model category

for ∞-groupoids

# Contents

## Idea

Extra axioms on a monoidal model category $\mathbf{S}$ that guarantee a particularly good homotopy theory of $\mathbf{S}$-enriched categories are referred to as excellent model category structure (Lurie).

## Definition

###### Definition

Let $\mathbf{S}$ be a monoidal model category. It is called excellent if

• it is a combinatorial model category;

• every monomorphism if a cofibration

• the collection of cofibrations is closed under products;

• it satisfies the invertibility hypothesis: for any equivalence $f$ in an $\mathbf{S}$-enriched category $C$, the localization functor $C \to C[f^{-1}]$ is an equivalence of $\mathbf{S}$-enriched categories.

This is (Lurie, def. A.3.2.16).

## References

Section A.3 in

Revised on April 5, 2015 17:31:33 by Adeel Khan (2.243.174.221)