# Definition A.3.1.5 of Higher Topos Theory

I am now reading the book Higher Topos Theory. In A.3.1.5, it gives the definition of a $$\mathbf{S}$$-enriched model category, where $$\mathbf{S}$$ is a monoidal model category. But in the book model structures are introduced only on $$\mathsf{Set}$$-enriched categories.

So, what does a model structure on a $$\mathbf{S}$$-enriched category mean?

Is it supposed to be that $$\mathbf{S}$$ obtains a forgetful functor to $$\mathsf{Set}$$, and the model structure is defined on the category with respect to the $$\mathsf{Set}$$ enrichment, or is it something else?

• I suspect Lurie is using implicitly the lax monoidal forgetful functor $\mathrm{Hom}_{\mathbf{S}}(1_{\mathbf{S}},-):\mathbf{S}→\mathrm{Set}$, although I'll confess I never really thought about it (but this gives the sensible choice of forgetful functor for both simplicial sets and chain complexes). – Denis Nardin Mar 11 at 9:09

## 1 Answer

Your guess is correct, indeed. In general, given any monoidal category $$(\mathbf V, \otimes, 1)$$, and any $$\mathbf V$$-enriched category $$\mathbf C$$, one can always consider the underlying category $$\mathbf C_0$$ as the ($$\mathbf{Set}$$-)category having as objects the same objects as $$\mathbf C$$, and as hom-sets $$\mathbf C_0(x,y):= \mathbf V(1,\mathbf{Hom}_{\mathbf C}(x,y))$$ You can easily work out how to define composition, after checking that $$\mathbf V(1,-)$$ is a lax monoidal functor, as pointed out by Denis in the comments.

Now, if $$\mathbf S$$ is a monoidal model category, and $$\mathbf A$$ is a $$\mathbf S$$-enriched category, "equipping $$\mathbf A$$ with a model structure" just means "equipping $$\mathbf A_0$$ with a model structure", whereas to talk about a $$\mathbf S$$-enriched model category one requires the two extra conditions spelled out in HTT A.3.1.5.