nLab
enriched Quillen adjunction
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
Enriched category theory
Contemts
Definition
In enriched model category theory, an enriched Quillen adjunction is an adjunction in the sense of enriched category theory whose underlying ordinary adjunction is a Quillen adjunction between ordinary model categories .

Here “underlying” refers to the underlying ordinary category $C_0$ of any $V$ -enriched category, defined by $C_0(x,y) = V(I,C(x,y))$ . (Recall that an enriched model category is an enriched category, together with a model structure on its underlying ordinary category, and some compatibility conditions.)

Special cases
A special role is playes by sSet -enriched Quillen adjunctions, for the standard model structure on simplicial sets . See simplicial Quillen adjunction for more on that

Revised on August 25, 2010 15:13:45
by

Urs Schreiber
(131.211.36.96)