nLab
cohesive site

Idea

A cohesive site is a small site whose topos of sheaves is a cohesive topos.

Definition

Definition

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

1. $C$ has a terminal object.

2. 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 $C/U$.

3. Every object $U\in C$ admits a global section $*\to U$.

4. $C$ is a cosifted category.

Notice that if $C$ has finite products then it is also cosifted.

Properties: sheaves on a cohesive site

Proposition

For $C$ a cohesive site, the category of sheaves $\mathrm{Sh}(C)$ on $C$ is a cohesive topos over Set for which cohesive pieces have points .

Proof

Following the notation at cohesive topos, we write

$$(\mathrm{Disc}⊣\Gamma ):=(L\mathrm{Const}⊣\Gamma ):\mathrm{Sh}(C)\to \mathrm{Set}$$(Disc \dashv \Gamma) := (L Const \dashv \Gamma) : Sh(C) \to Set

for the global section geometric morphism, where the inverse image $\mathrm{Disc}$ constructs discrete objects. We need to exhibit two more adjoints

$$({\Pi }_0⊣\mathrm{Disc}⊣\Gamma ⊣\mathrm{CoDisc}):\mathrm{Sh}(C)\to \mathrm{Set}$$(\Pi_0 \dashv Disc \dashv \Gamma \dashv CoDisc) : Sh(C) \to Set

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 $\mathrm{Disc}$ has a further left adjoint given by taking colimits over $C^{\mathrm{op}}$, which we denote ${\Pi }_0$. Hence $\mathrm{Sh}(C)$ is a locally connected topos.

Moreover, since $C$ is cosifted, ${\Pi }_0$ preserves finite products. In particular, $\mathrm{Sh}(C)$ 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 $\mathrm{Sh}(C)$ is a local topos. The right adjoint $\mathrm{Codisc}$ of $\Gamma$ is defined by

$$\mathrm{CoDisc}(A)(U)=A^{C(*,U)}=A^{\Gamma (U)}.$$CoDisc(A)(U) = A^{C(*,U)} = A^{\Gamma(U)} \,.

We now claim that the transformation $\mathrm{Disc}(A)\to \mathrm{Codisc}(A)$ 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 $\mathrm{Disc}(A)(U)\to \mathrm{Codisc}(A)(U)$ is just the diagonal

$$A\to A^{C(*,U)}$$A \to A^{C(*,U)}

which is monic since $C(*,U)$ is always inhabited (by assumption on $C$).

Examples

Cohesive presheaf sites

Consider a category $C$ equipped with the trivial coverage/topology. Then the category of sheaves on $C$ is the category of presheaves on $C$

$$\mathrm{Sh}(C)\simeq \mathrm{PSh}(C)$$Sh(C) \simeq PSh(C)

and trivially every constant presheaf is a sheaf. So we always have an adjoint triple of functors

$$({\Pi }_0⊣\mathrm{Disc}⊣\Gamma ):\mathrm{Sh}(C)\to \mathrm{Set},$$(\Pi_0 \dashv Disc \dashv \Gamma) : Sh(C) \to Set \,,

where

• ${\Pi }_0$ is the functor that takes colimits of functors $X:C^{\mathrm{op}}\to \mathrm{Set}$

  $${\Pi }_{0}X=\underrightarrow{\lim }X$$\Pi_0 X = {\lim_\to} X

• $\Gamma$ is the functor that takes limits;

  $$\Gamma X=\underleftarrow{\lim }X.$$\Gamma X = {\lim_\leftarrow} X \,.

The condition that ${\Pi }_0$ preserves finite products is precisely the condition that $C$ be a cosifted category.

In conclusion we have

Proposition

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 $\mathrm{Sh}(C)=\mathrm{PSh}(C)$ is a cohesive topos. The last condition implies that cohesive pieces have points in $\mathrm{PSh}(C)$.

Sites of open balls

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:

Proposition

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 $\mathrm{ThCartSp}$ is the Cahiers topos.

Proposition

The cohesive concrete objects of the cohesive topos $\mathrm{Sh}(\mathrm{CartSp})$ are precisely the diffeological spaces.

See cohesive topos for more on this.

Related concepts

• locally connected topos / locally ∞-connected (∞,1)-topos

  • connected topos / ∞-connected (∞,1)-topos

  • strongly connected topos / strongly ∞-connected (∞,1)-topos

  • totally connected topos / totally ∞-connected (∞,1)-topos

• local topos / local (∞,1)-topos.

• cohesive topos / cohesive (∞,1)-topos

and

• locally connected site / locally ∞-connected site

  • connected site / ∞-connected site

  • strongly connected site / strongly ∞-connected site

  • totally connected site / totally ∞-connected site

• local site / ∞-local site

• cohesive site / ∞-cohesive site