nLab
model structure for Segal categories

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

The nerve functor

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

has a left adjoint (“fundamental category” functor)

$${\tau }_1:\mathrm{PreSegalCat}\to \mathrm{Cat}.$$\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

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

$$\mathrm{compl}:\mathrm{PreSegalCat}\to \mathrm{PreSegalCat}$$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. $\mathrm{compl}(X)$ is a Segal category;

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

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

4. $\mathrm{compl}(i_X)$ is a categorical equivalence.

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

Definition

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

• a cofibration precisely if it is a monomorphism;

• a weak equivalence precisely if its completion $\mathrm{compl}(f)$ 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

$$\begin{array}{ccc}X& \underset{\simeq }{\overset{i_X}{\to }}& \mathrm{compl}(X)\\ \downarrow & & \downarrow \\ X& \underset{\simeq }{\overset{i_Y}{\to }}& \mathrm{compl}(Y)\end{array}$$\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

• left proper;

• cartesian monoidal;

• a Cisinski model structure.

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

Relation to other model structures

See table - models for (infinity,1)-categories.

Related concepts

• model structure for Segal operads

References

The model structure for Segal categories was introduced in

• André Hirschowitz, Carlos Simpson, Descente pour les $n$-champs (Descent for $n$-stacks) (arXiv:9807049)

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

• Julie Bergner, Three models for the homotopy theory of homotopy theories (arXiv:math/0504334)

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

• Julie Bergner, A characterization of fibrant Segal categories, Proceedings of the AMS (2007) (arXiv:0603400, pdf)

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

• Jacob Lurie, (infinity,2)-Categories and the Goodwillie Calculus

• pdf