# (Sh,Sh-map) represents the category of sheaves on a stack.

I'm trying to understand the following theorem, but I don't think I'm reading it correctly.

Let $(\mathcal{C},J)$ be a site (with a subcanonical topology). Write $\mathcal{C}/X$ for the groupoid of objects over $X\in \mathcal{C}$. Let $\mbox{Sh}:\mathcal{C}^{op} \rightarrow \mbox{Gpds}$ be the functor taking $X$ to the category of sheaves on $\mathcal{C}/X$ and isomorphisms of sheaves, and let $\mbox{Sh-map}:\mathcal{C}^{op} \rightarrow \mbox{Gpds}$ be the functor taking $X$ to the category whose objects are sheaf morphisms $\mathscr{F} \rightarrow \mathscr{G}$ and whose morphisms are commuting squares of sheaves determined by isomorphisms $\mathscr{F}_1 \stackrel{\sim}{\rightarrow} \mathscr{F}_2$ and $\mathscr{G}_1 \stackrel{\sim}{\rightarrow} \mathscr{G}_2$. These are in fact both stacks on $\mathcal{C}$, and moreover they determine a category-object $(\mbox{Sh},\mbox{Sh-map})$ in the category of stacks.

Theorem: The category of sheaves on a stack $\mathscr{M}$ is equivalent to the category of morphisms of stacks $\mathscr{M} \rightarrow (\mbox{Sh,Sh-map})$. That is, the objects are the 1-morphisms and the morphisms are the 2-morphisms.

I'd like to interpret this to mean that the objects of $Shv(\mathscr{M})$ are associated to 1-morphisms $\mathscr{M} \rightarrow \mbox{Sh}$, and that the morphisms of $Shv(\mathscr{M})$ are associated to 2-morphisms in $Hom_{Stacks}(\mathscr{M},\mbox{Sh})$, which in turn should be the same as 1-morphisms $\mathscr{M} \rightarrow \mbox{Sh-map}$. But there a number of problems with this.

First, given a sheaf $\mathcal{F} \in Shv(\mathscr{M})$ I'm having trouble constructing a natural transformation $\mathscr{M} \rightarrow \mbox{Sh}$. Perhaps I shouldn't, but to check this I'm using a test object $X\in \mathcal{C}$. By Yoneda, an object of $\mathscr{M}(X)$ is the same as a 1-morphism of stacks $f:X\rightarrow \mathscr{M}$, and so I obtain an object of $Sh(X)$ (i.e. a sheaf on $\mathcal{C}/X$) via $(\alpha:Y\rightarrow X) \mapsto \mathcal{F}(f\alpha:Y \rightarrow X \rightarrow \mathscr{M})$. That's natural enough. Again by Yoneda, a morphism in $\mathscr{M}(X)$ is a 2-morphism between maps $f,g:X\rightarrow \mathscr{M}$ of stacks, i.e. a section $s:X\rightarrow X\times_\mathscr{M} X$ of the projection from the 2-category fiber product. Out of this, I'm supposed to construct a natural transformation from the sheaf $(\alpha:Y\rightarrow X) \mapsto \mathcal{F}(f\alpha:Y \rightarrow X \rightarrow \mathscr{M})$ to the sheaf $(\alpha:Y\rightarrow X) \mapsto \mathcal{F}(g\alpha:Y \rightarrow X \rightarrow \mathscr{M})$. But the only structure in place to give me such a thing is a morphism in $Stacks/\mathscr{M}$ between $f\alpha$ and $g\alpha$, and I don't see how to construct this.

Second, a 2-morphism between 1-morphisms $f,g\in Hom_{Stacks}(\mathscr{M},\mbox{Sh})$ is a section $s:\mathscr{M} \rightarrow \mathscr{M} \times_{\mbox{Sh}} \mathscr{M}$. Thus for any $(\alpha:X\rightarrow \mathscr{M})\in \mathscr{M}(X)$, we get an object $(\alpha,\beta:X \rightarrow \mathscr{M},\varphi:f\alpha \stackrel{\sim}{\rightarrow} g\alpha)\in (\mathscr{M}\times_{\mbox{Sh}}\mathscr{M})(X)$. On the other hand, a 1-morphism $\mathscr{M} \rightarrow \mbox{Sh-map}$ is for each $\alpha:X \rightarrow \mathscr{M}$ an arbitrary morphism on sheaves on $\mathcal{C}/X$. These can't be the same.

By the way, I've tried to do (what I think is) the right thing and work out the sheaf in $Shv(\mbox{Sh})$ associated to the 1-morphism $\mbox{Id}:\mbox{Sh} \rightarrow \mbox{Sh}$, following Yoneda and all. From the above, it's easy to see what this sheaf should do to morphisms $X\rightarrow \mbox{Sh}$ from a representable stack. But it appears that I need to make choices if I want to say what it does to arbitrary morphisms of stacks $\mathscr{N} \rightarrow \mbox{Sh}$. Perhaps instead I should take a limit or colimit over its application to the full subcategory of representable stacks over $\mathscr{N}$?

