Notice that if has finite products then it is also cosifted.
Following the notation at cohesive topos, we write
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.
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 ).
and trivially every constant presheaf is a sheaf. So we always have an adjoint triple of functors
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
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 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.
cohesive site / ∞-cohesive site