nLab
excellent model category
Context
Model category theory
model category

Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$ -categories
Model structures
for $\infty$ -groupoids
for ∞-groupoids

for $n$ -groupoids
for $\infty$ -groups
for $\infty$ -algebras
general
specific
for stable/spectrum objects
for $(\infty,1)$ -categories
for stable $(\infty,1)$ -categories
for $(\infty,1)$ -operads
for $(n,r)$ -categories
for $(\infty,1)$ -sheaves / $\infty$ -stacks
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)