4452:
155:. A simplicial set contains vertices (known as "0-simplices" in this context) and arrows ("1-simplices") between some of these vertices. Two vertices may be connected by several arrows, and directed loops that connect a vertex to itself are also allowed. Unlike directed multigraphs, simplicial sets may also contain higher simplices. A 2-simplex, for instance, can be thought of as a two-dimensional "triangular" shape bounded by a list of three vertices
4699:
4719:
4709:
3272:
for simplicial sets and the usual homotopy category of CW complexes with homotopy classes of continuous maps between them. It is part of the general definition of a
Quillen adjunction that the right adjoint functor (in this case, the singular set functor) carries fibrations (resp. trivial fibrations)
219:-th vertex. The vertices of a simplex need not be distinct and a simplex is not determined by its vertices and faces: two different simplices may share the same list of faces (and therefore the same list of vertices), just like two different arrows in a multigraph may connect the same two vertices.
3749:
can be thought of as categories in which the composition of morphisms is defined only up to homotopy, and information about the composition of higher homotopies is also retained. Quasi-categories are defined as simplicial sets satisfying one additional condition, the weak Kan condition.
2538:
2952:
2400:
2710:
3443:, Quillen developed surprisingly efficient methods for manipulating infinite simplicial sets. These methods were used in other areas on the border between algebraic geometry and topology. For instance, the
1068:
675:
1548:
3450:
Both the algebraic K-theory and the André–Quillen homology are defined using algebraic data to write down a simplicial set, and then taking the homotopy groups of this simplicial set.
1992:
1927:
1843:
1767:
1628:
1438:
1223:
830:
1683:
2621:
2577:
2805:
70:, glued together according to the rules of the simplicial set. Indeed, one may view a simplicial set as a purely combinatorial construction designed to capture the essence of a "
3629:
1302:
909:
1101:
708:
109:
A simplicial set is a categorical (that is, purely algebraic) model capturing those topological spaces that can be built up (or faithfully represented up to homotopy) from
3544:
1362:
1256:
969:
863:
1329:
936:
1145:
752:
3567:
3498:
3732:
3712:
3692:
3669:
3649:
3587:
3518:
3475:
1165:
772:
570:
The simplex category Δ is generated by two particularly important families of morphisms (maps), whose images under a given simplicial set functor are called
4096:
2791:
To define the realization functor, we first define it on standard n-simplices Δ as follows: the geometric realization |Δ| is the standard topological
3694:
is not an abelian group, it can happen that it has a composition which is sufficiently commutative so that one can use the above idea to prove that
97:. A construction analogous to that of simplicial sets can be carried out in any category, not just in the category of sets, yielding the notion of
2780: + 1)-dimensional Euclidean space defined below) and these topological simplices are glued together in the fashion the simplices of
2533:{\displaystyle X_{n}=X()\cong \operatorname {Nat} (\operatorname {hom} _{\Delta }(-,),X)=\operatorname {hom} _{\textbf {sSet}}(\Delta ^{n},X)}
2378:+ 1) nonnegative integers. (In many texts, it is written instead as hom(,-) where the homset is understood to be in the opposite category Δ.)
139:
largely continue to prefer CW complexes, there is a growing contingent of researchers interested in using simplicial sets for applications in
3937:
3904:
117:
to modeling topological spaces, with the crucial difference that simplicial sets are purely algebraic and do not carry any actual topology.
128:
2649:
3215:
that the category of simplicial sets with these classes of morphisms becomes a model category, and indeed satisfies the axioms for a
3991:
2632:
985:
592:
3444:
4089:
1451:
4293:
4248:
3192:
227:
152:
3785:
3374:
4743:
4722:
4662:
3800:
2749:
223:
3191:
on the category of simplicial sets, one has to define fibrations, cofibrations and weak equivalences. One can define
3370:
and simplicial abelian groups also carry closed model structures induced by that of the underlying simplicial sets.
3232:
of spaces. With the model structure in place, a homotopy theory of simplicial sets can be developed using standard
4712:
4498:
4362:
4270:
3742:
3345:
3080:
Every order-preserving map φ:→ induces a continuous map |Δ|→|Δ| in a natural way, which by composition yields
555:
4671:
4315:
4253:
4176:
3222:
3200:
3159:. Intuitively, this adjunction can be understood as follows: a continuous map from the geometric realization of
3016:
3012:
2746:
1998:
that has these face and degeneracy maps. So the identities provide an alternative way to define simplicial sets.
478:
4753:
4748:
4702:
4658:
4263:
4082:
1932:
1867:
3948:
1782:
1706:
1567:
1377:
1170:
777:
4258:
4240:
2977:
2947:{\displaystyle |\Delta ^{n}|=\{(x_{0},\dots ,x_{n})\in \mathbb {R} ^{n+1}:0\leq x_{i}\leq 1,\sum x_{i}=1\}.}
2019:
1643:
551:
3216:
2598:
2554:
4465:
4231:
4211:
4134:
3955:
3738:
521:
94:
40:
4347:
4186:
3241:
3219:
2007:
44:
36:
3592:
1261:
868:
4159:
4154:
2548:
3960:
4503:
4451:
4381:
4377:
4181:
3790:
3770:
3424:
3353:
3233:
1073:
680:
441:
explicitly as part of the definition, the short and elegant modern definition uses the language of
3199:. A map of simplicial sets is defined to be a weak equivalence if its geometric realization is a
4357:
4352:
4334:
4216:
4191:
4006:
3837:
3436:
3341:
3237:
140:
136:
3523:
4758:
4666:
4603:
4591:
4493:
4418:
4413:
4371:
4367:
4149:
4144:
3987:
3933:
3900:
3795:
3420:
3325:
3269:
3097:
411:-th vertex. This description implicitly requires certain consistency relations among the maps
83:
63:
2044:. Every morphism φ:→ in Δ is an order preserving map, and via composition induces a map
1334:
1228:
941:
835:
250: = 0, 1, 2, ..., together with certain maps between these sets: the
4627:
4513:
4488:
4423:
4408:
4403:
4342:
4171:
4139:
4015:
3892:
3829:
3820:
Eilenberg, Samuel; Zilber, J. A. (1950). "Semi-Simplicial
Complexes and Singular Homology".
3367:
3333:
3321:
3228:
A key turning point of the theory is that the geometric realization of a Kan fibration is a
513:
458:
454:
56:
52:
48:
3914:
1307:
914:
4539:
4105:
3910:
3229:
3112:
1106:
713:
442:
132:
75:
4576:
3651:
is an abelian group, we can actually iterate this infinitely many times, and obtain that
3549:
3480:
3377:
which yields an equivalence of categories between simplicial abelian groups and bounded
4571:
4555:
4518:
4508:
4428:
3973:
3775:
3764:
3746:
3717:
3697:
3677:
3654:
3634:
3572:
3503:
3460:
3432:
3212:
3188:
2624:
1150:
757:
90:
79:
32:
3373:
The homotopy groups of simplicial abelian groups can be computed by making use of the
4737:
4566:
4398:
4275:
4201:
4019:
3378:
3361:
3196:
4054:
3419:
Simplicial sets were originally used to give precise and convenient descriptions of
2321:
Another important class of examples of simplicial sets is given by the singular set
4320:
4221:
4049:
4001:
3884:
3440:
3428:
3208:
2382:
71:
4581:
3336:, we are just talking about the simplicial sets that were defined above. Letting
3986:. Lecture Notes in Mathematics. Vol. 341. Springer-Verlag. pp. 85–147.
