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, a category with weak equivalences means the bare minimum of what may reasonably go by that name:

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

A morphism in WW is 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 Theorem 6.1 of

It is shown in Meier that categories of fibrant objects are fibrant in this model structure.

Nerve functors

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

References

Revised on July 19, 2017 14:59:20 by Todd Trimble (24.146.226.222)