nLab cohesive site

Cohesive site

Cohesive site


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



Let CC be a small site, i.e. a small category equipped with a coverage/Grothendieck topology. We say that CC is a cohesive site if

  1. CC has a terminal object.

  2. The coverage on CC makes it a locally connected site, i.e. every covering sieve on an object UCU\in C is connected as a subcategory of the slice category C/UC/U.

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

  4. CC is a cosifted category

    (for instance in that it has all finite products, see at categories with finite products are cosifted).


Sheaves on a cohesive site are cohesive


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


Following the notation at cohesive topos, we write

(DiscΓ)(LConstΓ):Sh(C)Set (Disc \dashv \Gamma) \coloneqq (L Const \dashv \Gamma) \;\colon\; Sh(C) \to Set

for the global section geometric morphism, where the inverse image DiscDisc constructs discrete objects. We need to exhibit two more adjoints

(Π 0DiscΓCoDisc):Sh(C)Set (\Pi_0 \dashv Disc \dashv \Gamma \dashv CoDisc) \;\colon\; Sh(C) \to Set

and show that Π 0\Pi_0 preserves finite products. Finally we need to show that ΓXΠ 0X\Gamma X \to \Pi_0 X is an epimorphism for all XX.

Firstly, since CC is a locally connected site, any constant presheaf is a sheaf. This implies that the functor DiscDisc has a further left adjoint given by taking colimits over C opC^{op}, which we denote Π 0\Pi_0. Hence Sh(C)Sh(C) is a locally connected topos.

Moreover, since CC is cosifted, Π 0\Pi_0 preserves finite products. In particular, Sh(C)Sh(C) is connected and even strongly connected.

Next, we claim that CC 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 CC 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 CC is a local site, and so Sh(C)Sh(C) is a local topos. The right adjoint CodiscCodisc of Γ\Gamma is defined by

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

We now claim that the transformation Disc(A)Codisc(A)Disc(A) \to Codisc(A) is monic. Since sheaves are closed under limits in presheaves, this condition can be checked pointwise at each object UCU\in C. But since constant presheaves are sheaves, the map Disc(A)(U)Codisc(A)(U)Disc(A)(U) \to Codisc(A)(U) is just the diagonal

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

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


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


Cohesive presheaf sites

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

Sh(C)PSh(C) Sh(C) \simeq PSh(C)

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

(Π 0DiscΓ):Sh(C)Set, (\Pi_0 \dashv Disc \dashv \Gamma) : Sh(C) \to Set \,,


  • Π 0\Pi_0 is the functor that takes colimits of functors X:C opSetX : C^{op} \to Set

    Π 0X=lim X \Pi_0 X = {\lim_\to} X
  • Γ\Gamma is the functor that takes limits;

    ΓX=lim X. \Gamma X = {\lim_\leftarrow} X \,.

The condition that Π 0\Pi_0 preserves finite products is precisely the condition that CC 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 Sh(C)=PSh(C)Sh(C) = PSh(C) is a cohesive topos. The last condition implies that cohesive pieces have points in PSh(C)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:


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 ThCartSpThCartSp is the Cahiers topos.


The cohesive concrete objects of the cohesive topos Sh(CartSp)Sh(CartSp) are precisely the diffeological spaces.

See cohesive topos for more on this.


Last revised on June 24, 2018 at 14:35:23. See the history of this page for a list of all contributions to it.