nLab
model structure on categories with weak equivalences

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-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

Contents

Idea

The model category structure on the category of categories with weak equivalences is a model for the (∞,1)-category of (∞,1)-categories.

Every category with weak equivalences CC presents under Dwyer-Kan simplicial localization a simplicially enriched category or alternatively under Charles Rezk’s simplicial nerve a Segal space, both of which are incarnations of a corresponding (∞,1)-category C\mathbf{C} with the same objects of CC, at least the 1-morphisms of CC and such that every weak equivalence in CC becomes a true equivalence (homotopy equivalence) in C\mathbf{C}.

Details

For the purposes of the present entry, we understand under a category with weak equivalences the absolute minimum structure that may deserve to go by that name, namely a relative category:

Definition A relative category (C,W)(C,W) is a category CC equipped with a choice of wide subcategory WW.

A morphism in WW are called a weak equivalence in CC. Notice that we do not require here that these weak equivalence satisfy 2-out-of-3, nor even that they contain all isomorphisms.

A morphism (C 1,W 1)(C 2,W 2)(C_1,W_1) \to (C_2,W_2) of relative catgeories is a functor C 1C 2C_1 \to C_2 that preserves weak equivalences.

Write RelCatRelCat for the category of relative categories and such morphisms between them.

Model category structure

The model category structure on RelCatRelCat is obtained from that on bisimplicial sets modelling complete Segal spaces in section 6.1 of

  • Clark Barwick and Dan Kan, Relative categories; another model for the homotopy theory of homotopy theories – Part I: the model structure (pdf)

Nerve functors

The compatibility of the various nerve and simplicial localization functors is in section 1.11 of

  • Clark Barwick and Dan Kan, Relative categories; another model for the homotopy theory of homotopy theories – Part II: the weak equivalences (pdf)

References

  • Clark Barwick and Dan Kan,

    • Relative categories; another model for the homotopy theory of homotopy theories – Part I: the model structure (pdf)

    • Relative categories; another model for the homotopy theory of homotopy theories – Part II: the weak equivalences (pdf)

    • Partial model categories and their simplicial nerves, a relative Yoneda embedding, and a Quillen theorem B nB_n for homotopy pullbacks (pdf)

Revised on July 30, 2010 07:21:43 by Urs Schreiber (134.100.32.207)