nLab model structure on sSet-categories

model category

Model structures

for ∞-groupoids

for $(\infty,1)$-sheaves / $\infty$-stacks

$(\infty,1)$-Category theory

(∞,1)-category theory

Contents

Idea

Depending on the chosen model category structure, the category sSet of simplicial sets may model ∞-groupoids (for the standard model structure on simplicial sets) or (∞,1)-categories in the form of quasi-categories (for the Joyal model structure) on SSet.

Accordingly, there are model category structures on sSet-categories that similarly model (n,r)-categories with $r$ shifted up by 1:

• there is a model structure on $SSet$-enriched categories whose fibrant objects are ∞-groupoid/Kan complex-enriched categories and which models (∞,1)-categories;

This we discuss below.

Both are special cases of a model structure on enriched categories.

Model for $(\infty,1)$-categories

Here we describe the model category structure on SSet Cat that makes it a model for the (∞,1)-category of (∞,1)-categories.

Definition

An sSet-enriched functor $F : C \to D$ between sSet-categories is called a weak equivalence precisely if

Such a morphism is also called a Dwyer-Kan weak equivalence after the work by Dwyer-Kan on simplicial localization.

Proposition

A Quillen equivalence $C \stackrel{\leftarrow}{\to} D$ between model categories induces a Dwyer-Kan-equivalence $L C \leftrightarrow L D$ between their simplicial localizations.

Proposition

The category SSet Cat of small simplicially enriched categories carries the structure of a model category with

• weak equivalences the Dwyer-Kan equivalences;

• fibrations those sSet-enriched functors $F : C \to D$ such that

1. for all $x, y \in C$ the morphism $F_{x,y} : C(x,y) \to D(F(x), F(y))$ is a fibration in the standard model structure on simplicial sets;

2. the induced functor $\pi_0(F) : Ho(C) \to Ho(D)$ on homotopy categories is an isofibration.

Remark

In particular, the fibrant objects in this structure are the Kan complex-enriched categories, i.e. the strictly ∞-groupoid-enriched ones (see (n,r)-category).

Properties

Proposition

The Bergner model structure of prop. 2 is a right proper model category.

The entries of the following table display models, model categories, and Quillen equivalences between these that present the (∞,1)-category of (∞,1)-categories (second table), of (∞,1)-operads (third table) and of $\mathcal{O}$-monoidal (∞,1)-categories (fourth table).

general pattern
strict enrichment(∞,1)-category/(∞,1)-operad
$\downarrow$$\downarrow$
enriched (∞,1)-category$\hookrightarrow$internal (∞,1)-category
(∞,1)Cat
SimplicialCategories$-$homotopy coherent nerve$\to$SimplicialSets/quasi-categoriesRelativeSimplicialSets
$\downarrow$simplicial nerve$\downarrow$
SegalCategories$\hookrightarrow$CompleteSegalSpaces
(∞,1)Operad
SimplicialOperads$-$homotopy coherent dendroidal nerve$\to$DendroidalSetsRelativeDendroidalSets
$\downarrow$dendroidal nerve$\downarrow$
SegalOperads$\hookrightarrow$DendroidalCompleteSegalSpaces
$\mathcal{O}$Mon(∞,1)Cat
DendroidalCartesianFibrations

References

A model category structure on the category of $sSet$-categories with a fixed set of objects was first given in

• William Dwyer, Dan Kan, Simplicial localization of categories , J. Pure and Applied Algebra 17 (3) (1980),

Dywer, Spalinski and later Rezk then pointed out that there ought to exist a model category structure on the collection of all $sSet$-categories that models the (∞,1)-category of (∞,1)-categories. This was then constructed in

A survey is in section 3 of

See also section A.3.2 of

and

Recall the slight but crucial difference between the two notions of “simplicial categories”, the other being an internal category in sSet. But also for this latter concept there is a model category structure which presents (infinity,1)-categories, see

• Geoffroy Horel, A model structure on internal categories (arXiv:1403.6873).

Revised on January 7, 2015 12:34:04 by Adeel Khan (77.182.72.204)