structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
A cohesive site is a small site whose topos of sheaves is a cohesive topos.
Let $C$ be a small site, i.e. a small category equipped with a coverage/Grothendieck topology. We say that $C$ is a cohesive site if
$C$ has a terminal object.
The coverage on $C$ makes it a locally connected site, i.e. every covering sieve on an object $U\in C$ is connected as a subcategory of the slice category $C/U$.
Every object $U\in C$ admits a global section $*\to U$.
$C$ is a cosifted category.
Notice that if $C$ has finite products then it is also cosifted.
For $C$ a cohesive site, the category of sheaves $Sh(C)$ on $C$ is a cohesive topos over Set for which cohesive pieces have points .
Following the notation at cohesive topos, we write
for the global section geometric morphism, where the inverse image $Disc$ constructs discrete objects. We need to exhibit two more adjoints
and show that $\Pi_0$ preserves finite products. Finally we need to show that $\Gamma X \to \Pi_0 X$ is an epimorphism for all $X$.
Firstly, since $C$ is a locally connected site, any constant presheaf is a sheaf. This implies that the functor $Disc$ has a further left adjoint given by taking colimits over $C^{op}$, which we denote $\Pi_0$. Hence $Sh(C)$ is a locally connected topos.
Moreover, since $C$ is cosifted, $\Pi_0$ preserves finite products. In particular, $Sh(C)$ is connected and even strongly connected.
Next, we claim that $C$ 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 $C$ 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 $C$ is a local site, and so $Sh(C)$ is a local topos. The right adjoint $Codisc$ of $\Gamma$ is defined by
We now claim that the transformation $Disc(A) \to Codisc(A)$ is monic. Since sheaves are closed under limits in presheaves, this condition can be checked pointwise at each object $U\in C$. But since constant presheaves are sheaves, the map $Disc(A)(U) \to Codisc(A)(U)$ is just the diagonal
which is monic since $C(*,U)$ is always inhabited (by assumption on $C$).
Consider a category $C$ equipped with the trivial coverage/topology. Then the category of sheaves on $C$ is the category of presheaves on $C$
and trivially every constant presheaf is a sheaf. So we always have an adjoint triple of functors
where
$\Pi_0$ is the functor that takes colimits of functors $X : C^{op} \to Set$
$\Gamma$ is the functor that takes limits;
The condition that $\Pi_0$ preserves finite products is precisely the condition that $C$ 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 $U$ has a global element $* \to U$.
The first two conditions ensure that $Sh(C) = PSh(C)$ is a cohesive topos. The last condition implies that cohesive pieces have points in $PSh(C)$.
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 $ThCartSp$ is the Cahiers topos.
The cohesive concrete objects of the cohesive topos $Sh(CartSp)$ are precisely the diffeological spaces.
See cohesive topos for more on this.
and
cohesive site / ∞-cohesive site