Deduce that a morphism of sheaves of sets on X is an epimorphism if and only if it is onto on stalks.

I don't know how I should prove it. ]]>

Consider the presheaf Sh on S valued in the bicategory of categories

that sends s∈S to the category of sheaves over the slice site S/s.

One can show that Sh is actually a sheaf of categories (or rather a stack in bicategories, to be precise).

In other words, a sheaf can be specified locally on the elements of some cover and then glued together,

and the same is true for morphisms of sheaves.

Is this fact somewhere in the literature?

I am also interested in the versions for ∞-sheaves on ∞-sites (any model will do). ]]>

The article local model structure on simplicial presheaves states that for a site with enough points stalkwise weak equivalences of simplicial presheaves coincide with weak equivalences of simplicial presheaves in the Bousfield localization of componentwise weak equivalences with respect to all hypercovers (i.e., weak equivalences in a local model structure).

Is a proof of this statement written up somewhere? (The article cited above gives a reference to Jardine, which claims, but does not prove this statement.)

Also, is it possible to formulate an analog of this statement for sites that do not have enough points? (Presumably we would have to talk about sufficiently refined (hyper)covers instead of points.)

]]>Consider a site S (I am mostly interested in hypercomplete sites, e.g., the site of smooth manifolds). The category of simplicial presheaves on S can be equipped with the local projective model structure given by the left Bousfield localization of the global projective model structure (weak equivalences and fibrations are componentwise) with respect to all hypercovers (for hypercomplete sites Čech covers are sufficient).

Fibrant objects in the resulting model structures are precisely those objects that are fibrant in the global projective model structure (i.e., componentwise fibrant) and satisfy descent with respect to all hypercovers (Čech covers for hypercomplete sites).

Now assume we are working in the category of simplicial *sheaves*,
i.e., simplicial objects in the category of sheaves of sets on S.

Presumably the global projective model structure restricts to simplicial sheaves.
Is there a reference for this claim? I was only able to find references
for the *injective* local model structure (i.e., the Joyal-Jardine structure),
but nothing for the projective case.

My main question concerns descent condition for the case of simplicial *sheaves*.
Assume we have a simplicial sheaf that is globally fibrant.
Can we somehow exploit the fact that individual simplicial components are sheaves
(and not merely presheaves) to simplify the general descent condition?
Or is it just as complicated as the descent for presheaves?
(Again we can assume the site to be hypercomplete if it helps.)