3981:
3927:
3734:-theory of a ring, considered as a topological space, is an infinite loop space.
4561:
4433:
4303:
4035:
3780:
3204:
20:
4062:
3175:
in such a fashion that these maps are compatible with the way the simplices in
4613:
4551:
4164:
3977:
3923:
3896:
3454:
3453:
Simplicial methods are often useful when one wants to prove that a space is a
3431:'s idea of considering classifying spaces of categories, and in particular by
3236:
methods. Furthermore, the geometric realization and singular functors give a
2757:
114:
2999:
of topological spaces, as the target category of geometric realization: like
31:
in a specific way. Simplicial sets are higher-dimensional generalizations of
4607:
4298:
4039:
3759:
4676:
4308:
4206:
3714:
is an infinite loop space. In this way, one can prove that the algebraic
2295:
lengthen the sequence by inserting an identity morphism at position
1994:
that satisfy the simplicial identities, there is a unique simplicial set
3447:
of a ring is a "non-abelian homology", defined and studied in this way.
3171:
a continuous map from the corresponding standard topological simplex to
151:
Simplicial sets can be viewed as a higher-dimensional generalization of
4646:
4636:
4285:
4196:
3841:
2796:
2640:
2393:
stand in 1–1 correspondence with the natural transformations from Δ to
517:
492:
124:
110:
86:
is equivalent to the familiar homotopy category of topological spaces.
67:
4641:
516:. (Alternatively and equivalently, one may define simplicial sets as
3833:
2784:
hang together. In this process the orientation of the simplices of
113:
and their incidence relations. This is similar to the approach of
66:, known as its geometric realization. This realization consists of
4523:
4074:
3100:
of "probing" a target topological space with standard topological
2995:
of compactly-generated
Hausdorff spaces, rather than the category
2338:
consists of all the continuous maps from the standard topological
2318:; in this sense simplicial sets generalize posets and categories.
559:
78:. Specifically, the category of simplicial sets carries a natural
135:
are generalized by analogous results for simplicial sets. While
4066:
2705:{\displaystyle X\cong \varinjlim _{\Delta ^{n}\to X}\Delta ^{n}}
4463:
4116:
4078:
407:+1)-simplex which arises from the given one by duplicating the
1063:{\displaystyle \sigma ^{n,0},\dotsc ,\sigma ^{n,n}\colon \to }
670:{\displaystyle \delta ^{n,0},\dotsc ,\delta ^{n,n}\colon \to }
2715:
where the colimit is taken over the category of simplices of
3115:
to the geometric realization functor described above, i.e.:
2957:
The definition then naturally extends to any simplicial set
550:, whose objects are simplicial sets and whose morphisms are
2587:
and whose morphisms are natural transformations Δ → Δ over
2366:, denoted Δ, is a simplicial set defined as the functor hom
2148:
A similar construction can be performed for every category
382:-th face, the face "opposite to" (i.e. not containing) the
336: = 0, 1, 2, ... and 0 ≤
289: = 1, 2, 3, ... and 0 ≤
3167:
is uniquely specified if we associate to every simplex of
2245:
drops the first morphism from such a list, the face map
2168:, where we consider as a category with objects 0,1,...,
3096:(). This definition is analogous to a standard idea in
982:
are the images in that simplicial set of the morphisms
589:
are the images in that simplicial set of the morphisms
3891:. Progress in Mathematics. Vol. 174. Birkhäuser.
2132:-th element from such a list, and the degeneracy maps
3720:
3700:
3680:
3657:
3637:
3595:
3575:
3569:
itself is a group, we can iterate the procedure, and
3552:
3526:
3506:
3483:
3463:
2808:
2652:
2601:
2557:
2403:
2230:. (In particular, the 0-simplices are the objects of
1935:
1870:
1785:
1709:
1646:
1570:
1454:
1380:
1337:
1310:
1264:
1231:
1173:
1153:
1109:
1076:
988:
944:
917:
871:
838:
780:
760:
716:
683:
595:
120:
To get back to actual topological spaces, there is a
2745:
to its corresponding realization in the category of
199: + 1 vertices (which are 0-simplices) and
4626:
4590:
4538:
4531:
4482:
4391:
4333:
4284:
4239:
4230:
4127:
3737:In recent years, simplicial sets have been used in
1543:{\displaystyle d_{n-1,i}d_{n,j}=d_{n-1,j-1}d_{n,i}}
3726:
3706:
3686:
3663:
3643:
3623:
3581:
3561:
3538:
3512:
3492:
3469:
3211:of simplicial sets. It is a difficult theorem of
2946:
2704:
2615:
2571:
2532:
1986:
1921:
1837:
1761:
1677:
1622:
1542:
1432:
1356:
1323:
1296:
1250:
1217:
1159:
1139:
1095:
1062:
963:
930:
903:
857:
824:
766:
746:
702:
669:
566:Face and degeneracy maps and simplicial identities
554:between them. This is nothing but the category of
3589:is homotopy equivalent to the double loop space
2370:(-, ) where denotes the ordered set {0, 1, ... ,
2286: + 1)th morphisms. The degeneracy maps
546:Simplicial sets form a category, usually denoted
43:. Formally, a simplicial set may be defined as a
16:Mathematical construction used in homotopy theory
207: − 1)-simplices). The vertices of the
3967:(An elementary introduction to simplicial sets)
3203:. A map of simplicial sets is defined to be a
2346:. The singular set is further explained below.
4004:(1974). "Categories and cohomology theories".
1167:twice. Let us denote these degeneracy maps by
4090:
4040:"A leisurely introduction to simplicial sets"
3865:
2984:. The geometric realization is functorial on
195:-simplex is an object made up from a list of
55:. Simplicial sets were introduced in 1950 by
8:
3854:
2938:
2832:
2026:, as follows: for every object of Δ we set
222:Simplicial sets should not be confused with
131:. Most classical results on CW complexes in
62:Every simplicial set gives rise to a "nice"
3976:(1973). "Higher algebraic K-theory: I". In
2991:It is significant that we use the category
143:where CW complexes do not naturally exist.
4718:
4708:
4535:
4479:
4460:
4236:
4124:
4113:
4097:
4083:
4075:
3949:"Simplicial Sets and van Kampen's Theorem"
3273:to fibrations (resp. trivial fibrations).
