is a
4243:η
4237:γ
4146:groupoids
4108:≥
4046:⊗
4006:⋯
3984:→
3962:⊗
3959:⋯
3956:⊗
3934:⊗
3902:∘
3786:∗
3762:∗
3715:θ
3702:…
3687:θ
3674:∘
3668:θ
3644:θ
3637:…
3625:θ
3618:∘
3615:θ
3577:θ
3570:…
3558:θ
3537:θ
3462:∈
3439:→
3385:→
3364:Morphisms
3273:⋯
3219:…
3165:…
3146:∗
3131:θ
3124:…
3112:θ
3105:∘
3102:θ
3080:∗
3071:θ
3064:…
3048:∗
3039:θ
3032:∘
3029:θ
2990:∈
2758:⋯
2739:…
2689:⋯
2616:∗
2592:θ
2585:…
2564:θ
2557:∘
2554:θ
2536:θ
2529:…
2517:θ
2510:∘
2501:∗
2498:θ
2459:∈
2429:∗
2404:∈
2262:θ
2255:…
2237:θ
2230:…
2205:θ
2198:…
2180:θ
2173:∘
2154:θ
2147:…
2135:θ
2128:∘
2125:θ
2073:θ
2066:…
2048:θ
2041:∘
2032:θ
2025:…
1997:θ
1990:…
1972:θ
1965:∘
1956:θ
1945:∘
1942:θ
1906:θ
1903:∘
1894:θ
1879:…
1867:∘
1864:θ
1818:θ
1811:…
1799:θ
1792:∘
1789:θ
1786:↦
1770:θ
1763:…
1751:θ
1744:θ
1718:⋯
1696:→
1670:×
1667:⋯
1664:×
1642:×
1624:∘
1579:…
1437:∈
1385:Σ
1342:∗
1322:∘
1296:∈
1280:…
1237:∈
1202:∈
1157:−
1141:…
1119:−
1081:…
1056:∗
968:∗
721:⋯
669:⋯
647:∈
628:…
609:∘
564:∈
548:…
523:∈
481:∈
385:→
323:∈
288:Intuition
199:in 1972.
10912:(2002).
10712:16838440
10369:(1996).
10203:(1972).
9830:See also
8803:, where
8309:algebras
5751:Examples
5055:a priori
4488:-module
4370:, where
4368:-objects
4160:Given a
3852:category
3807:category
2650:′
2638:(where
2623:′
1854:identity
1608:function
1533:identity
292:Suppose
274:homology
11141:at the
11088:3643404
11044:Bibcode
10883:Bibcode
10849:3239126
10692:Bibcode
10532:(2005)
10515:3239126
10387:1423619
10333:1301191
9772:clonoid
8510:of all
7497:-cubes
7489:or the
6457:, ...,
5329:versus
4903:(apply
4258:in the
4152:, etc.
4136:over a
4134:modules
4038:(where
3195:(where
1375:, or a
980:of the
247:duality
175:History
11139:operad
11124:
11109:
11094:
11086:
11076:
11054:
11015:
10922:
10893:
10847:
10837:
10802:
10710:
10651:
10568:
10539:
10513:
10503:
10393:
10385:
10347:
10339:
10331:
10323:
10239:
10229:
10211:
10185:
10175:
10141:
9768:minion
9764:Clones
9759:Clones
9560:where
8988:, and
7866:, and
7721:, for
7404:, for
6650:
6637:
6621:
6580:
5611:
5608:
5403:
5400:
4211:operad
4075:spaces
3484:that:
2312:operad
265:, the
25:operad
11034:arXiv
11005:(PDF)
10970:(PDF)
10949:arXiv
10873:arXiv
10845:S2CID
10800:S2CID
10782:arXiv
10739:arXiv
10708:S2CID
10682:arXiv
10620:arXiv
10511:S2CID
10483:arXiv
10441:arXiv
10420:arXiv
10337:S2CID
10271:(PDF)
9867:Notes
7860:Po Hu
7833:disks
7499:PROPs
7098:disks
6907:over
6161:over
6153:over
6147:field
6145:is a
5857:. If
4484:free
4302:monad
4213:over
4209:. An
1400:plain
271:graph
269:, or
228:monad
165:group
23:, an
11122:ISBN
11107:ISBN
11074:ISBN
11052:ISBN
11013:ISBN
10988:2008
10940:link
10920:ISBN
10891:ISBN
10835:ISBN
10649:ISBN
10566:ISBN
10537:ISBN
10501:ISBN
10402:2018
10321:ISSN
10283:2018
10237:ISSN
10227:ISBN
10183:ISSN
10173:ISBN
10139:ISSN
9770:(or
9063:The
8567:for
7823:The
7815:The
7505:and
7059:The
2393:for
1604:, a
1377:non-
1334:and
1259:and
1191:for
280:and
241:and
191:and
11146:Lab
11092:Zbl
10827:doi
10792:doi
10749:doi
10735:142
10700:doi
10599:doi
10493:doi
10391:Zbl
10345:Zbl
10313:doi
10219:doi
10165:doi
10129:doi
9817:In
9801:.
9774:).
8477:In
7104:of
7032:.)
7008:End
6851:If
6247:of
6141:If
5053:is
4966:by
4868:to
4702:or
4627:not
4464:Set
4454:Set
4413:Set
4262:on
4034:in
2364:on
1502:in
1398:or
1010:on
447:to
261:of
187:by
19:In
11162::
11090:,
11084:MR
11082:,
11072:,
11050:,
11042:,
11011:,
10980:51
10978:.
10972:.
10936:}}
10932:{{
10908:,
10889:.
10881:.
10843:.
10833:.
10821:.
10798:.
10790:.
10778:20
10776:.
10772:.
10747:.
10733:.
10729:.
10706:.
10698:.
10690:.
10678:48
10676:.
10672:.
10595:83
10593:.
10589:.
10543:,
10509:.
10499:.
10491:.
10477:.
10410:^
10389:.
10383:MR
10381:.
10377:.
10373:.
10343:.
10335:.
10329:MR
10327:.
10319:.
10309:76
10307:.
10303:.
10273:.
10266:.
10235:.
10225:.
10217:.
10181:.
10171:.
10137:.
10125:74
10123:.
10119:.
9755:.
9465:E.
9434:,
8984:,
8311:.
8307:k-
8292:.
7874:.
7862:,
7579:n.
7482:.
7442:.
7382:i-
6723:id
6391:,
6163:k.
6133:.
4526:.
4144:,
4140:,
4062:.
3842:.
3019:,
1856::
1558:,
1367:A
1354:.
1311:.
1226:,
505:,
467:.
363::=
284:.
237:,
171:.
11144:n
11128:.
11113:.
11046::
11036::
10990:.
10957:.
10951::
10942:)
10928:.
10899:.
10885::
10875::
10851:.
10829::
10823:5
10806:.
10794::
10784::
10757:.
10751::
10741::
10714:.
10702::
10694::
10684::
10657:.
10628:.
10622::
10607:.
10601::
10574:.
10517:.
10495::
10485::
10479:5
10463:.
10449:.
10443::
10428:.
10422::
10404:.
10355:.
10315::
10285:.
10243:.
10221::
10189:.
10167::
10145:.
10131::
10089:.
10073:n
10069:i
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5732:=
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2012:1
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2004:,
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1960:1
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151:L
127:L
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