model category

for ∞-groupoids

# 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 $C$ 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 $\mathbf{C}$ with the same objects of $C$, at least the 1-morphisms of $C$ and such that every weak equivalence in $C$ becomes a true equivalence (homotopy equivalence) in $\mathbf{C}$.

## 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 $(C,W)$ is a category $C$ equipped with a choice of wide subcategory $W$.

A morphism in $W$ are called a weak equivalence in $C$. 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 $(C_1,W_1) \to (C_2,W_2)$ of relative catgeories is a functor $C_1 \to C_2$ that preserves weak equivalences.

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

### Model category structure

The model category structure on $RelCat$ 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

• 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 $B_n$ for homotopy pullbacks (pdf)

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