1103:is the only (order-preserving) surjection
477:≥0. The morphisms in Δ are (non-strictly)
3959:
3719:
3699:
3679:
3656:
3636:
3600:
3594:
3574:
3551:
3525:
3520:is homotopy equivalent to the loop space
3505:
3482:
3462:
3019:is defined differently in the categories
2926:
2904:
2879:
2875:
2874:
2861:
2842:
2824:
2818:
2809:
2807:
2696:
2675:
2670:
2660:
2651:
2608:
2600:
2564:
2556:
2515:
2498:
2497:
2451:
2408:
2402:
2234:and the 1-simplices are the morphisms of
2088:(), can be thought of as ordered length-(
1972:
1959:
1940:
1934:
1907:
1894:
1875:
1869:
1829:
1813:
1800:
1790:
1784:
1747:
1737:
1724:
1714:
1708:
1670:
1661:
1651:
1645:
1614:
1598:
1585:
1575:
1569:
1528:
1500:
1481:
1459:
1453:
1424:
1408:
1395:
1385:
1379:
1342:
1336:
1315:
1309:
1282:
1269:
1263:
1236:
1230:
1203:
1178:
1172:
1152:
1108:
1081:
1075:
1018:
993:
987:
949:
943:
922:
916:
889:
876:
870:
843:
837:
810:
785:
779:
759:
715:
710:is the only (order-preserving) injection
688:
682:
625:
600:
594:
4055:Simplicial Objects in Algebraic Topology
2980:is taken over the n-simplex category of
2056:(). It is straightforward to check that
1987:{\displaystyle s_{n,i}:X_{n}\to X_{n+1}}
1922:{\displaystyle d_{n,i}:X_{n}\to X_{n-1}}
1304:. If the first index is clear, we write
911:. If the first index is clear, we write
74:" topological space for the purposes of
3812:
2164:() is the set of all functors from to
1838:{\displaystyle s_{i}s_{j}=s_{j+1}s_{i}}
1762:{\displaystyle d_{i}s_{j}=s_{j}d_{i-1}}
1623:{\displaystyle d_{i}s_{j}=s_{j-1}d_{i}}
1433:{\displaystyle d_{i}d_{j}=d_{j-1}d_{i}}
1367:The defined maps satisfy the following
1218:{\displaystyle s_{n,0},\dotsc ,s_{n,n}}
825:{\displaystyle d_{n,0},\dotsc ,d_{n,n}}
2354:-simplex and the category of simplices
2040:), the order-preserving maps from to
1678:{\displaystyle d_{i}s_{j}={\text{id}}}
2616:{\displaystyle \Delta \downarrow {X}}
2572:{\displaystyle \Delta \downarrow {X}}
2060:is a contravariant functor from Δ to
1857:Conversely, given a sequence of sets
7:
3767:, a generalization of simplicial set
3306:or equivalently a covariant functor
2756:is the topological space (in fact a
2014:,≤), we can define a simplicial set
129:compactly generated Hausdorff spaces
3439:. In this work, which earned him a
3427:. This idea was vastly extended by
3415:History and uses of simplicial sets
3201:weak homotopy equivalence of spaces
2499:
774:. Let us denote these face maps by
89:Simplicial sets are used to define
3597:
3527:
3477:is a group with classifying space
3183:Homotopy theory of simplicial sets
2815:
2752:. Intuitively, the realization of
2693:
2672:
2602:
2558:
2512:
2452:
2199:can be thought of as sequences of
230:rather than directed multigraphs.
14:
2254:drops the last, and the face map
211:-th face are the vertices of the
203: + 1 faces (which are (
127:which turns simplicial sets into
4717:
4707:
4698:
4697:
4450:
4059:University of Chicago Press 1967
3624:{\displaystyle \Omega ^{2}B(BG)}
1297:{\displaystyle X_{n}\to X_{n+1}}
904:{\displaystyle X_{n}\to X_{n-1}}
457:. The objects of Δ are nonempty
2389:-simplices of a simplicial set
2092:+1) sequences of elements from
344:). We think of the elements of
3929:Methods of Homological Algebra
3618:
3609:
3104:-simplices. Furthermore, the
2867:
2835:
2825:
2810:
2681:
2605:
2561:
2527:
2508:
2487:
2478:
2475:
2469:
2460:
2444:
2432:
2429:
2423:
2420:
1965:
1900:
1275:
1134:
1128:
1125:
1122:
1110:
1057:
1051:
1048:
1045:
1033:
882:
741:
735:
732:
729:
717:
664:
658:
655:
652:
640:
437:. Rather than requiring these
1:
2799:in general position given by
2768:is replaced by a topological
2583:natural transformations) Δ →
2221: → ... →
1096:{\displaystyle \sigma ^{n,i}}
703:{\displaystyle \delta ^{n,i}}
224:abstract simplicial complexes
4020:10.1016/0040-9383(74)90022-6
3457:. The basic idea is that if
2750:Hausdorff topological spaces
2635:shows that a simplicial set
4392:Constructions on categories
3801:Abstract simplicial complex
3671:is an infinite loop space.
3348:, we obtain the categories
3292:is a contravariant functor
3035:via geometric realization.
2172:and a single morphism from
233:Formally, a simplicial set
4775:
4499:Higher-dimensional algebra
3889:Simplicial Homotopy Theory
3743:derived algebraic geometry
3346:category of abelian groups
3320:where Δ still denotes the
3155:and any topological space
3031:corresponds to the one in
2579:, whose objects are maps (
528:.) Given a simplicial set
479:order-preserving functions
4693:
4472:
4459:
4448:
4123:
4112:
3897:10.1007/978-3-0348-8707-6
3866:Goerss & Jardine 1999
3539:{\displaystyle \Omega BG}
3381:and is given by functors
3223:simplicial model category
2302:We can recover the poset
403:-simplex the degenerate (
27:is an object composed of
3855:Gelfand & Manin 2013
3244:inducing an equivalence
3039:Singular set for a space
2741:taking a simplicial set
2727:There is a functor |•|:
2203:composable morphisms in
2195:-simplices of the nerve
2071:-simplices of the nerve
578:of that simplicial set.
237:is a collection of sets
228:simple undirected graphs
82:, and the corresponding
4309:Cokernels and quotients
4232:Universal constructions
3947:Allegretti, Dylan G.L.
3786:Dold–Kan correspondence
3375:Dold–Kan correspondence
3242:closed model categories
3151:for any simplicial set
3076:) for each object ∈ Δ.
3047:of a topological space
2776:dimensional subset of (
2325:of a topological space
2075:, i.e. the elements of
1357:{\displaystyle s_{n,i}}
1251:{\displaystyle s_{n,i}}
964:{\displaystyle d_{n,i}}
858:{\displaystyle d_{n,i}}
558:on Δ. As such, it is a
552:natural transformations
4466:Higher category theory
4212:Natural transformation
3739:higher category theory
3728:
3708:
3688:
3665:
3645:
3625:
3583:
3563:
3540:
3514:
3494:
3471:
3445:André–Quillen homology
3051:is the simplicial set
2948:
2706:
2617:
2573:
2534:
2152:, to obtain the nerve
1988:
1923:
1839:
1763:
1679:
1624:
1544:
1434:
1358:
1325:
1298:
1252:
1225:respectively, so that
1219:
1161:
1141:
1097:
1064:
978:of the simplicial set
965:
932:
905:
859:
832:respectively, so that
826:
768:
748:
704:
671:
95:higher category theory
59:and Joseph A. Zilber.
