Could not include topos theory - contents
category object in an (∞,1)-category, groupoid object
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 $F : \bar \mathbb{C}^{op} \to \bar \mathcal{S}$.
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 an 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
But the resulting notion can of course be expressed entirely in terms of data in the ambient topos. This we spell out in
Let $\mathcal{S}$ be a topos and let $\mathbb{C}$ be an internal category in $\mathcal{S}$:
Write $\bar \mathbb{C}$ for the 2-sheaf on $\mathcal{S}$
that is represented by $\mathbb{C}$. More explicitly, this is the pseudofunctor which to an object $X \in \mathcal{S}$ assigns
Write
for the 2-sheaf that classifies the codomain fibration of $\mathcal{S}$, the pseudofunctor which sends an object to the corresponding slice topos and morphisms to base change
(also called the “self-indexing of $\mathcal{S}$”).
This is the incarnation of $\mathcal{S}$ itself, regard internal to its 2-sheaf 2-topos $\hat Sh_{2}(\mathcal{T}, can)$.
An internal presheaf on $\mathbb{C}$ (internal to $\mathcal{T}$) is a morphism
of 2-sheaves on $\mathcal{S}$. (Also called an “indexed functor” between indexed categories“.)
Suppose moreover that $\mathbb{C}$ is equipped with the structure of an internal site. Then $F$ above is an internal sheaf on $\mathbb{C}$ if it satisfies the evident descent condition.
A morphism of internal presheaves is simply a 2-morphism in $\hat Sh_2(\mathcal{S}, can)$ (also called an “indexed natural transformation”). This yields a category
of internal presheaves. Accordingly we have the full subcategory
of internal sheaves.
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 $\mathbb{C}^{op}$.
An internal presheaf on $\mathbb{C}$ is an internal diagram $F \in \mathcal{S}^{\mathbb{C}^{op}}$.
An internal sheaf on $\mathbb{C}$ (with respect to $J$) is an internal presheaf on $\mathbb{C}$ satisfying one of the following equivalent conditions:
Let $\mathcal{S}$ and $\mathbb{C}$ be as above.
The category of internal presheaves $PSh(\mathbb{C}, \mathcal{S})$ is a topos.
This appears as (Johnstone, cor. B.2.3.17).
Let $f : \mathbb{C} \to \mathbb{D}$ be an internal functor. Write $\bar f : \bar \mathbb{C} \to \bar \mathbb{D}$ for the corresponding morphism in $\hat Sh_2(\mathcal{S}, can)$. Precomposition with this morphism induces a functor of internal presheaf catgeories
This is the inverse image of a geometric morphism of toposes
This appears as (Johnstone, cor. B.2.3.22).
Internal presheaves in a Grothendieck topos are discussed in Section V.7 of
and in section B2.3 of
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).