## S-categories, S-groupoids, Segal categories and quasicategories (2008)

### BibTeX

@MISC{Porter08s-categories,s-groupoids,,

author = {Timothy Porter},

title = {S-categories, S-groupoids, Segal categories and quasicategories},

year = {2008}

}

### OpenURL

### Abstract

The notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Laguña, the Canary Islands, in September, 2003. They assume the audience knows some abstract homotopy theory and as Heiner Kamps was in the audience in Hagen, it is safe to assume that the notes assume a reasonable knowledge of our book, [26], or any equivalent text if one can be found! What do the notes set out to do? “Aims and Objectives! ” or should it be “Learning Outcomes”? • To revisit some oldish material on abstract homotopy and simplicially enriched categories, that seems to be being used in today’s resurgence of interest in the area and to try to view it in a new light, or perhaps from new directions; • To introduce Segal categories and various other tools used by the Nice-Toulouse group of abstract homotopy theorists and link them into some of the older ideas; • To introduce Joyal’s quasicategories, (previously called weak Kan complexes but I agree with André that his nomenclature is better so will adopt it) and show how that theory links in with some old ideas of Boardman and Vogt, Dwyer and Kan, and Cordier and myself; • To ask lots of questions of myself and of the reader. The notes include some material from the ‘Cubo ’ article, [35], which was itself based on notes for a course at the Corso estivo Categorie e Topologia in 1991, but the overlap has been kept as small as is feasible as the purpose and the audience of the two sets of notes are different and the abstract homotopy theory has ‘moved on’, in part, to try the new methods out on those same ‘old ’ problems and to attack new ones as well. As usual when you try to specify ‘learning outcomes ’ you end up asking who has done the learning, the audience? Perhaps. The lecturer, most certainly! 1

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Citation Context ...ings like this for any splitting of {0, 1, . . ., n} of the form {0.1. . . .,k} and {k, . . .,n}. 4 Dwyer-Kan Hammock Localisation: more simplicially enriched categories. In his original contribution =-=[36]-=- to abstract homotopy theory, Quillen introduced the notion of a model category. Such a context is a category, C, together with three classes of maps: weak equivalences, W = Cw.e.; fibrations, fib = C... |

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Citation Context ...e constructions, to form spectra for studying corresponding cohomology theories and to help ‘delooping’ spaces where appropriate. Various approaches had been tried, notably that of Boardman and Vogt, =-=[2]-=-. In each case the idea was to mirror the homotopy coherent algebraic structures that occurred in loop spaces, etc. As an example of the problem, Segal mentions the following: Suppose C is a category ... |

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Citation Context ...just the underlying simplicial space, but suggests that the classifying space might be thought of as a bisimplicial space. 23(ii) It is worth recalling the structure of Lawvere’s algebraic theories, =-=[28]-=-, for comparison. One way to view these is to use some elementary ideas from topos theory. The category of finite sets is given the structure of a basic algebraic site T0 by taking epimorphic families... |

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Citation Context ... paper, [44]. Segal’s student Leitch used a similar construction to describe a homotopy commutative cube (actually a homotopy coherent cube), cf. [29], and this was used by Segal in his famous paper, =-=[37]-=-, under the name of the ‘explosion’ of A. All this was still in the topological framework and the link with the comonad resolution was still not in evidence. Although it seems likely that Kan knew of ... |

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Citation Context ...and the link with the comonad resolution was still not in evidence. Although it seems likely that Kan knew of this link between homotopy coherence and the comonadic resolutions by at least 1980, (cf. =-=[15]-=-), the construction does not seem to appear in his work with Dwyer as being linked with coherence until much later. Cordier made the link explicit in [8] and showed how Leitch and Segal’s work fitted ... |

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Citation Context ...ructure between different levels of the model. They are there in, say, the Conduché model (2-crossed modules) as the Peiffer lifting, (cf. Conduché, [7]) and in the Loday model, (crossed squares, cf. =-=[30]-=-), as the h-map. In a general dimension, n, there will be pairings like this for any splitting of {0, 1, . . ., n} of the form {0.1. . . .,k} and {k, . . .,n}. 4 Dwyer-Kan Hammock Localisation: more s... |

