Recall that if $\mathscr{C}$ is a site (which, for me, means a category equipped with a Grothendieck pretopology—not sieves) that it’s not in general true that the representable presheaf $h_X$ associated to an object $X$ of $\mathscr{C}$ is a sheaf.
In particular, one can consider the composition of functors
$$\mathscr{C}\to \mathsf{PSh}(\mathscr{C})\xrightarrow{(-)^\#}\mathsf{Sh}(\mathscr{C})$$
(where $(-)^\#$ denotes sheafification) and think of the full image—the full subcategory of $\mathsf{Sh}(\mathscr{C})$ whose objects are sheaves isomorphic to $h_X^\#$ for an object $X$ of $\mathscr{C}$. Let me denote this category with the (probably poorly chosen) notation $\mathscr{C}^\#$.
My question is then the following
Question: What is an explicit description of $\mathscr{C}^\sharp$?
To give an idea of what type of description I’m looking for, I think that one can write that there is a bijection
$$\mathrm{Hom}(h_X^\#,h_Y^\#)=\varinjlim_{\substack{\{U_i\to X\}\\ \mathscr{C}\text{-coverings}}}\check{H}^0_\mathrm{loc}(\{U_i\to X\}, h_Y)$$
where
$$\check{H}^0_\mathrm{loc}(\{U_i\to X\}, h_Y):=\left\{(f_i)\in\prod_i \mathrm{Hom}_{\mathscr{C}}(U_i,Y): f_i\text{ and }f_j\text{ agree locally on overlaps for all }i,j\right\}$$
where $f_i$ and $f_j$ agreeing locally on overlaps means that there exists a cover $\{V_k\to U_i\times_X U_j\}$ such that $(f_i)\mid_{V_k}=(f_j)\mid_{V_k}$.
This is literally, I believe, just because one has natural bijections
$$\mathrm{Hom}(h_X^\#,h_Y^\#)=\mathrm{Hom}(h_X,h_Y^\sharp)=h_Y^\#(X)$$
the first from the definition of sheafification, and the second from Yoneda’s lemma. The claimed equality then follows from the explicit description of sheafification.
Refined question: With the above description of $\mathrm{Hom}(h_X^\#,h_Y^\#)$ (or a slightly modified, but still
explicit version)
- What is the actual map $h_X^\sharp(Z)\to h_Y^\#(Z)$ for $Z$ an object of $\mathscr{C}$?
- What is the composition $$\mathrm{Hom}(h_X^\#,h_Y^\#)\times \mathrm{Hom}(h_Y^\#,h_Z^\#)\to \mathrm{Hom}(h_X^\#,h_Z^\#)$$ look
like?
NB: I’m mostly looking for a reference. Obviously you can answer this question by tracing through all the definitions, but I’d like to save myself the time/writing and just cite something. I’d be a little surprised if a reference didn’t exist.
Thanks!
Best Answer
I don't have the reference you want, but here's the concrete description of composition you ask for.
An element of $h_Y^\sharp(X)$ is witnessed by some cover $\{U_i\to X\}$ and maps $U_i\to Y$ agreeing on overlaps. An element of $h_Z^\sharp(Y)$ is witnessed by a cover $\{V_j\to Y\}$ and maps $V_j\to Z$ agreeing on overlaps. The composition in $h_Z^\sharp(X)$ is then witnessed by the cover $\{U_i\times_Y V_j \to U_i \to X\}$ and the maps $U_i\times_Y V_j \to V_j \to Z$.