# 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).

| $\downarrow$ | | $\downarrow$ | | | | enriched (∞,1)-category | $\hookrightarrow$ | internal (∞,1)-category | | | | | | | | | | | (∞,1)Cat | | | | | SimplicialCategories | $-$homotopy coherent nerve$\to$ | SimplicialSets/quasi-categories | | RelativeSimplicialSets | | $\downarrow$simplicial nerve | | $\downarrow$ | | | | SegalCategories | $\hookrightarrow$ | CompleteSegalSpaces | | | | | | | | | | | (∞,1)Operad | | | | | SimplicialOperads | $-$homotopy coherent dendroidal nerve$\to$ | DendroidalSets | | RelativeDendroidalSets | | $\downarrow$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