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Definition
Presentation over a site
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A cohesive site is a small site whose topos of sheaves is a cohesive topos.
Let be a small site, i.e. a small category equipped with a coverage/Grothendieck topology. We say that is a cohesive site if
has a terminal object.
The coverage on makes it a locally connected site, i.e. every covering sieve on an object is connected as a subcategory of the slice category .
Every object admits a global section .
is a cosifted category
(for instance in that it has all finite products, see at categories with finite products are cosifted).
For a cohesive site, the category of sheaves on is a cohesive topos over Set for which pieces have points .
Following the notation at cohesive topos, we write
for the global section geometric morphism, where the inverse image constructs discrete objects. We need to exhibit two more adjoints
and show that preserves finite products. Finally we need to show that is an epimorphism for all .
Firstly, since is a locally connected site, any constant presheaf is a sheaf. This implies that the functor has a further left adjoint given by taking colimits over , which we denote . Hence is a locally connected topos.
Moreover, since is cosifted, preserves finite products. In particular, is connected and even strongly connected.
Next, we claim that is a local site. This means that its terminal object is cover-irreducible, i.e. any covering sieve of must contain its identity map. But since is a locally connected site, every covering family is inhabited, and since every object has a global section, every covering sieve must include a global section. In the case of , the only global section is an identity map; hence is a local site, and so is a local topos. The right adjoint of is defined by
We now claim that the transformation is monic. Since sheaves are closed under limits in presheaves, this condition can be checked pointwise at each object . But since constant presheaves are sheaves, the map is just the diagonal
which is monic since is always inhabited (by assumption on ).
A cohesive topos over a cohesive site satisfies Aufhebung of the moments of becoming. See at Aufhebung the section Aufhebung of becoming – Over cohesive sites
Consider a category equipped with the trivial coverage/topology. Then the category of sheaves on is the category of presheaves on
and trivially every constant presheaf is a sheaf. So we always have an adjoint triple of functors
where
The condition that preserves finite products is precisely the condition that be a cosifted category.
In conclusion we have
A small category equipped with the trivial coverage/topology is a cohesive site if
it is cosifted;
has a terminal object .
every object has a global element .
The first two conditions ensure that is a cohesive topos. The last condition implies that cohesive pieces have points in .
Any full small subcategory of Top on connected topological spaces with the canonical induced open cover coverage is a cohesive site. If a subcategory on contractible spaces, then this is also an (∞,1)-cohesive site.
Specifically we have:
The categories CartSp and ThCartSp equipped with the standard open cover coverage are cohesive sites.
The axioms are readily checked.
Notice that the cohesive topos over is the Cahiers topos.
The cohesive concrete objects of the cohesive topos are precisely the diffeological spaces.
See cohesive topos for more on this.
and
Last revised on June 24, 2018 at 14:35:23. See the history of this page for a list of all contributions to it.