nLab
enriched Quillen adjunction

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\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 0C_0 of any VV-enriched category, defined by C 0(x,y)=V(I,C(x,y))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)