37:partially ordered sets
3822:Annals of Mathematics
3729:
3709:
3689:
3666:
3646:
3626:
3584:
3564:
3541:
3515:
3495:
3472:
3187:In order to define a
2949:
2739:geometric realization
2723:Geometric realization
2707:
2633:following isomorphism
2618:
2574:
2549:category of simplices
2535:
2263:for 0 <
2008:partially ordered set
1989:
1924:
1840:
1764:
1680:
1625:
1545:
1448:. (This is short for
1435:
1369:simplicial identities
1359:
1326:
1324:{\displaystyle s_{i}}
1299:
1253:
1220:
1162:
1142:
1098:
1065:
966:
933:
931:{\displaystyle d_{i}}
906:
860:
827:
769:
749:
705:
672:
493:contravariant functor
439:simplicial identities
374:assigns to each such
137:algebraic topologists
122:geometric realization
45:contravariant functor
4335:Algebraic categories
3922:Gelfand, Sergei I.;
3718:
3698:
3678:
3655:
3635:
3593:
3573:
3550:
3524:
3504:
3481:
3461:
2806:
2760:) obtained if every
2650:
2599:
2555:
2401:
2064:: a simplicial set.
1933:
1868:
1783:
1707:
1644:
1568:
1452:
1378:
1335:
1308:
1262:
1229:
1171:
1151:
1140:{\displaystyle \to }
1107:
1074:
986:
942:
915:
869:
836:
778:
758:
747:{\displaystyle \to }
714:
681:
593:
585:of a simplicial set
481:between these sets.
386:-th vertex. The map
153:directed multigraphs
93:, a basic notion of
4504:Homotopy hypothesis
4182:Commutative diagram
3791:Simplicial homotopy
3771:Simplicial presheaf
3238:Quillen equivalence
3234:homotopical algebra
3017:categorical product
2772:simplex (a certain
2747:compactly-generated
2591:arising from maps
2110: ≤ ... ≤
1864:together with maps
226:, which generalize
215:-simplex minus the
68:geometric simplices
4744:Algebraic topology
4217:Universal property
3724:
3704:
3684:
3661:
3641:
3621:
3579:
3562:{\displaystyle BG}
3559:
3536:
3510:
3493:{\displaystyle BG}
3490:
3467:
3437:algebraic K-theory
3421:classifying spaces
3342:category of groups
3277:Simplicial objects
2944:
2702:
2688:
2668:
2643:of its simplices:
2613:
2569:
2530:
1984:
1919:
1835:
1759:
1675:
1620:
1540:
1430:
1354:
1321:
1294:
1248:
1215:
1157:
1137:
1093:
1060:
961:
928:
901:
855:
822:
764:
744:
700:
667:
518:covariant functors
141:algebraic geometry
99:simplicial objects
4731:
4730:
4689:
4688:
4685:
4684:
4667:monoidal category
4622:
4621:
4494:Enriched category
4446:
4445:
4442:
4441:
4419:Quotient category
4414:Opposite category
4329:
4328:
3983:Higher K-Theories
3939:978-3-662-12492-5
3906:978-3-7643-6064-1
3883:Goerss, Paul G.;
3796:Simplicial sphere
3727:{\displaystyle K}
3707:{\displaystyle X}
3687:{\displaystyle X}
3664:{\displaystyle G}
3644:{\displaystyle G}
3582:{\displaystyle G}
3513:{\displaystyle G}
3470:{\displaystyle G}
3368:Simplicial groups
3326:opposite category
3283:simplicial object
3270:homotopy category
3098:singular homology
3027:, and the one in
2661:
2659:
2501:
2310:and the category
2278:and composes the
1673:
1160:{\displaystyle i}
767:{\displaystyle i}
522:opposite category
461:sets of the form
453:Let Δ denote the
449:Formal definition
191:. In general, an
167:and three arrows
84:homotopy category
64:topological space
4766:
4721:
4720:
4711:
4710:
4701:
4700:
4536:
4514:Simplex category
4489:Categorification
4480:
4461:
4454:
4424:Product category
4409:Kleisli category
4404:Functor category
4249:Terminal objects
4237:
4172:Adjoint functors
4125:
4114:
4099:
4092:
4085:
4076:
4046:
4044:
4023:
4002:Segal, Graeme B.
3997:
3965:
3963:
3953:
3943:
3918:
3885:Jardine, John F.
3869:
3863:
3857:
3852:
3846:
3845:
3817:
3747:Quasi-categories
3733:
3731:
3730:
3725:
3713:
3711:
3710:
3705:
3693:
3691:
3690:
3685:
3670:
3668:
3667:
3662:
3650:
3648:
3647:
3642:
3630:
3628:
3627:
3622:
3605:
3604:
3588:
3586:
3585:
3580:
3568:
3566:
3565:
3560:
3545:
3543:
3542:
3537:
3519:
3517:
3516:
3511:
3499:
3497:
3496:
3491:
3476:
3474:
3473:
3468:
3364:, respectively.
3334:category of sets
3322:simplex category
3106:singular functor
3013:cartesian closed
2953:
2951:
2950:
2945:
2931:
2930:
2909:
2908:
2890:
2889:
2878:
2866:
2865:
2847:
2846:
2828:
2823:
2822:
2813:
2711:
2709:
2708:
2703:
2701:
2700:
2687:
2680:
2679:
2669:
2622:
2620:
2619:
2614:
2612:
2595:in Δ. That is,
2578:
2576:
2575:
2570:
2568:
2547:gives rise to a
2539:
2537:
2536:
2531:
2520:
2519:
2504:
2503:
2502:
2456:
2455:
2413:
2412:
2374:} of the first (
2267: <
2238:.) The face map
2191:Concretely, the
2119:). The face map
2067:Concretely, the
2048:(φ) :
1993:
1991:
1990:
1985:
1983:
1982:
1964:
1963:
1951:
1950:
1928:
1926:
1925:
1920:
1918:
1917:
1899:
1898:
1886:
1885:
1844:
1842:
1841:
1836:
1834:
1833:
1824:
1823:
1805:
1804:
1795:
1794:
1768:
1766:
1765:
1760:
1758:
1757:
1742:
1741:
1729:
1728:
1719:
1718:
1684:
1682:
1681:
1676:
1674:
1671:
1666:
1665:
1656:
1655:
1629:
1627:
1626:
1621:
1619:
1618:
1609:
1608:
1590:
1589:
1580:
1579:
1549:
1547:
1546:
1541:
1539:
1538:
1523:
1522:
1492:
1491:
1476:
1475:
1439:
1437:
1436:
1431:
1429:
1428:
1419:
1418:
1400:
1399:
1390:
1389:
1363:
1361:
1360:
1355:
1353:
1352:
1330:
1328:
1327:
1322:
1320:
1319:
1303:
1301:
1300:
1295:
1293:
1292:
1274:
1273:
1257:
1255:
1254:
1249:
1247:
1246:
1224:
1222:
1221:
1216:
1214:
1213:
1189:
1188:
1166:
1164:
1163:
1158:
1146:
1144:
1143:
1138:
1102:
1100:
1099:
1094:
1092:
1091:
1069:
1067:
1066:
1061:
1029:
1028:
1004:
1003:
970:
968:
967:
962:
960:
959:
937:
935:
934:
929:
927:
926:
910:
908:
907:
902:
900:
899:
881:
880:
864:
862:
861:
856:
854:
853:
831:
829:
828:
823:
821:
820:
796:
795:
773:
771:
770:
765:
753:
751:
750:
745:
709:
707:
706:
701:
699:
698:
676:
674:
673:
668:
636:
635:
611:
610:
514:category of sets
459:linearly ordered
455:simplex category
399:assigns to each
91:quasi-categories
57:Samuel Eilenberg
53:category of sets
49:simplex category
4774:
4773:
4769:
4768:
4767:
4765:
4764:
4763:
4754:Simplicial sets
4749:Homotopy theory
4734:
4733:
4732:
4727:
4681:
4651:
4618:
4595:
4586:
4543:
4527:
4478:
4468:
4455:
4438:
4387:
4325:
4294:Initial objects
4280:
4226:
4119:
4108:
4106:Category theory
4103:
4042:
4034:
4031:
4029:Further reading
4026:
4000:
3994:
3974:Quillen, Daniel
3972:
3961:10.1.1.539.7411
3951:
3946:
3940:
3921:
3907:
3882:
3878:
3873:
3872:
3864:
3860:
3853:
3849:
3834:10.2307/1969364
3819:
3818:
3814:
3809:
3756:
3716:
3715:
3696:
3695:
3676:
3675:
3653:
3652:
3633:
3632:
3596:
3591:
3590:
3571:
3570:
3548:
3547:
3522:
3521:
3502:
3501:
3479:
3478:
3459:
3458:
3417:
3405:
3394:
3379:chain complexes
3279:
3230:Serre fibration
3189:model structure
3185:
3179:hang together.
