nLab enriched model category



Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

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



An enriched model category is an enriched category CC together with the structure of a model category on the underlying category C 0C_0 such that both structures are compatible in a reasonable way.


Let VV be a monoidal model category.

A VV-enriched model category is

The last two conditions here are equivalent to the fact that the copower

:C×VC \otimes : C \times V \to C

is a Quillen bifunctor.


Change of enrichment




(monoidal model category is enriched model over itself)
Every monoidal model category is an enriched model category over itself, via the enrichment of its underlying closed monoidal category.


One just needs to see that the pullback-power axiom is implied by (in fact it is equivalent to) the pushout-product axiom. This equivalence is an instance of Joyal-Tierney calculus (see this Prop.):



we have the following logical equivalences.

CC C CC FW C FW C FW C FW CCW CW CCW F C F CW F CW FW CW F C F C F \begin{aligned} \mathrm{C} \Box \mathrm{C} & \,\subset\, & \mathrm{C} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{C} \Box \mathrm{C} &\,\;⧄\;\,& \mathrm{FW} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{C} & \,⧄\, & \mathrm{FW}^{\Box \mathrm{C}} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{FW}^{\Box \mathrm{C}} & \,\subset\, & \mathrm{FW} \\ {\phantom{-}} \\ \mathrm{C} \Box \mathrm{CW} & \,\subset\, & \mathrm{CW} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{C} \Box \mathrm{CW} &\,\;⧄\;\,& \!\!\mathrm{F}\;\;\; & \;\;\; \Leftrightarrow \;\;\; & \mathrm{C} & \;⧄\; & \mathrm{F}^{\Box \mathrm{CW}} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{F}^{\Box \mathrm{CW}} & \,\subset\, & \mathrm{FW} \\ & & & & && & \;\;\; \Leftrightarrow \;\;\; & \mathrm{CW} & \;⧄\; & \mathrm{F}^{\Box \mathrm{C}} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{F}^{\Box \mathrm{C}} & \,\subset\, & \mathrm{F} \end{aligned}

Here the outer equivalences are by definition of the lifting properties in a model category, while the middle equivalences are by Joyal-Tierney calculus. The statements on the far left constitute the pushout-product axiom, while those on the far right constiture the pullback-power axiom.


Since the model structure on compactly-generated topological spaces as well as the classical model structure on simplicial sets are monoidal model categories, they are, by by Exp. , also enriched modal categories over themselves.


A model category enriched over the classical model structure on simplicial sets is called a simplicial model category.


The general definition of enriched model categories has its origin in

where it is formulated in the special case of simplicial model categories (but the generalization is immediate).

Textbook account:

See also:

On enriched Reedy model categories:

Last revised on December 17, 2023 at 12:32:21. See the history of this page for a list of all contributions to it.