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model structure on sSet-categories

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Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (∞,1)-categories

Model structures

for ∞-groupoids

for ∞-groupoids

for n-groupoids

for ∞-groups

for ∞-algebras

general

specific

for stable/spectrum objects

for (∞,1)-categories

for stable (∞,1)-categories

for (∞,1)-operads

for (n,r)-categories

for (∞,1)-sheaves / ∞-stacks

(∞,1)-Category theory

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

Contents

Idea

Depending on the chosen model category structure, the category sSet of simplicial sets may model ∞-groupoids (for the standard model structure on simplicial sets) or (∞,1)-categories in the form of quasi-categories (for the Joyal model structure) on SSet.

Accordingly, there are model category structures on sSet-categories that similarly model (n,r)-categories with r shifted up by 1:

Both are special cases of a model structure on enriched categories.

Model for (∞,1)-categories

Here we describe the model category structure on SSet Cat that makes it a model for the (∞,1)-category of (∞,1)-categories.

Definition An sSet-enriched functor F:C→D between sSet-categories is called a weak equivalence precisely if

Such a morphism is also called a Dwyer-Kan weak equivalence after the work by Dwyer-Kan on simplicial localization.

Propositon A Quillen equivalence C→←D between model categories induces a Dwyer-Kan-equivalence LC↔LD between their simplicial localizations.

Proposition The category SSet Cat of small simplicially enriched categories carries the structure of a model category with

In particualar, the fibrant objects in this structure are the Kan complex-enriched categories, i.e. the strictly ∞-groupoid-enriched ones (see (n,r)-category).

The entries of the following table display models, model categories, and Quillen equivalences between these that present the (∞,1)-category of (∞,1)-categories (second table), of (∞,1)-operads (third table) and of 𝒪-monoidal (∞,1)-categories (fourth table).

general pattern
strict enrichment(∞,1)-category/(∞,1)-operad
↓↓
enriched (∞,1)-category↪internal (∞,1)-category
(∞,1)Cat
SimplicialCategories−homotopy coherent nerve→SimplicialSets/quasi-categoriesRelativeSimplicialSets
↓simplicial nerve↓
SegalCategories↪CompleteSegalSpaces
(∞,1)Operad
SimplicialOperads−homotopy coherent dendroidal nerve→DendroidalSetsRelativeDendroidalSets
↓dendroidal nerve↓
SegalOperads↪DendroidalCompleteSegalSpaces
𝒪Mon(∞,1)Cat
DendroidalCartesianFibrations

References

A model category structure on the category of sSet-categories with a fixed set of objects was first given in

  • William Dwyer, Dan Kan, Simplicial localization of categories , J. Pure and Applied Algebra 17 (3) (1980),

Dywer, Spalinski and later Rezk then pointed out that there ought to exist a model category structure on the collection of all sSet-categories that models the (∞,1)-category of (∞,1)-categories. This was then constructed in

A survey is in section 3 of

See also section A.3.2 of

Model for (∞,2)-categories

for the moment see (∞,2)-category for more on this

References

Section A.3 of

Also

Revised on June 8, 2013 14:47:52 by Urs Schreiber (66.46.90.198)
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