3138:
3124:
3071:
3041:
3007:, the category
2971:
2922:
2900:
2873:
2857:
2838:
2814:
2804:
2803:
2725:
2692:
2671:
2648:
2647:
2597:
2596:
2553:
2552:
2511:
2493:
2447:
2404:
2399:
2398:
2369:
2356:
2337:
2314:from the nerve
2306:from the nerve
2294:
2276:
2262:
2253:
2244:
2229:
2220:
2213:
2141:duplicates the
2140:
2127:
2118:
2109:
2102:
2083:
2035:
2004:
1968:
1955:
1936:
1931:
1930:
1903:
1890:
1871:
1866:
1865:
1862:
1825:
1809:
1796:
1786:
1781:
1780:
1777: + 1.
1743:
1733:
1720:
1710:
1705:
1704:
1701: + 1.
1657:
1647:
1642:
1641:
1610:
1594:
1581:
1571:
1566:
1565:
1524:
1496:
1477:
1455:
1450:
1449:
1420:
1404:
1391:
1381:
1376:
1375:
1338:
1333:
1332:
1311:
1306:
1305:
1278:
1265:
1260:
1259:
1232:
1227:
1226:
1199:
1174:
1169:
1168:
1149:
1148:
1105:
1104:
1077:
1072:
1071:
1014:
989:
984:
983:
976:degeneracy maps
945:
940:
939:
918:
913:
912:
885:
872:
867:
866:
839:
834:
833:
806:
781:
776:
775:
756:
755:
712:
711:
684:
679:
678:
621:
596:
591:
590:
576:degeneracy maps
568:
537:
532:we often write
451:
443:category theory
436:
423:
398:
373:
352:
331:
321:
312:
299:degeneracy maps
284:
274:
265:
245:
149:
133:homotopy theory
107:
80:model structure
76:homotopy theory
33:directed graphs
17:
12:
11:
5:
4772:
4770:
4762:
4761:
4756:
4751:
4746:
4736:
4735:
4729:
4728:
4726:
4725:
4715:
4705:
4694:
4691:
4690:
4687:
4686:
4683:
4682:
4680:
4679:
4674:
4669:
4655:
4649:
4644:
4639:
4633:
4631:
4624:
4623:
4620:
4619:
4617:
4616:
4611:
4600:
4598:
4593:
4588:
4587:
4585:
4584:
4579:
4574:
4569:
4564:
4559:
4548:
4546:
4541:
4533:
4529:
4528:
4526:
4521:
4519:String diagram
4516:
4511:
4509:Model category
4506:
4501:
4496:
4491:
4486:
4484:
4477:
4476:
4473:
4470:
4469:
4464:
4457:
4456:
4449:
4447:
4444:
4443:
4440:
4439:
4437:
4436:
4431:
4429:Comma category
4426:
4421:
4416:
4411:
4406:
4401:
4395:
4393:
4389:
4388:
4386:
4385:
4375:
4365:
4363:Abelian groups
4360:
4355:
4350:
4345:
4339:
4337:
4331:
4330:
4327:
4326:
4324:
4323:
4318:
4313:
4312:
4311:
4301:
4296:
4290:
4288:
4282:
4281:
4279:
4278:
4273:
4268:
4267:
4266:
4256:
4251:
4245:
4243:
4234:
4228:
4227:
4225:
4224:
4219:
4214:
4209:
4204:
4199:
4194:
4189:
4184:
4179:
4174:
4169:
4168:
4167:
4162:
4157:
4152:
4147:
4142:
4131:
4129:
4121:
4120:
4117:
4110:
4109:
4104:
4102:
4101:
4094:
4087:
4079:
4073:
4072:
4063:simplicial set
4060:
4047:
4030:
4027:
4025:
4024:
4014:(3): 293–312.
3998:
3992:
3970:
3944:
3938:
3924:Manin, Yuri I.
3919:
3905:
3879:
3877:
3874:
3871:
3870:
3858:
3847:
3828:(3): 499–513.
