In detail this means that is
with the property that
it has a terminal object ;
for every covering family in
the simplicial set of points in is weakly equivalent to the set of points of :
These conditions are stronger than for a cohesive site, which only guarantees cohesiveness of the 1-topos over it.
This definition is supposed to model the following ideas:
every object has an underlying set of points . We may think of each as specifying one way in which there can be cohesion on this underlying set of points;
in view of the nerve theorem the condition that is contractible means that itself is contractible, as seen by the Grothendieck topology on . This reflects the local aspect of cohesion: we only specify cohesive structure on contractible lumps of points;
in view of this, the remaining condition that is contractible is the -analog of the condition on a concrete site that is surjective. This expresses that the notion of topology on and its concreteness over Set are consistent.
The following sites are -cohesive:
we can take the morphisms in to be
or smooth maps
This is a site of definition for the Cahiers topos.
Since every star-shaped region in is diffeomorphic to an open ball (see there for details) we have that the covers on CartSp by convex subsets are good open covers in the strong sense that any finite non-empty intersection is diffeomorphic to an open ball and hence diffeomorphic to a Cartesian space. Therefore these are good open covers in the strong sense of the term and their Cech nerves are degreewise coproducts of representables.
If moreover for all objects of we have that is inhabited, then the axiom “pieces have points” also holds.
Since the (n,1)-topos over a site for any arises as the full sub-(∞,1)-category of the -topos on the -truncated objects and since the definition of cohesive -topos is compatible with such truncation, it follows that
Let be an -cohesive site. Then for all the (n,1)-topos is cohesive.
To prove this, we need to show that
This follows with the discussion at ∞-connected site.
is a local (∞,1)-topos.
This follows with the discussion at ∞-local site.
The fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos preserves finite (∞,1)-products.
If is not empty for all , then pieces have points in .
The last two conditions we demonstrate now.
The functor whose existence is guaranteed by the above proposition preserves products:
Finally we prove that pieces have points in if all objects of have points.
By the above discussion both and are presented by left Quillen functors on the projective model structure . By Dugger’s cofibrant replacement theorem (see model structure on simplicial presheaves) we have for any simplicial presheaf that a cofibrant replacement is given by an object that in the lowest two degrees is
where the coproduct is over all morphisms out of representable presheaves as indicated.
The model for sends this to
whereas the model for sends this to
The morphism from the first to the latter is the evident one that componentwise sends to the point. Since by assumption each is nonempty, this is componentwise an epi. Hence the whole morphism is an epi on .
cohesive site, ∞-cohesive site