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:

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.

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).

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
$\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
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

Model for $(\infty,2)$-categories

for the moment see (∞,2)-category for more on this

References

Section A.3 of

Also

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 sturcture which presents (infinity,1)-categories, see

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

Revised on March 29, 2014 03:06:03 by Urs Schreiber (185.37.147.12)