3811:
3810:
3808:
3805:
3804:
3803:
3798:
3793:
3788:
3783:
3778:
3776:Quasi-category
3773:
3768:
3765:Dendroidal set
3762:
3755:
3752:
3723:
3703:
3683:
3660:
3640:
3620:
3617:
3614:
3611:
3608:
3603:
3599:
3578:
3558:
3555:
3535:
3532:
3529:
3509:
3489:
3486:
3466:
3416:
3413:
3412:
3411:
3403:
3396:
3395:
3392:
3362:abelian groups
3360:of simplicial
3352:of simplicial
3318:
3317:
3304:
3303:
3288:in a category
3278:
3275:
3266:
3265:
3213:Daniel Quillen
3197:Kan fibrations
3184:
3181:
3149:
3148:
3134:
3120:
3078:
3077:
3064:
3040:
3037:
2974:
2973:
2966:
2955:
2954:
2943:
2940:
2937:
2934:
2929:
2925:
2921:
2918:
2915:
2912:
2907:
2903:
2899:
2896:
2893:
2888:
2885:
2882:
2877:
2872:
2869:
2864:
2860:
2856:
2853:
2850:
2845:
2841:
2837:
2834:
2831:
2827:
2821:
2817:
2812:
2724:
2721:
2713:
2712:
2699:
2695:
2691:
2686:
2683:
2678:
2674:
2667:
2664:
2658:
2655:
2625:slice category
2611:
2607:
2604:
2567:
2563:
2560:
2529:
2526:
2523:
2518:
2514:
2510:
2507:
2496:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2454:
2450:
2446:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2419:
2416:
2411:
2407:
2367:
2355:
2348:
2333:
2290:
2274:
2258:
2249:
2242:
2225:
2218:
2211:
2136:
2123:
2114:
2107:
2100:
2079:
2031:
2003:
2000:
1981:
1978:
1975:
1971:
1967:
1962:
1958:
1954:
1949:
1946:
1943:
1939:
1916:
1913:
1910:
1906:
1902:
1897:
1893:
1889:
1884:
1881:
1878:
1874:
1860:
1855:
1854:
1832:
1828:
1822:
1819:
1816:
1812:
1808:
1803:
1799:
1793:
1789:
1778:
1756:
1753:
1750:
1746:
1740:
1736:
1732:
1727:
1723:
1717:
1713:
1702:
1669:
1664:
1660:
1654:
1650:
1639:
1617:
1613:
1607:
1604:
1601:
1597:
1593:
1588:
1584:
1578:
1574:
1563:
1537:
1534:
1531:
1527:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1499:
1495:
1490:
1487:
1484:
1480:
1474:
1471:
1468:
1465:
1462:
1458:
1427:
1423:
1417:
1414:
1411:
1407:
1403:
1398:
1394:
1388:
1384:
1351:
1348:
1345:
1341:
1318:
1314:
1291:
1288:
1285:
1281:
1277:
1272:
1268:
1245:
1242:
1239:
1235:
1212:
1209:
1206:
1202:
1198:
1195:
1192:
1187:
1184:
1181:
1177:
1156:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1090:
1087:
1084:
1080:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1027:
1024:
1021:
1017:
1013:
1010:
1007:
1002:
999:
996:
992:
958:
955:
952:
948:
925:
921:
898:
895:
892:
888:
884:
879:
875:
852:
849:
846:
842:
819:
816:
813:
809:
805:
802:
799:
794:
791:
788:
784:
763:
754:that "misses"
743:
740:
737:
734:
731:
728:
725:
722:
719:
697:
694:
691:
687:
666:
663:
660:
657:
654:
651:
648:
645:
642:
639:
634:
631:
628:
624:
620:
617:
614:
609:
606:
603:
599:
567:
564:
535:
506:
505:
486:simplicial set
471:
470:
465:= {0, 1, ...,
450:
447:
428:
415:
390:
365:
357:-simplices of
348:
326:
317:
304:
279:
270:
257:
241:
148:
145:
106:
103:
25:simplicial set
15:
13:
10:
9:
6:
4:
3:
2:
4771:
4760:
4757:
4755:
4752:
4750:
4747:
4745:
4742:
4741:
4739:
4724:
4716:
4714:
4706:
4704:
4696:
4695:
4692:
4678:
4675:
4673:
4670:
4668:
4664:
4660:
4656:
4654:
4652:
4645:
4643:
4640:
4638:
4635:
4634:
4632:
4629:
4625:
4615:
4612:
4609:
4605:
4602:
4601:
4599:
4597:
4589:
4583:
4580:
4578:
4575:
4573:
4570:
4568:
4567:Tetracategory
4565:
4563:
4560:
4557:
4556:pseudofunctor
4553:
4550:
4549:
4547:
4545:
4537:
4534:
4530:
4525:
4522:
4520:
4517:
4515:
4512:
4510:
4507:
4505:
4502:
4500:
4497:
4495:
4492:
4490:
4487:
4485:
4481:
4475:
4474:
4471:
4467:
4462:
4458:
4453:
4435:
4432:
4430:
4427:
4425:
4422:
4420:
4417:
4415:
4412:
4410:
4407:
4405:
4402:
4400:
4399:Free category
4397:
4396:
4394:
4390:
4383:
4382:Vector spaces
4379:
4376:
4373:
4369:
4366:
4364:
4361:
4359:
4356:
4354:
4351:
4349:
4346:
4344:
4341:
4340:
4338:
4336:
4332:
4322:
4319:
4317:
4314:
4310:
4307:
4306:
4305:
4302:
4300:
4297:
4295:
4292:
4291:
4289:
4287:
4283:
4277:
4276:Inverse limit
4274:
4272:
4269:
4265:
4262:
4261:
4260:
4257:
4255:
4252:
4250:
4247:
4246:
4244:
4242:
4238:
4235:
4233:
4229:
4223:
4220:
4218:
4215:
4213:
4210:
4208:
4205:
4203:
4202:Kan extension
4200:
4198:
4195:
4193:
4190:
4188:
4185:
4183:
4180:
4178:
4175:
4173:
4170:
4166:
4163:
4161:
4158:
4156:
4153:
4151:
4148:
4146:
4143:
4141:
4138:
4137:
4136:
4133:
4132:
4130:
4126:
4122:
4115:
4111:
4107:
4100:
4095:
4093:
4088:
4086:
4081:
4080:
4077:
4071:
4069:
4064:
4061:
4058:
4056:
4051:
4050:May, J. Peter
4048:
4041:
4037:
4033:
4032:
4028:
4021:
4017:
4013:
4009:
4008:
4003:
3999:
3995:
3993:3-540-06434-6
3989:
3985:
3984:
3979:
3975:
3971:
3968:
3962:
3957:
3950:
3945:
3941:
3935:
3931:
3930:
3925:
3920:
3916:
3912:
3908:
3902:
3898:
3894:
3890:
3886:
3881:
3880:
3875:
3867:
3862:
3859:
3856:
3851:
3848:
3843:
3839:
3835:
3831:
3827:
3823:
3816:
3813:
3806:
3802:
3799:
3797:
3794:
3792:
3789:
3787:
3784:
3782:
3779:
3777:
3774:
3772:
3769:
3766:
3763:
3761:
3758:
3757:
3753:
3751:
3748:
3744:
3740:
3735:
3721:
3701:
3681:
3672:
3658:
3638:
3615:
3612:
3606:
3601:
3576:
3556:
3553:
3533:
3530:
3507:
3487:
3484:
3464:
3456:
3451:
3448:
3446:
3442:
3438:
3434:
3430:
3426:
3422:
3414:
3409:
3401:
3400:
3399:
3390:
3387:
3384:
3383:
3382:
3380:
3376:
3371:
3369:
3365:
3363:
3359:
3355:
3351:
3347:
3343:
3339:
3335:
3331:
3327:
3323:
3316:
3312:
3309:
3308:
3307:
3302:
3298:
3295:
3294:
3293:
3291:
3287:
3284:
3276:
3274:
3271:
3263:
3259:
3255:
3251:
3247:
3246:
3245:
3243:
3239:
3235:
3231:
3226:
3224:
3221:
3218:
3214:
3210:
3206:
3202:
3198:
3194:
3190:
3182:
3180:
3178:
3174:
3170:
3166:
3162:
3158:
3154:
3146:
3142:
3137:
3133:) ≅ hom
3132:
3128:
3123:
3118:
3117:
3116:
3114:
3113:right adjoint
3110:
3107:
3103:
3099:
3095:
3091:
3087:
3083:
3075:
3070:
3069:
3062:
3058:
3057:
3056:
3054:
3050:
3046:
3038:
3036:
3034:
3030:
3026:
3022:
3018:
3014:
3010:
3006:
3002:
2998:
2994:
2989:
2987:
2983:
2979:
2970:
2964:
2963:
2962:
2960:
2941:
2935:
2932:
2927:
2923:
2919:
2916:
2913:
2910:
2905:
2901:
2897:
2894:
2891:
2886:
2883:
2880:
2870:
2862:
2858:
2854:
2851:
2848:
2843:
2839:
2829:
2819:
2802:
2801:
2800:
2798:
2794:
2789:
2787:
2783:
2779:
2775:
2771:
2767:
2763:
2759:
2755:
2751:
2748:
2744:
2740:
2736:
2733:
2730:
2722:
2720:
2718:
2697:
2689:
2684:
2676:
2665:
2662:
2656:
2653:
2646:
2645:
2644:
2642:
2638:
2634:
2630:
2626:
2609:
2594:
2590:
2586:
2582:
2565:
2551:, denoted by
2550:
2546:
2543:Furthermore,
2541:
2524:
2521:
2516:
2505:
2494:
2490:
2484:
2481:
2472:
2466:
2463:
2457:
2448:
2441:
2438:
2435:
2426:
2417:
2414:
2409:
2405:
2396:
2392:
2388:
2384:
2379:
2377:
2373:
2365:
2363:
2353:
2350:The standard
2349:
2347:
2345:
2341:
2336:
2332:
2328:
2324:
2319:
2317:
2313:
2309:
2305:
2300:
2298:
2293:
2289:
2285:
2281:
2277:
2270:
2266:
2261:
2257:
2252:
2248:
2241:
2237:
2233:
2228:
2224:
2217:
2214: →
2210:
2206:
2202:
2198:
2194:
2189:
2187:
2184: ≤
2183:
2179:
2175:
2171:
2167:
2163:
2159:
2155:
2151:
2146:
2145:-th element.
