Backround
Definition
Presentation over a site
Models
An -cohesive site is a site such that the (∞,1)-category of (∞,1)-sheaves over it is a cohesive (∞,1)-topos.
A site is -cohesive over ∞Grpd if it is
and an ∞-local site.
In detail this means that is
a site – a small category equipped with a coverage;
with the property that
it has a terminal object ;
it is a cosifted category
(for instance in that it has all finite products, see at categories with finite products are cosifted);
for every covering family in
the Cech nerve is degreewise a coproduct of representables;
the simplicial set obtained by replacing each copy of a representable by a point is contractible (weakly equivalent to the point in the classical model structure on simplicial sets)
the simplicial set of points in is weakly equivalent to the set of points of :
These conditions are stronger than for a cohesive site, as the latter 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 site for a presheaf topos, hence with trivial topology, is -cohesive, def. , if it has finite products.
All covers consist of only the identity morphism . The Cech nerve is then the simplicial object constant on and hence satisfies its two conditions above trivially.
The following sites are -cohesive, def. :
the category CartSp with covering families given by open covers by convex subsets ;
we can take the morphisms in to be
– in which case the sheaf topos over it models generalized topological spaces, the 2-sheaf 2-topos contains for instance topological stacks;
or smooth maps
– in which case the sheaf topos models generalized smooth spaces such as diffeological spaces, the (∞,1)-sheaf (∞,1)-topos is that of ∞-Lie groupoids;
the site ThCartSp of smooth loci consisting of smooth loci of the form with the second factor infinitesimal, where covering families are those of the form with a covering family in as above.
This is a site of definition for the Cahiers topos.
More discussion of these two examples is at ∞-Lie groupoid and ∞-Lie algebroid.
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.
The fact that follows from the nerve theorem, using that a Cartesian space regarded as a topological space is contractible.
Let be an -cohesive site. Then the (∞,1)-sheaf (∞,1)-topos over is a cohesive (∞,1)-topos that satisfies the axiom “discrete objects are concrete” .
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
is a locally ∞-connected (∞,1)-topos and a ∞-connected (∞,1)-topos.
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:
By the discussion at ∞-connected site we have that is given by the (∞,1)-colimit . By the assumption that is a cosifted (∞,1)-category, it follows that this operation preserves finite 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 .
A cohesive (∞,1)-topos over an -cohesive site satisfies Aufhebung of the moments of becoming. See at Aufhebung the section Aufhebung of becoming – Over cohesive sites.
and
Last revised on June 14, 2018 at 10:22:32. See the history of this page for a list of all contributions to it.