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 quasi-category corresponding to a simplicial category CC is its homotopy coherent nerve NN

sSetCatN||sSet. sSet Cat \stackrel{\overset{|-|}{\leftarrow}}{\underset{N}{\to}} sSet \,.

Relations

Via homotopy coherent nerve

For CC any SSet-enriched category, the canonical morphism

|N(C)|C |N(C)| \to C

is an equivalence in that it is essentially surjective on the underlying homotopy categories and a weak equivalence of simplicial sets hom-wise (…details/links…)

For SS any simplicial set, the canonical morphism

SN(|S|) S \to N(|S|)

is a categorical equivalence of simplicial sets.

Via W¯\bar W-construction

We have an evident inclusion

sSetCatCat Δ sSet 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 : 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)

Model category structures

The above relations constitute arrange into a Quillen equivalence between model category structures on quasicategories and simplicially enriched categories.

There is the

The homotopy coherent nerve

sSetCatNsSet Joyal sSet Cat \stackrel{N}{\to} sSet_{Joyal}

is the right adjoint part of a Quillen equivalence between these model structures.

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)

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

Revised on April 4, 2017 15:27:25 by Tim Porter (95.144.140.105)