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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?

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  • $\begingroup$ 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). $\endgroup$ – Denis Nardin Mar 11 at 9:09
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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.

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