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model structure on sSet-enriched presheaves

Idea

The model structure $\mathrm{sPSh}(C)$ on sSet-enriched presheaves is supposed to be a presentation of the (∞,1)-category of (∞,1)-sheaves on an sSet-site $C$.

It generalizes the model structure on simplicial presheaves which is the special case obtained when $C$ happens to be just an ordinary category.

This means that in as far as the model structure on simplicial presheaves models ∞-stacks, the model structure on sSet-enriched categories models things like derived stacks.

Definition

The construction of the model structure on sSet-enriched categories closely follows the discussion of the model structure on simplicial presheaves, only that everything now takes place in enriched category theory.

To define the model structure first consider the ordinary category

$$\mathrm{sPSh}(C):=\mathrm{sSet}\mathrm{Cat}(C^{\mathrm{op}},\mathrm{sSet})$$sPSh(C) := sSet Cat(C^{op}, sSet)

of sSet-enriched functors.

A morphism $f:A\to B$ in $\mathrm{sPSh}(C)$ is a transformation given by a collection of morphisms $f_c:A(c)\to B(c)$ in sSet.

The global projective model structure on $\mathrm{sPSh}(C)$ has (as the global model structure on simplicial presheaves) as fibrations and weak equivalences the objectwise fibrations and weak equivalences, i.e. those transformations for which all $f_c$ are a fibration or weak equivalence, respectively.

But $\mathrm{sPSh}(C)$ is naturally enriched to an sSet-enriched functor category,

$$\mathrm{sPSh}(C{)}_{\mathrm{proj}}:=[C^{\mathrm{op}},\mathrm{SSet}]$$sPSh(C)_{proj} := [C^{op}, SSet]

and with the above model structure this extends to the structure of a simplicial model category, called the global projective simplicial model structure on SSet-presheaves.

The model structure that we are after is the left Bousfield localization $\mathrm{sPSh}(C{)}_{\mathrm{proj}}^{l\mathrm{loc}}$ of $\mathrm{sPSh}(C{)}_{\mathrm{proj}}$ at the following set of morphisms (…see TV page 14…)

It seems that the claim is that, indeed, in the special case that $C$ happens to be an ordinary category, the model structure on $\mathrm{sPSh}(C{)}_{\mathrm{proj}}^{l\mathrm{loc}}$ reproduces the projective local model structure on simplicial presheaves.

References

The theory of model structures on SSet-enriched presheaf categories was developed in

• Bertrand Toen, Gabriele Vezzosi, Topos Theory (arXiv)