# nLab model structure for Segal categories

model category

## Model structures

for ∞-groupoids

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

#### $(\infty,1)$-Category theory

(∞,1)-category theory

# Contents

## Idea

A model category structure whose fibrant objects are Segal categories. This is a presentation of the (∞,1)-category (∞,1)Cat from the point of view of (weakly) ∞Grpd-enriched categories.

## Definition

Write $PreSegalCat \hookrightarrow [\Delta^{op}, sSet]$ for the full subcategory on those bisimplicial sets $X$ for which $X_0$ is a discrete simplicial set (the “precategories”).

The nerve functor

$N : Cat \to PreSegalCat$

has a left adjoint (“fundamental category” functor)

$\tau_1 : PreSegalCat \to Cat \,.$
###### Definition

Say a morphism $f : X \to Y$ in $PreSegalCat$ is

• full and faithful if for all $a,b \in X_0$ the induced morphism

$X(a,b) \to X(f(a),f(b))$

is a weak homotopy equivalence of simplicial sets;

• essentially surjective if $\tau_1(f)$ is essentially surjective.

• a categorical equivalence if it is both full and faithful as well as essentially surjective.

###### Proposition

There is an essentially unique completion functor

$compl \colon PreSegalCat \to PreSegalCat$

equipped with a natural transformation

$i \colon id_{PreSegalCat} \to compl$

such that for all pre-Segal categories $X$

1. $compl(X)$ is a Segal category;

2. $i_X \colon X \to compl(X)$ is an isomorphism on the sets of objects;

3. $i_X$ is a categorical equivalence if $X$ is already a Segal category;

4. $compl(i_X)$ is a categorical equivalence.

This is (HS, def. 2.1, lemma 2.2).

###### Definition

Say a morphism $f : X \to Y$ in $PreSegalCat$ is

• a cofibration precisely if it is a monomorphism;

• a weak equivalence precisely if its completion $compl(f)$ by prop. 1 is a categorical equivalence.

(…)

###### Proposition

This defines a model category structure for Segal categories (…)

(…)

###### Remark

It follows that a map $X \to Y$ between Segal categories is a weak equivalence precisely if it is a categorical equivalence.

Because by prop. 1 we have a commuting square of the form

$\array{ X &\underoverset{\simeq}{i_X}{\to}& compl(X) \\ \downarrow && \downarrow \\ X &\underoverset{\simeq}{i_Y}{\to}& compl(Y) }$

where the horizontal morphisms are categorical equivalences, and by prop. 2 these satisfy 2-out-of-3.

## Properties

### General

###### Proposition

Equipped with the classes of maps defined in def. 2, $PreSegalCat$ is a model category which is

Cartesian closure is shown in (Pellissier). The fact that it is a Cisinski model structure follows from the result in (Bergner).

## References

The model structure for Segal categories was introduced in

(even for Segal n-categories). An alternative proof of the existence of this model structure is given in section 5 of

The cartesian closure of the model structure was established in

• Regis Pellissier. Catégories enrichies faibles. Thèse, Université de Nice-Sophia Antipolis (2002), (arXiv:math/0308246)

The fact that the fibrant Segal categories in this model structure are precisely the Reedy fibrant bisimplicial sets is due to

Model structures for Segal categories enriched over more general (∞,1)-categories are discussed in section 2 of

Revised on November 3, 2013 20:50:41 by Tim Porter (95.148.109.31)