nLab
enriched Quillen adjunction
Contemts
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 enriched adjunction 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 played by sSet -enriched Quillen adjunctions, for the standard model structure on simplicial sets . See simplicial Quillen adjunction for more on that.

Last revised on September 20, 2018 at 13:20:19.
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