Vogt's theorem on categories of homotopy coherent diagrams

Cordier, Jean-Marc; Porter, Timothy

doi:10.1017/s0305004100065877

Cited by 69 publications

(69 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark. This corollary is the result mentioned in [21] for extending that version of Vogt's theorem to the case where B is a locally Kan full sub S-category of a complete S-category and the rectification used in the proof (which was an example of a coherent end) involves fibrant objects, since B is locally Kan. This use of subcategories of fibrant objects mirrors that given in Quillen's theory, the idea being that on fibrant (cofibrant) objects, the homotopy theory is more easy to manipulate.…”

Section: Proposition 21 If a Is An S-category And Tsupporting

confidence: 73%

“…simplicial descriptions of homotopy coherence, [16]; 2. Vogt's theorem, [52], interpreting homotopy categories of diagrams as categories of coherent diagrams [21], see also [20] and more recently, [7]; 3. rectifications of coherent diagrams, [18] and [23]; 4. simplicial formulation of homotopy limits, [10] and [18]; 5. descriptions of Steenrod homology, [19] and [27]; 6. ideas independently developed by Heller [33] and others; 7. geometric constructions in strong shape theory, cf. [37] and [32], and of course, 8.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Homotopy coherent category theory

Cordier

Porter

1997

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. This article is an introduction to the categorical theory of homotopy coherence. It is based on the construction of the homotopy coherent analogues of end and coend, extending ideas of Meyer and others. The paper aims to develop homotopy coherent analogues of many of the results of elementary category theory, in particular it handles a homotopy coherent form of the Yoneda lemma and of Kan extensions. This latter area is linked with the theory of generalised derived functors.In homotopy theory, one often needs to "do" universal algebra "up to homotopy" for instance in the theory of homotopy everything H-spaces. Universal algebra nowadays is most easily expressed in categorical terms, so this calls for a form of category theory "up to homotopy" (cf. Heller [34]).In studying the homotopy theory of compact metric spaces, there is the classically known complication that the assignment of the nerve of an open cover to a cover is only a functor up to homotopy. Thus in shape theory, [39], one does not have a very rich underlying homotopy theory. Strong shape (cf. Lisica and Mardešić, [37]) and Steenrod homotopy (cf. Edwards and Hastings [27]) have a richer theory but at the cost of much harder proofs. Shape theory has been interpreted categorically in an elegant way. This provides an overview of most aspects of the theory, as well as the basic proobject formulation that it shares withétale homotopy and the theory of derived categories. Can one perform a similar process with strong shape and hence provide tools for enriching that theory, rigidifyingétale homotopy and enriching derived categories? We note, in particular, Grothendieck's plan for the theory of derived categories, sketched out in 'Pursuing Stacks', [31], which bears an uncanny ressemblance to the view of homotopy theory put forward by Heller in [34].Grothendieck's 'Pursuing Stacks' program, in fact, again raises the spectre of doing categorical construction 'up to homotopy', as his image of a stack is as a sheaf 'up to homotopy' in which the stalks are algebraic models of homotopy types and the whole object has geometric meaning.Problems of homotopy coherence also arise naturally in studying equivariant homotopy theory. If G is a group then the equivariant homotopy of G-complexes can be studied in a useful way by translating to a category of diagrams indexed by the orbit category of G, that is the full subcategory of G-sets determined by If however G is a topological group then this category is more naturally considered to be simplicially enriched and the equivariant homotopy theory of the G-complexes ends up benefitting from results on homotopy coherence to handle the bar-resolutions etc. used to translate from the equivariant setting to the category of diagrams; see [24] and the references it contains. Again the equivariant theory looks like the discrete case but with the categorical arguments done 'up to homotopy'.In this paper we try to lay some of the foundations of such a theory of categories 'up to homotopy' or more exactly 'up to coherent ...

show abstract

Section: Proposition 21 If a Is An S-category And Tsupporting

confidence: 73%

mentioning

confidence: 99%

Homotopy coherent category theory

Cordier

Porter

1997

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Quasi-categories have been extensively studied by Cordier and Porter [6], by Joyal [11; 12], and by Lurie [13]. If K is a quasi-category and x and y are two objects of K , then one may associate a "mapping space" K.x; y/ which is a simplicial set.…”

Section: Introductionmentioning

confidence: 99%

Rigidification of quasi-categories

Dugger

Spivak

2011

Algebr. Geom. Topol.

View full text Add to dashboard Cite

We give a new construction for rigidifying a quasi-category into a simplicial category, and prove that it is weakly equivalent to the rigidification given by Lurie. Our construction comes from the use of necklaces, which are simplicial sets obtained by stringing simplices together. As an application of these methods, we use our model to reprove some basic facts from Lurie [13] about the rigidification process.

show abstract

“…A special case of this for simplicial categories was proved by Cordier and Porter [7]. Remark 7.3 Inner Kan simplicial sets and inner Kan dendroidal sets are related as follows.…”

Section: The Inner Kan Condition For Dendroidal Setsmentioning

confidence: 93%

Dendroidal sets

Moerdijk

Weiss

2007

Algebr. Geom. Topol.

View full text Add to dashboard Cite

We introduce the concept of a dendroidal set. This is a generalization of the notion of a simplicial set, specially suited to the study of (coloured) operads in the context of homotopy theory. We define a category of trees, which extends the category used in simplicial sets, whose presheaf category is the category of dendroidal sets. We show that there is a closed monoidal structure on dendroidal sets which is closely related to the Boardman-Vogt tensor product of (coloured) operads. Furthermore, we show that each (coloured) operad in a suitable model category has a coherent homotopy nerve which is a dendroidal set, extending another construction of Boardman and Vogt. We also define a notion of an inner Kan dendroidal set, which is closely related to simplicial Kan complexes. Finally, we briefly indicate the theory of dendroidal objects in more general monoidal categories, and outline several of the applications and further theory of dendroidal sets.

show abstract

Vogt's theorem on categories of homotopy coherent diagrams

Cited by 69 publications

References 16 publications

Homotopy coherent category theory

Homotopy coherent category theory

Rigidification of quasi-categories

Dendroidal sets

Contact Info

Product

Resources

About