nLab
model structure for Segal categories

Idea

A model category structure whose fibrant objects are precisely the Reedy fibrant 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. 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. 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. these satisfy 2-out-of-3.

Properties

General

Proposition

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

• left proper;

• cartesianmonoidal;

• 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 Segal categories 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