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Citation Context ... (C, W) satisfies any of the usual ‘calculus of fractions’ type conditions, then the homotopy type of those nerves already stabilises early on in the process (i.e. for small n). The argument given in =-=[16]-=- is indirect, so let us briefly see why one of these claims is true. Suppose that (C, W) satisfies a calculus of left fractions, then (i) whenever there is a diagram X ′ u ← X f → Y in C with u ∈ W, t... |

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Citation Context ... to notions of ‘thinness’ as used by Brown and Higgins in the study of crossed complexes and their relationship to ω-groupoids, see, for instance, [4], and also to Duskin’s ‘hypergroupoid’ condition, =-=[14]-=-. Another result that is sometimes useful, is a refinement of ‘groupoids give Kan complexes’:. The proof is ‘the same’: 12Lemma 4 Let A = Ner(C), the nerve of a category C. (i) Any (n, 0)-horn f : Λ ... |

34 |
T.: Abstract homotopy and simple homotopy theory
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Citation Context ...er, 2003. They assume the audience knows some abstract homotopy theory and as Heiner Kamps was in the audience in Hagen, it is safe to assume that the notes assume a reasonable knowledge of our book, =-=[26]-=-, or any equivalent text if one can be found! What do the notes set out to do? “Aims and Objectives!” or should it be “Learning Outcomes”? • To revisit some oldish material on abstract homotopy and si... |

33 |
Quasi-categories and Kan complexes
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(Show Context)
Citation Context ...iss out the zeroth or last faces. More exactly: Definition A simplicial set K is a weak Kan complex or quasi-category if for any n and 0 < k < n, any (n, k)-horn in K has a filler. Remarks (i) Joyal, =-=[25]-=-, uses the term inner horn for any (n, k)-horn in K with 0 < k < n. The two remaining cases are then conveniently called outer horns. (ii) For any space X, its singular complex, Sing(X) is given by Si... |

30 |
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Citation Context ...ese interchange squares give part of the pairing structure between different levels of the model. They are there in, say, the Conduché model (2-crossed modules) as the Peiffer lifting, (cf. Conduché, =-=[7]-=-) and in the Loday model, (crossed squares, cf. [30]), as the h-map. In a general dimension, n, there will be pairings like this for any splitting of {0, 1, . . ., n} of the form {0.1. . . .,k} and {k... |

26 |
R.W.: A homotopy 2–groupoid of a hausdorff space
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Citation Context ...-groupoid could be adapted, as can this bigroupoid case, to allow for a notion of thinness followed by a quotient to, say, a Gray groupoid, analogously to the way in which the HKK bigroupoid leads in =-=[23]-=-, to a 2-groupoid, or a double groupoid with connections. 7 Conclusion? We have looked at the way in which S-categories, their homotopy coherent form, Segal 1categories, and, perhaps, their iterated f... |

22 | Homotopy Coherent Category Theory - Cordier, Porter |

21 | A cellular nerve for higher categories
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Citation Context ...d as a category in the usual way. The cases n = 0 and n = 1 give no surprises. S[0] has one object 0 and S[0](0, 0 is isomorphic to ∆[0], as the only simplices are degenerate copies of the identity. S=-=[1]-=- likewise has a trivial simplicial structure, being just the category [1] considered as an S-category. Things do get more interesting at n = 2. The key here is the identification of S[2](0, 2). There ... |

20 |
T.: Vogt’s theorem on categories of homotopy coherent diagrams
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- 1986
(Show Context)
Citation Context ...c.(B), and so we are left with the first part. The following theorem was proved by Cordier and myself, but the idea was essentially in Boardman and Vogt’s lecture notes, like so much else! Theorem 1 (=-=[9]-=-) If B is a locally Kan S-category then Nerh.c.(B) is a quasi-category. � It seems to be the case that if B is only locally weakly Kan, then Nerh.c.(B) need not be a quasi-category. 16�� �� �� �� �� ... |

