nLab enriched Quillen adjunction

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.