It is standard to define the tensor product $M\otimes_R N$ of $R$-modules as a universal object of bilinear maps from $M\times N$. Now, suppose that $\mathscr{F}$, $\mathscr{G}$ are sheaves of $\mathscr{O}$-modules on a topological space $X$. I'm trying to give a categorical definition of $\mathscr{F}\otimes_{\mathscr{O}}\mathscr{G}$ as an object in the category of sheaves on $X$, satisfying some universal property. Presumably it should begin with some kind of "bilinear morphism" from $\mathscr{F}\times\mathscr{G}$, but how should I define "bilinear" (or "balanced") morphism in category theory language? (i.e., without reference to elements of sets, or open subsets of $X$.)
[Math] Categorical definition of tensor product
category-theorysheaf-theorytensor-products
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I figured out how to prove it.
Let $(X, \mathscr{O})$ be a ringed space. Let $\mathscr{F}, \mathscr{G}$ be sheaves of $\mathscr{O}$-modules. Define $\mathscr{H}(U) = \mathscr{F}(U) \otimes_{\mathscr{O}(U)} \mathscr{G}(U)$. Fix $p \in X$. Assume $U$ is an open n.h of $p$.
The $\mathscr{O}_p$-module structure on $\mathscr{F}_p \otimes_{\mathscr{O}_p} \mathscr{G}_p$ induces a $\mathscr{O}(U)$-module structure on $\mathscr{F}_p \otimes_{\mathscr{O}_p} \mathscr{G}_p$ .
Define $\alpha_U : \mathscr{F}(U) \times \mathscr{G}(U) \to \mathscr{F}_p \otimes_{\mathscr{O}_p} \mathscr{G}_p, \quad (s,t) \mapsto s_p \otimes t_p$. This map is $2$-linear over $\mathscr{O}(U)$. Therefore $\alpha_U$ induces an $\mathscr{O}(U)$-module homomorphism from $\mathscr{F}(U) \otimes_{\mathscr{O}(U)} \mathscr{G}(U)$ to $\mathscr{F}_p \otimes_{\mathscr{O}_p} \mathscr{G}_p$. We shall abuse notation and also call this map $\alpha_U$.
Now forget about the $\mathscr{O}(U)$ module structure on the sections of $\mathscr{H}$. The $\alpha_U$s form a co-cone over the $\mathscr{H}(U)$s with $p \in U$ (that is they make the appropriate diagrams commute), therefore they induce a homomorphism of abelian groups $h : \mathscr{H}_p \to \mathscr{F}_p \otimes_{\mathscr{O}_p} \mathscr{G}_p$
Define $\psi : \mathscr{F}_p \times \mathscr{G}_p \to \mathscr{H}_p$ by $(s_p, t_p) \mapsto (s|_{U \cap V} \otimes t|_{U \cap V})_p$ where $s$ is a section of $\mathscr{F}$ over $U$ and $t$ is a section of $\mathscr{G}$ over $V$. It is easily verified that $\psi$ is 2-linear over $\mathscr{O}_p$. Therefore $\psi$ induces a $\mathscr{O}_p$ homomorphism from $\mathscr{F}_p \otimes_{p} \mathscr{G}_p$ to $\mathscr{H}_p$. It is easily verified that this map and $h$ are inverses.
That categorical definition is for pre-sheaves, the topological definition is for sheaves.
In topological pre-sheaves, a map is surjective if it is epimorphic for each open set $U$ in $X$.
In topological sheaves, however, we instead have to "sheaf-ify" the definition, and we say that the map is "surjective" if the sheaf-ification of the cokernel map is zero.
Basically, in both cases, you have two categories, $\mathcal{Sh}$ and $\mathcal{PSh}$, and in $\mathcal{PSh}$, the "surjective" maps are the ones that are epimorphisms on each $U$, but in the $\mathcal{Sh}$ catageory, you have a more complicated definition of "surjective" (or "epimorphism.")
Consider, instead, two categories, $\mathcal{Ab}$ the category of abelian groups, and $\mathcal{AbTF}$, the full subcategory of "torsion-free" abelian groups - that is, the abelian groups, $A$, where for any $n\in\mathbb Z$ and $a\in A$, $na=0$ iff $n=0$ or $a=0$.
There is the natural inclusion functor $\mathcal{AbTF}\to\mathcal{Ab}$ and a natural adjoint sending $A\to A/N(A)$ where $N(A)$ is the subgroup of nilpotent elements of $A$.
But in $\mathcal{AbTF}$, the "epimorphisms" are not the ones with cokernel (in $\mathcal{Ab}$) $0$, they are the ones with cokerkels which are nilpotent. So, for example, in $\mathcal{Ab}$, the morphism $\mathbb Z\to\mathbb Z$ sending $x\to 2x$ is not an epimorphism, that same map, when considered as a map in $\mathcal{AbTF}$, is an epimorphism.
So consider the "sheafification" functor $\mathcal{PSh}\to \mathcal{Sh}$ to be much like the functor $\mathcal{Ab}\to\mathcal{AbTF}$.
(I believe, but don't quote me, that $f:A\to B$ in $\mathcal{AbTF}$ is an epimorphism if and only if $f\otimes \mathbb Q:A\otimes \mathbb Q\to B\otimes\mathbb Q$ is an epimorphism in $\mathcal{Ab}$.)
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I'm pretty tired, as I normally would have been asleep for over an hour by now, but I'll take a stab at this. We need to basically use $\mathscr{O}$-bilinear morphisms(I haven't defined them and won't! Not publically anyway...), and we unfortunately use sets in this. I don't really think there's a way around this, as there is no convenient categorical definition of the tensor product for modules. The tensor product certainly does have one for algebras, but this is not the case we are interested in here.
So this should have some sort of property(starting with the presheaf tensor product) that any $\mathscr{O}$-bilinear natural transformation from the presheaf of sets $\mathscr{F}\times \mathscr{G}$ to a presheaf $\mathscr{H}$ should factor through(via the 'natural' projection) the presheaf tensor product $\mathscr{F}\otimes_\mathscr{O} \mathscr{G}$. When $\mathscr{H}$ is a sheaf, it factors through the sheafification of the presheaf tensor product(by property of sheafification!) so we only need to show this universal property for presheaves. We see first that $\mathscr{F}\otimes \mathscr{G}: U \mapsto \mathscr{F}(U)\otimes_{\mathscr{O}(U)} \mathscr{G}(U)$ at least gives an abelian group structure on it, and the restriction morphism sending $\rho_\mathscr{F} \otimes \rho_\mathscr{G}: f\otimes g \mapsto (\rho_\mathscr{F})\otimes(\rho_\mathscr{G})$ gives a compatible $\mathscr{O}$-module structure.
Now let's say we're given a $\mathscr{O}$-bilinear natural transformation of presheaves from $\mathscr{F} \times \mathscr{G}$ to $\mathscr{H}$. In particular for each $U$ we get a bilinear map from $\mathscr{F}(U) \times \mathscr{G}(U)$ to $\mathscr{H}(U)$, so we get a morphism $\mathscr{F}(U)\otimes\mathscr{G}(U) \rightarrow \mathscr{H}(U)$. The commutativity of the bilinear natural transformation diagram guarantees that this is a natural transformation of functors, i.e. a morphism of presheaves.
Again, if we are working in the category of sheaves then the sheafification of this construction will have the universal property.