# nLab model structure for Segal categories

model category

## Model structures

for ∞-groupoids

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

#### $\left(\infty ,1\right)$-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 $\mathrm{PreSegalCat}↪\left[{\Delta }^{\mathrm{op}},\mathrm{sSet}\right]$ for the full subcategory on those bisimplicial sets $X$ for which ${X}_{0}$ is a discrete simplicial set (the “precategories”).

The nerve functor

$N:\mathrm{Cat}\to \mathrm{PreSegalCat}$N : Cat \to PreSegalCat

has a left adjoint (“fundamental category” functor)

${\tau }_{1}:\mathrm{PreSegalCat}\to \mathrm{Cat}\phantom{\rule{thinmathspace}{0ex}}.$\tau_1 : PreSegalCat \to Cat \,.
###### Definition

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

• full and faithful if for all $a,b\in {X}_{0}$ the induced morphism

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

is a weak homotopy equivalence of simplicial sets;

• essentially surjective if ${\tau }_{1}\left(f\right)$ 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

$\mathrm{compl}:\mathrm{PreSegalCat}\to \mathrm{PreSegalCat}$compl \colon PreSegalCat \to PreSegalCat

equipped with a natural transformation

$i:{\mathrm{id}}_{\mathrm{PreSegalCat}}\to \mathrm{compl}$i \colon id_{PreSegalCat} \to compl

such that for all pre-Segal categories $X$

1. $\mathrm{compl}\left(X\right)$ is a Segal category;

2. ${i}_{X}:X\to \mathrm{compl}\left(X\right)$ is an isomorphism on the sets of objects;

3. ${i}_{X}$ is a categorical equivalence if $X$ is already a Segal category;

4. $\mathrm{compl}\left({i}_{X}\right)$ is a categorical equivalence.

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

###### Definition

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

• a cofibration precisely if it is a monomorphism;

• a weak equivalence precisely if its completion $\mathrm{compl}\left(f\right)$ by prop. 1 is a categorical equivalence.

(…)

###### Proposition

This defines a model category structure fro 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

$\begin{array}{ccc}X& \underset{\simeq }{\overset{{i}_{X}}{\to }}& \mathrm{compl}\left(X\right)\\ ↓& & ↓\\ X& \underset{\simeq }{\overset{{i}_{Y}}{\to }}& \mathrm{compl}\left(Y\right)\end{array}$\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, $\mathrm{PreSegalCat}$ 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 April 8, 2013 15:50:08 by Urs Schreiber (82.113.106.75)