Contents

Idea

The notions of presheaf, site and sheaf can be formulated internal to any topos. The ordinary such notions are recovered by internalization into Set.

More precisely, the direct internalization of these notions is into the very large 2-topos $\hat Sh_2(\mathcal{S},can)$ of the given ambient topos $\mathcal{S}$, since an internal presheaf is to be an $\mathcal{S}$-valued internal functor, but $\mathcal{S}$ does not quite sit inside itself. It does, however, sit inside $\hat Sh_2(\mathcal{S},can)$, incarnated as the 2-sheaf $\bar \mathcal{S}$ corresponding to its codomain fibration.

Therefore, regarding $\hat Sh_2(\mathcal{S}, can)$ as the 2-category of internal categories in $\mathcal{S}$, an internal site in $\mathcal{S}$ is an object $\bar \mathbb{C}$ of $\hat Sh_2(\mathcal{S}, can)$ and an internal presheaf is a morphism $F : \bar \mathbb{C}^{op} \to \bar \mathcal{S}$.

Definition

While it is straightforward to define an internal site, hence the domain of an internal (pre)sheaf, the definition of the codomain is slightly more subtle, for that needs to be a copy of the ambient universe internalized into itself. One way to naturally say this is by passing to the external 2-sheaves 2-topos. This version of the definition we state in

• In terms of external 2-sheaves.

But the resulting notion can of course be expressed entirely in terms of data in the ambient topos. This we spell out in

• Explicit definition.

In terms of external 2-sheaves

Let $\mathcal{S}$ be a topos and let $\mathbb{C}$ be an internal category in $\mathcal{S}$:

Explicit definition

We unwind what the above amounts to more explicitly.

Let $(\mathbb{C},J)$ be an internal site in $\mathcal{S}$, i.e. an internal category $\mathbb{C}$ equipped with an internal coverage $J$. Let $\mathcal{S}^{\mathbb{C}^{op}}$ be the topos of internal diagrams on $\mathbb{C}^{op}$.

Properties

Let $\mathcal{S}$ and $\mathbb{C}$ be as above.

This appears as (Johnstone, cor. B.2.3.17).

This appears as (Johnstone, cor. B.2.3.22).

The global section functors of internal sheaf toposes in $\mathcal{S}$ are bounded geometric morphisms over $\mathcal{S}$.

Related concepts

• internal diagram

References

On internal presheaves in elementary toposes:

• Marta Bunge, Internal Presheaf Toposes, Cahiers,Cahiers de topologie et géométrie différentielle catégoriques? 18 3 (1977) 291-330 [numdam:CTGDC_1977__18_3_291_0]

On internal presheaves in a Grothendieck toposes:

• Saunders MacLane, Ieke Moerdijk, Section V.7 of Sheaves in Geometry and Logic (1992)

and in section B2.3 of

• Peter Johnstone, Sketches of an Elephant

In these references internal presheaves are introduced in components as in the explicit definition above. The equivalence to the abstract formulation above, in terms of morphisms between 2-sheaves, follows for instance with (Johnstone, lemma B.2.3.13).