Categorical definition of tensor product

Question

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 $X$.)

Answer 1

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.