nLab
model structure for Segal categories

Context

Model category theory

model category, model \infty -category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

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

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

Contents

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[Δ op,sSet]PreSegalCat \hookrightarrow [\Delta^{op}, sSet] for the full subcategory on those bisimplicial sets XX for which X 0X_0 is a discrete simplicial set (the “precategories”).

The nerve functor

N:CatPreSegalCat N : Cat \to PreSegalCat

has a left adjoint (“fundamental category” functor)

τ 1:PreSegalCatCat. \tau_1 : PreSegalCat \to Cat \,.
Definition

Say a morphism f:XYf : X \to Y in PreSegalCatPreSegalCat is

  • full and faithful if for all a,bX 0a,b \in X_0 the induced morphism

    X(a,b)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 τ 1(f)\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:PreSegalCatPreSegalCat compl \colon PreSegalCat \to PreSegalCat

equipped with a natural transformation

i:id PreSegalCatcompl i \colon id_{PreSegalCat} \to compl

such that for all pre-Segal categories XX

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

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

  3. i Xi_X is a categorical equivalence if XX is already a Segal category;

  4. compl(i X)compl(i_X) is a categorical equivalence.

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

Definition

Say a morphism f:XYf : X \to Y in PreSegalCatPreSegalCat is

  • a cofibration precisely if it is a monomorphism;

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

(…)

Proposition

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

(…)

Remark

It follows that a map XYX \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

X i X compl(X) X i Y compl(Y) \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. , PreSegalCatPreSegalCat 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).

Relation to other model structures

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

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

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

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