model structure for Segal categories


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



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.


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 \,.

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.


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).


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.



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



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.




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.


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