2144:
2139:
2135:
2131:
2126:
2122:
2117:
2113:
2106:
2103: ≤
2099:
2095:
2091:
2087:
2082:
2078:
2074:
2070:
2065:
2063:
2059:
2055:
2051:
2047:
2043:
2039:
2034:
2029:
2025:
2021:
2017:
2013:
2009:
2001:
1999:
1997:
1979:
1976:
1973:
1969:
1960:
1956:
1952:
1947:
1944:
1941:
1937:
1914:
1911:
1908:
1904:
1895:
1891:
1887:
1882:
1879:
1876:
1872:
1863:
1852:
1848:
1830:
1826:
1820:
1817:
1814:
1810:
1806:
1801:
1797:
1791:
1787:
1779:
1776:
1772:
1754:
1751:
1748:
1744:
1738:
1734:
1730:
1725:
1721:
1715:
1711:
1703:
1700:
1696:
1692:
1688:
1667:
1662:
1658:
1652:
1648:
1640:
1637:
1633:
1615:
1611:
1605:
1602:
1599:
1595:
1591:
1586:
1582:
1576:
1572:
1564:
1561:
1557:
1553:
1535:
1532:
1529:
1525:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1497:
1493:
1488:
1485:
1482:
1478:
1472:
1469:
1466:
1463:
1460:
1456:
1447:
1443:
1425:
1421:
1415:
1412:
1409:
1405:
1401:
1396:
1392:
1386:
1382:
1374:
1373:
1372:
1370:
1365:
1349:
1346:
1343:
1339:
1316:
1312:
1289:
1286:
1283:
1279:
1270:
1266:
1243:
1240:
1237:
1233:
1210:
1207:
1204:
1200:
1196:
1193:
1190:
1185:
1182:
1179:
1175:
1154:
1131:
1119:
1116:
1113:
1088:
1085:
1082:
1078:
1054:
1042:
1039:
1036:
1030:
1025:
1022:
1019:
1015:
1011:
1008:
1005:
1000:
997:
994:
990:
981:
977:
972:
956:
953:
950:
946:
923:
919:
896:
893:
890:
886:
877:
873:
850:
847:
844:
840:
817:
814:
811:
807:
803:
800:
797:
792:
789:
786:
782:
761:
738:
726:
723:
720:
695:
692:
689:
685:
661:
649:
646:
643:
637:
632:
629:
626:
622:
618:
615:
612:
607:
604:
601:
597:
588:
584:
579:
577:
573:
565:
563:
561:
557:
553:
549:
544:
542:
538:
531:
527:
523:
519:
515:
511:
504:
500:
497:
496:
495:
494:
490:
487:
482:
480:
476:
468:
464:
463:
462:
460:
456:
448:
446:
444:
440:
435:
431:
427:
422:
418:
414:
410:
406:
402:
397:
393:
389:
385:
381:
378:-simplex its
377:
372:
368:
364:
360:
356:
351:
347:
343:
340: ≤
339:
335:
329:
325:
320:
316:
313: :
311:
307:
303:
300:
296:
293: ≤
292:
288:
282:
278:
275: →
273:
269:
266: :
264:
260:
256:
253:
249:
244:
240:
236:
231:
229:
225:
220:
218:
214:
210:
206:
202:
198:
194:
190:
187: →
186:
182:
179: →
178:
174:
171: →
170:
166:
162:
158:
154:
146:
144:
142:
138:
134:
130:
126:
123:
118:
116:
112:
104:
102:
100:
96:
92:
87:
85:
81:
77:
73:
69:
65:
60:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
4647:
4628:Categorified
4532:n-categories
4483:Key concepts
4321:Direct limit
4304:Coequalizers
4222:Yoneda lemma
4128:Key concepts
4118:Key concepts
4067:
4053:
4036:Riehl, Emily
4011:
4005:
3982:
3966:
3932:. Springer.