20 |
Homotopy theory and simplicial groupoids, Nederl
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Citation Context ...ced (that is, if K0 was a singleton), then the free group functor applied to K in a subtle way, gave a simplicial group whose homotopy groups were those of K, with a shift of dimension. With Dwyer in =-=[18]-=-, he gave the necessary variant of that construction to enable it to apply to the non-reduced case. This gives a ‘simplicial groupoid’ G(K) as follows: The object set of all the groupoids G(K)n will b... |

20 |
Homotopy limits and colimits
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Citation Context ...mark The history of this construction is interesting. A variant of it, but with topologically enriched categories as the end result, is in the work of Boardman and Vogt, [2] and also in Vogt’s paper, =-=[44]-=-. Segal’s student Leitch used a similar construction to describe a homotopy commutative cube (actually a homotopy coherent cube), cf. [29], and this was used by Segal in his famous paper, [37], under ... |

19 |
Vers une interprétation Galoisienne de la théorie de l’homotopie, Cahiers de top. et geom. diff. cat
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Citation Context ...l 1-category should be a weakened version of that of S-category and we have discussed them informally earlier. A more formal treatment is given in several places. The following is adapted from Toen’s =-=[42]-=-. Definitions • A Segal 1-precategory is specified by a functor A : ∆ op → S (i.e. a bisimplicial set) such that A0 is a constant simplicial set called the set of objects of A. • A morphism between tw... |

15 | Theory and applications of crossed complexes
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- 1994
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Citation Context ... get Cat(X, Y ), Cat(X, Y )n = Cat([n] × X, Y ). 3We leave the other structure up to the reader. (iv) Crs, the category of crossed complexes: see [26] for background and other references, and Tonks, =-=[43]-=- for a more detailed treatment of the simplicially enriched category structure; Crs(A, B) := Crs(π(n) ⊗ C, D) Composition has to be defined using an approximation to the identity, again see [43]. (v) ... |

13 |
la notion de diagramme homotopiquement cohérent. Cahiers Topologie Géom. Différentielle 23
- Cordier
- 1982
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Citation Context ...maps remove or insert brackets, but care must be taken when removing innermost brackets as the compositions that can then take place can result in chains with identities which then need removing, see =-=[8]-=-, that is why the comandic description is so much simpler, as it manages all that itself. To understand S(A) in general it pays to examine the simplest few cases. The key cases are when A = [n], the o... |

13 |
La théorie de l’homotopie de Grothendieck
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- 2005
(Show Context)
Citation Context ...s, and more generally derived Kan extensions, in a model category setting is still central to much of the work on abstract homotopy theory, (cf. Les Dérivateurs, by George Maltsiniotis, [31] see also =-=[32]-=-, Denis-Charles Cisinski’s thesis, and subsequent work, (cf. [5,6] and related papers), the resumé of Thomason’s note books published by Chuck Weibel, [46] and Carlos Simpson’s, [39], for example). In... |

10 |
R.W.: A homotopy bigroupoid of a topological space
- Hardie, Kamps, et al.
(Show Context)
Citation Context ...recisely a homotopy relative to the boundary of the squares in dimension (1, 1). This makes it look as if the Tamsamani bigroupoid is essentially the same as that of Hardie, Kamps and Kieboom, (HKK), =-=[24]-=-. It would be interesting to see if Tamsamani’s weak 3-groupoid could be adapted, as can this bigroupoid case, to allow for a notion of thinness followed by a quotient to, say, a Gray groupoid, analog... |

10 | Applications of Peiffer pairing in the Moore complex of a simplicial group
- Mutlu, Porter
- 1998
(Show Context)
Citation Context ... the way in which maps from S[4] to a S-category produce that interchange square? (The work of Ali Mutlu and myself on higher order Peiffer pairings in simplicial groups may be of relevance here, cf, =-=[33,34]-=-.) 2. The Tamsamani method starts with a space X and produces a multi-simplicial singular complex. That method could be applied to other types of object, for example, a simplicial group, a category, a... |

