nLab
relation between quasi-categories and simplicial categories

Contents

Idea

As a model for an (∞,1)-category a simplicially enriched category may be thought of as a semi-strictification of a quasi-category: composition along 0-cells is strictly associative and unital.

The Quillen equivalence between the Joyal model structure and the Dwyer–Kan–Bergner model structure

Our notation follows (Bergner).

There is a Quillen equivalence (due to Joyal (unpublished) and Lurie)

SCN˜sSet.SC \stackrel{\overset{\mathfrak{C}}{\leftarrow}}{\underset{\tilde N}{\to}} sSet.

Here SCSC is the model category of simplicial categories equipped with the Dwyer–Kan–Bergner model structure and sSetsSet denotes the Joyal model structure on simplicial sets.

The functor N˜\tilde N is the homotopy coherent nerve functor. The functor \mathfrak{C} is its left adjoint functor.

In particular, for CC a fibrant SSet-enriched category, the canonical morphism

(N˜(C))C\mathfrak{C}(\tilde N(C)) \to C

given by the counit of the above adjunction is derived, hence a Dwyer–Kan weak equivalence of simplicial categories.

For SS any simplicial set, the canonical morphism

SN˜(R((S)))S \to \tilde N(R(\mathfrak{C}(S)))

is a categorical equivalence of simplicial sets, where RR denotes the fibrant replacement functor in the Joyal model structure.

For more details, see, for example, \cite[§7.8]{Bergner} or Dugger–Spivak [DuggerSpivak.Rigidification], [DuggerSpivak.Mapping].

Relations

Via W¯\bar W-construction

We have an evident inclusion

sSetCatCat ΔsSet Cat \hookrightarrow Cat^{\Delta}

of simplicially enriched categories into simplicial objects in Cat.

On the latter the W¯\bar W-functor is defined as the composite

W¯:Cat ΔN ΔsSet ΔsSet \bar W \colon Cat^\Delta \stackrel{N^\Delta}{\to} sSet^\Delta \stackrel{}{\to} sSet

where first we degreewise form the ordinary nerve of categories and then take the total simplicial set of bisimplicial sets (the right adjoint of pullback along the diagonal Δ nΔ n×Δ n\Delta^n \to \Delta^n \times \Delta^n).

Proposition

For CC a simplicial groupoid there is a weak homotopy equivalence

𝒩(C)W¯(C)\mathcal{N}(C) \to \bar W(C)

from the homotopy coherent nerve

(Hinich)

There is an operadic analog of the relation between quasi-categories and simplicial categories, involving, correspondingly dendroidal sets and simplicial operads.

References

The idea of a homotopy coherent nerve has been around for some time. It was first made explicit by Cordier in 1980, and the link with quasi-categories was then used in the joint work of him with Porter. That work owed a lot to earlier ideas of Boardman and Vogt about seven years earlier who had used a more topologically based approach. Precise references are given and the history discussed more fully at the entry, homotopy coherent nerve.

The Quillen equivalence between the model structure for quasi-categories and the model structure on sSet-categories is described in

A detailed discussion of the map from quasi-categories to SSetSSet-categories is in

More along these lines is in

  • Emily Riehl, On the structure of simplicial categories associated to quasi-categories (pdf)

An expository account is in Section 7.8

  • Julie Bergner, The homotopy theory of (∞,1)-categories, London Mathematical Society Student Texts 90, 2018.

See also

An introduction and overview of the relation between quasi-categories and simplicial categories is in section 1.1.5 of

The details are in section 2.2

Last revised on November 26, 2020 at 04:30:35. See the history of this page for a list of all contributions to it.