The notes you are reading seem to disagree with more commonly accepted language (cf. SGA1 Exp 13, Vistoli's notes, or the Stacks project). Some of this seems to be an attempt at expository ease, e.g., the parenthetical remark in example 8.2 ("We will mention the following technical difficulties but will ignore them for now:") where "for now" really means forever. Oddly enough, one of the mentioned technical difficulties is more or less what prevents $\text{Sh}$ and $\text{Sh-map}$ from having natural stack structures in the sense of the notes - pullback is not strictly functorial. This un-naturality is why the common definition of stack is different - the notion of stack in the notes corresponds to the usual notion of stack in groupoids equipped with a splitting (or cleavage).

The use of the category object $(\text{Sh}, \text{Sh-map})$ is a kludge to replace the usual stack $Sh/\mathcal{C}$ (in categories rather than groupoids) whose objects are sheaves over comma categories, and whose morphisms over any $f: U \to V$ in $\mathcal{C}$ are $f$-maps of sheaves - see Examples 3.20 and 4.11 in Vistoli. The author of the notes employs $\text{Sh-map}$ in order to add non-invertible sheaf maps, because the 2-morphisms in $Hom_{Stacks}(\mathcal{M}, \text{Sh})$ are all invertible. In other words, you have to throw away the 2-morphisms that are given to you by $\text{Sh}$, and use the larger collection of possibly non-invertible two-morphisms afforded by $\text{Sh-map}$.

Once you have done that, I think your main problems are resolved. You've already worked out the object part of getting from a sheaf on $Stacks/\mathcal{M}$ to a natural transformation from $\mathcal{M}$ to $\text{Sh}$. If you have a morphism $\beta: X \to Z$ in $\mathcal{C}$, and $f: Z \to \mathcal{M}$, then $\beta$ induces a morphism of stacks over $\mathcal{M}$. If I'm not mistaken, the sheaf $\mathcal{F}$ takes this to the map in $\text{Sh}$ given by base change: $$\left( (\alpha: Y \mapsto Z) \mapsto \mathcal{F}(f \circ \alpha) \right) \mapsto \left( \beta^* \alpha: Y \times_Z X \to X) \mapsto \mathcal{F}(f \circ \beta \circ \beta^*\alpha) \right)$$

Similarly, you can get from a sheaf map on $Stacks/\mathcal{M}$ to a natural transformation from $\mathcal{M}$ to $\text{Sh-map}$. There seems to be a lot of additional checking necessary for proving the equivalence, which I don't feel like doing for you (sorry).

Let me try to strip off all the stack language, which confuses me, and recast what I think is your question just in terms of category theory. (If I have misinterpreted which part is your question, I apologize. The only question mark in your post is in the very last line, but I don't think that's the main question.)

You are, I believe, in the following situation. You have some ambient Cartesian category ($Stacks(M)$). You have a test object $M$ in your category, which happens to be the terminal object, if I'm not mistaken, but I don't think this matters. You have a category object $C_1 \rightrightarrows C_0$ internal to your category. Then in particular for any test object $M$, there is a category (in SET) whose morphisms are the arrows $M \to C_1$, and whose objects are the arrows $M \to C_0$ --- indeed, being a category object is equivalent to being a representable presheaf valued in categories. Then you also have a theorem that recognizing this category as some other more interesting category ($Sheaves(M)$).

This almost sums up your paragraph after the theorem. But complicating the matter is that you don't have some ambient Cartesian category. Rather, $Stacks(M)$ is a (Cartesian) 2-category. So now we have two separate notions. Indeed, as a 2-category, it is among other things enriched in categories, so that you have in fact a category of morphisms $M \to C_0$, for example. In the previous paragraph, I was only considering this as a set.

So then perhaps your question is the following. You have an ambient (Cartesian) 2-category, and let's assume it to be strict, a test object $M$, and an object $C_0$. Then $\hom(M,C_0)$ is a 1-category. The objects of this 1-category is the unenriched hom, which I will denote as $|\hom(M,C_0)|$. You'd like to recognize the morphisms of $\hom(M,C_0)$ as the set $|\hom(M,C_1)|$ for some particular $C_1$. Suppose that you have good Cartesian-closedness conditions, and an "inner hom" $\underline{\hom}$. Then what you'd like is a "walking arrow object" $Arr$ (which you do have with even stronger conditions, with buzzwords like "tensored over CAT"), and then $C_1 = \underline{\hom}(Arr,C_0)$.

My guess is that the category of stacks on $M$ does have all of these strong closedness and tensored conditions. Moreover, my guess is that the correctly-implemented inner-hom construction in the previous paragraph, applied when $C_0 = Sh(M)$, yields $C_1 = ShMap(M)$. Maybe these are originally the versions with target CAT, not GPD, but then the trick is to post-compose with the 2-functor CAT$\to$GPD that keeps only the invertible morphisms, and make sure that this doesn't change too much.

I hope this recasting helps, and that I didn't utterly misinterpret your question. This is about the limit of my category theory, and well past what I know about stacks properly.