9 |
Images directes cohomologiques dans les catégories de modèles
- Cisinski
(Show Context)
Citation Context ...rious! A 4-simplex has 5 vertices, and hence 5 tetrahedral faces. Each of the 5 tetrahedral faces will contribute a square to the above diagram, yet a cube has 6 square faces! (Things get ‘worse’ in S=-=[5]-=-(0, 5), which is a 4-cube, so has 8 cubes as its faces, but ∆[5] has only 6 faces.) Back to the extra face, this is (01)(12)(24) (01)(12)(234) � � (01)(12)(23)(34) (012)(24) (012)(234) (012)(23)(34) (... |

7 | Propriétés universelles et extensions de Kan dérivées
- Cisinski
(Show Context)
Citation Context ... setting is still central to much of the work on abstract homotopy theory, (cf. Les Dérivateurs, by George Maltsiniotis, [31] see also [32], Denis-Charles Cisinski’s thesis, and subsequent work, (cf. =-=[5,6]-=- and related papers), the resumé of Thomason’s note books published by Chuck Weibel, [46] and Carlos Simpson’s, [39], for example). In a series of articles [15–17] published in 1980, Dwyer and Kan pro... |

6 |
2-groupoid enrichments in homotopy theory and algebra. K-Theory 25
- Kamps, H, et al.
(Show Context)
Citation Context ...omotopies ‘sliding’ over each other, but that homotopy commutative square is not a trivial one and does encode structure. One solution is to used Gray groupoids, (see the discussion and references in =-=[27]-=-, for instance). Another related one is to use 25tri-groupoids, the next level up from bicategories and bigroupoids. Tamsamani’s approach uses Segal category ideas to encode this necessary lack of ‘s... |

6 |
les notions de n-catégorie et n-groupoïde non strictes via des ensembles multi-simpliciaux, K-Theory
- Tamsamani, Sur
- 1999
(Show Context)
Citation Context ...f homotopy theory or of homotopy types that incorporate both geometric and algebraic aspects of the theory. Based on Segal-categories and his delooping machine of [37], one gets the Tamsamani models, =-=[41]-=-, for weak n-categories and weak n-groupoids for any n. 6 Tamsamani weak n-categories The problem of finding good n-categorical models for the geometric data encoded in a homotopy type came to the for... |

5 |
Homotopy homomorphisms and the hammock localization
- Schwänzl, Vogt
- 1992
(Show Context)
Citation Context ...calisation of B A with respect to level homotopy equivalences is ‘closely related to’ S(S(A), Nerh.c.(B)) for B locally Kan? If so ‘how close’? A lot of light on this problem has been shed by Vogt in =-=[45]-=- in the topological setting. The question would seem to be of particular importance given the upsurge of interest in A∞-categories resulting from new approaches to quantum deformation. 4. Can one cons... |

4 | Categorical Aspects of Equivariant Homotopy - Cordier, Porter - 1996 |

4 |
Pursuing stacks, manuscript
- Grothendieck
- 1983
(Show Context)
Citation Context ...nt could be substituted for it and so would still allow for a description of homotopy coherent diagrams in that context. This important viewpoint can also be traced to Grothendieck’s Pursuing Stacks, =-=[22]-=-. The DKS extension of the construction, [19], although simple to do, is often useful and so will be outlined next. If A is already a S-category, think of it as a simplicial category, then applying th... |

4 |
Abstract homotopy theory: the interaction of category theory and homotopy theory
- Porter
(Show Context)
Citation Context ...ks in with some old ideas of Boardman and Vogt, Dwyer and Kan, and Cordier and myself; • To ask lots of questions of myself and of the reader. The notes include some material from the ‘Cubo’ article, =-=[35]-=-, which was itself based on notes for a course at the Corso estivo Categorie e Topologia in 1991, but the overlap has been kept as small as is feasible as the purpose and the audience of the two sets ... |