3928:
3888:
3861:
3850:
3825:
3821:
3815:
3736:
3673:
3452:
3449:
3441:Fields Medal
3429:Grothendieck
3418:
3407:
3397:
3388:
3385:
3372:
3366:
3357:
3349:
3337:
3329:
3319:
3314:
3310:
3305:
3300:
3299: : Δ →
3296:
3289:
3285:
3282:
3280:
3268:between the
3267:
3261:
3257:
3253:
3249:
3227:
3209:monomorphism
3186:
3176:
3172:
3168:
3164:
3160:
3156:
3152:
3150:
3144:
3140:
3135:
3130:
3126:
3121:
3108:
3105:
3101:
3093:
3089:
3085:
3081:
3079:
3073:
3067:
3065:
3060:
3055:defined by
3052:
3048:
3045:singular set
3044:
3042:
3032:
3028:
3024:
3020:
3008:
3004:
3000:
2996:
2992:
2990:
2985:
2981:
2975:
2968:
2958:
2956:
2792:
2790:
2785:
2781:
2777:
2773:
2769:
2765:
2761:
2753:
2742:
2738:
2734:
2731:
2728:
2726:
2716:
2714:
2636:
2628:
2592:
2588:
2584:
2580:
2544:
2542:
2394:
2390:
2386:
2383:Yoneda lemma
2380:
2375:
2371:
2361:
2359:
2357:
2351:
2343:
2342:-simplex to
2339:
2334:
2330:
2326:
2322:
2320:
2315:
2311:
2307:
2303:
2301:
2296:
2291:
2287:
2283:
2279:
2272:
2268:
2264:
2259:
2255:
2250:
2246:
2239:
2235:
2231:
2226:
2222:
2215:
2208:
2204:
2200:
2196:
2192:
2190:
2185:
2181:
2177:
2173:
2169:
2165:
2161:
2157:
2153:
2149:
2147:
2142:
2137:
2133:
2129:
2124:
2120:
2115:
2111:
2104:
2097:
2093:
2089:
2085:
2080:
2076:
2072:
2068:
2066:
2061:
2057:
2053:
2049:
2045:
2041:
2037:
2032:
2027:
2023:
2015:
2011:
2005:
1995:
1858:
1856:
1850:
1846:
1774:
1770:
1698:
1694:
1690:
1686:
1635:
1631:
1559:
1555:
1551:
1445:
1441:
1368:
1366:
1147:that "hits"
979:
975:
973:
586:
582:
580:
575:
571:
569:
547:
545:
540:
533:
529:
525:
509:
507:
502:
501: : Δ →
498:
488:
485:
483:
474:
472:
466:
452:
438:
433:
429:
425:
420:
416:
412:
408:
404:
400:
395:
391:
387:
383:
379:
375:
370:
366:
362:
358:
354:
349:
345:
341:
337:
333:
327:
323:
318:
314:
309:
305:
301:
298:
294:
290:
286:
280:
276:
271:
267:
262:
258:
254:
251:
247:
242:
238:
234:
232:
221:
216:
212:
208:
204:
200:
196:
192:
188:
184:
180:
176:
172:
168:
164:
160:
156:
150:
121:
119:
115:CW complexes
108:
98:
88:
72:well-behaved
61:
28:
24:
18:
4596:-categories
4572:Kan complex
4562:Tricategory
4544:-categories
4434:Subcategory
4192:Exponential
4160:Preadditive
4155:Pre-abelian
3978:Bass, Hyman
3868:, p. 7
3781:Kan complex
3631:. In case
3435:'s work of
3207:if it is a
3205:cofibration
3163:to a space
3003:and unlike
2961:by setting
2764:simplex of
2737:called the
1331:instead of
938:instead of
539:instead of
21:mathematics
4738:Categories
4614:3-category
4604:2-category
4577:∞-groupoid
4552:Bicategory
4299:Coproducts
4259:Equalizers
4165:Bicategory
3876:References
3455:loop space
3402:Γ: Ch
3193:fibrations
2976:where the
2758:CW complex
2627:of Δ over
2128:drops the
556:presheaves
361:. The map
105:Motivation
41:categories
4663:Symmetric
4608:2-functor
4348:Relations
4271:Pullbacks
3956:CiteSeerX
3760:Delta set
3598:Ω
3528:Ω
3324:and the
3088:) :
3063:)() = hom
2965:|X| = lim
2920:∑
2911:≤
2898:≤
2871:∈
2852:…
2816:Δ
2788:is lost.
2694:Δ
2690:
2682:→
2673:Δ
2666:→
2657:≅
2606:↓
2603:Δ
2562:↓
2559:Δ
2513:Δ
2506:
2464:−
2458:
2453:Δ
2442:
2436:≅
2360:standard
2180:whenever
1966:→
1912:−
1901:→
1752:−
1603:−
1517:−
1505:−
1464:−
1413:−
1276:→
1258:is a map
1194:…
1126:→
1079:σ
1049:→
1031::
1016:σ
1009:…
991:σ
894:−
883:→
865:is a map
801:…
733:→
724:−
686:δ
656:→
647:−
638::
623:δ
616:…
598:δ
583:face maps
572:face maps
520:from the
252:face maps
147:Intuition
111:simplices
47:from the
29:simplices
4759:Functors
4723:Glossary
4703:Category
4677:n-monoid
4630:concepts
4286:Colimits
4254:Products
4207:Morphism
4150:Concrete
4145:Additive
4135:Category
4007:Topology
3926:(2013).
3887:(1999).
3754:See also
3674:Even if
3406:→
2364:-simplex
2282:th and (
2160:. Here,
2030:() = hom
2006:Given a
2002:Examples
1070:, where
677:, where
4713:Outline
4672:n-group
4637:2-group
4592:Strict
4582:∞-topos
4378:Modules
4316:Pushout
4264:Kernels
4197:Functor
4140:Abelian
4065:at the
3980:(ed.).
3915:1711612
3842:1969364
3500:, then
3433:Quillen
3340:be the
3332:is the
3328:. When
2978:colimit
2797:simplex
2641:colimit
2381:By the
2329:. Here
1550:if 0 ≤
512:is the
353:as the
125:functor
51:to the
4659:Traced
4642:2-ring
4372:Fields
4358:Groups
4353:Magmas
4241:Limits
3990:
3958:
3936:
3913:
3903:
3840:
3425:groups
3354:groups
3313:: Δ →
3220:closed
3217:proper
3195:to be
3086:φ
3072:(|Δ|,
3029:CGHaus
3025:CGHaus
3015:; the
3009:CGHaus
2993:CGHaus
2735:CGHaus
2631:. The
2385:, the
2271:drops
2033:po-set
2018:, the
508:where
297:) and
4653:-ring
4540:Weak
4524:Topos
4368:Rings
4043:(PDF)
3952:(PDF)
3838:JSTOR
3807:Notes
3546:. If
3248:|•|:
3092:() →
2639:is a
2623:is a
2397:i.e.
2052:() →
2036:( ,
2020:nerve
1773:>
1634:<
1554:<
1444:<
560:topos
524:Δ to
491:is a
473:with
4343:Sets
3988:ISBN
3934:ISBN
3901:ISBN
3741:and
3398:and
3391:→ Ch
3356:and
3350:sGrp
3256:) ↔
3254:sSet
3136:sSet
3043:The
3033:sSet
3023:and
3001:sSet
2986:sSet
2972:| Δ|
2967:Δ →
2729:sSet
2581:i.e.
2500:sSet
2358:The
2096:: (
1929:and
974:The
581:The
574:and
548:sSet
543:().
424:and
183:and
39:and
23:, a
4187:End
4177:CCC
4070:Lab
4016:doi
3893:doi
3830:doi
3423:of
3408:sAb
3389:sAb
3358:sAb
3344:or
3262:Top
3240:of
3129:|,
3122:Top
3119:hom
3111:is
3021:Top
3011:is
3005:Top
2997:Top
2663:lim
2495:hom
2449:hom
2439:Nat
2176:to
2156:of
2062:Set
2022:of
1845:if
1769:if
1693:or
1685:if
1630:if
1440:if
526:Set
510:Set
503:Set
19:In
4740::
4665:)
4661:)(
4052:.
4038:.
4012:13
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