model category

## Model structures

for ∞-groupoids

### for $(\infty,1)$-sheaves / $\infty$-stacks

#### Enriched category theory

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)