I have been thinking about multisets for a while. These are sets where elements can repeat, so $S =\{ a,a,b,c,b\}$ is a multiset on the set $A = \{a,b,c\}$.

I have also been looking into morphisms between multisets. Take two multisets $S_A, S_B$ with underlying sets $A, B$. I would like to define a morphism between multisets $S_A, S_B$ as a span on the underlying sets, so $f = A \leftarrow C \rightarrow B$, and $f: S_A \rightarrow S_B$. I am not sure how to define span composition. I am thinking a lot about David Spivak's work these days, so I am going to suggest a definition from his text, see section 2.5.2.3, which I cannot reproduce here. He uses fiber products.

I have two questions. Firstly, does my definition of the objects and morphisms define a category? Secondly, is it the case that the definition of morphism, as given, also constitutes a tensor product between multisets in the category as defined?

I have only an intuition that tells me that this definition of morphism is both a morphism of multisets in the category defined but is also a tensor product. We know that the Eilenberg-Moore category for the multiset monad is actually $\mathbb{N}$-modules. We should find a notion of tensor product there. I am guessing that we can define a monad on Set that maps a set to its set of multisets and spans as defined. There should be a similar category of modules, and thus a tensor product. I can only guess that what I am thinking means that the spans, as defined, map to both morphisms in the category of modules and the tensor product in the category of modules.