3 | Maps between homotopy coherent diagrams - Cordier - 1988 |

3 |
The homotopy commutative cube
- Leitch
- 1974
(Show Context)
Citation Context ... the work of Boardman and Vogt, [2] and also in Vogt’s paper, [44]. Segal’s student Leitch used a similar construction to describe a homotopy commutative cube (actually a homotopy coherent cube), cf. =-=[29]-=-, and this was used by Segal in his famous paper, [37], under the name of the ‘explosion’ of A. All this was still in the topological framework and the link with the comonad resolution was still not i... |

3 | Homotopy ends and Thomason model categories
- Weibel
- 2001
(Show Context)
Citation Context ...teurs, by George Maltsiniotis, [31] see also [32], Denis-Charles Cisinski’s thesis, and subsequent work, (cf. [5,6] and related papers), the resumé of Thomason’s note books published by Chuck Weibel, =-=[46]-=- and Carlos Simpson’s, [39], for example). In a series of articles [15–17] published in 1980, Dwyer and Kan proposed a neat solution to this problem, simplicial localisations. We will limit ourselves ... |

1 |
Galois Theories, Cambridge Studies in Advanced Mathematics
- Borceux, Janelidze
- 1987
(Show Context)
Citation Context ...s are all ∆[0] or empty. It thus is possible to visualise S[2] as a copy of [2] with a 2-cell going towards the bottom: 1 � �� �� �� �� � ⇓ � �� � 0 � 2 The next case n = 3 is even more interesting. S=-=[3]-=-(i, j) will be (i) empty if j < i, (ii) isomorphic to ∆[0] if i = j or i = j − 1, (iii) isomorphic to ∆[1] by the same reasoning as we just saw for j = i + 2 and that leaves S[3](0, 3). This is a squa... |

1 | Fibrant diagrams, rectifiactions and a construction of Loday - Cordier, Porter - 1990 |

1 |
Homotopy Commutative Diagrams and their
- Dwyer, Kan, et al.
- 1989
(Show Context)
Citation Context ...diagrams of a given type. (We will return to this aspect a bit later in these notes, but an elementary introduction to this theory can be found in [26].) Finally Bill Dwyer, Dan Kan and Justin Smith, =-=[19]-=-, introduced a similar construction for an A which is an Scategory to start with, and motivated it by saying that S-functors with domain this S-category corresponded to ∞-homotopy commutative A-diagra... |

1 |
Introduction à la théorie des Dérivateurs, available at http://www.math.jussieu.fr/~maltsin
- Maltsiniotis
(Show Context)
Citation Context ...ts and colimits, and more generally derived Kan extensions, in a model category setting is still central to much of the work on abstract homotopy theory, (cf. Les Dérivateurs, by George Maltsiniotis, =-=[31]-=- see also [32], Denis-Charles Cisinski’s thesis, and subsequent work, (cf. [5,6] and related papers), the resumé of Thomason’s note books published by Chuck Weibel, [46] and Carlos Simpson’s, [39], fo... |

1 |
Limits in n-categories, available at: arXiv
- Simpson
(Show Context)
Citation Context ...is, [31] see also [32], Denis-Charles Cisinski’s thesis, and subsequent work, (cf. [5,6] and related papers), the resumé of Thomason’s note books published by Chuck Weibel, [46] and Carlos Simpson’s, =-=[39]-=-, for example). In a series of articles [15–17] published in 1980, Dwyer and Kan proposed a neat solution to this problem, simplicial localisations. We will limit ourselves to one of the two versions ... |

1 |
Effective generalized Siefert-van Kampen: how to calculate ΩX, available at:arXiv
- Simpson
(Show Context)
Citation Context ...s they do is best sought in the paper [37] by Segal, although it is not there but rather in [19] that they were introduced, but not named as such. (In fact their first naming seems to be in Simpson’s =-=[40]-=-.) In [37], one of the main aims was to get ‘up-to-homotopy’ models for algebraic structures so as to be able to iterate classifying space constructions, to form spectra for studying corresponding coh... |