on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
An enriched model category is an enriched category $C$ together with the structure of a model category on the underlying category $C_0$ such that both structures are compatible in a reasonable way.
Let $V$ be a monoidal model category.
A $V$-enriched model category is
an V-enriched category $C$
with the structure of a model category on the underlying category $C_0$
such that
(pullback-power axiom) for every cofibration $i \colon A \to B$ and fibration $p \colon X \to Y$ in $C_0$ the pullback powering morphism (dual to the pushout product) with respect to the powering in $V$
is a fibration with respect to the model structure on $V$;
and is an acyclic fibration whenever $i$ or $p$ are acyclic.
The last two conditions here are equivalent to the fact that the copower
is a Quillen bifunctor.
(…)
The most familiar examples of enriched model categories are simplicial model categories.
For a pointed model category or even stable model category there is enrichment in pointed simplicial sets with the smash product as tensor product.
Textbook account:
See also:
Last revised on September 13, 2021 at 06:45:55. See the history of this page for a list of all contributions to it.