nLab
model structure for Segal categories

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

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general

specific

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

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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 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. 1 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. 1 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. 2 these satisfy 2-out-of-3.

Properties

General

Proposition

Equipped with the classes of maps defined in def. 2, 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

  • 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)