# ct.category theory – the category of multisets and spans

I've been thinking about multisets for a while. These are sets where the elements can be repeated, so that $$S = {a, a, b, c, b }$$ it's a multiset on the set $$A = {a, b, c }$$.

I've also been looking at morphisms among multisets. Take two multisets $$S_A, S_B$$ with underlying sets $$A, B$$. I would like to define a map between multisets. $$S_A, S_B$$ as a lapse in the underlying sets, so that $$f = A leftarrow C igh B$$Y $$f: S_A rightarrow S_B$$.

This is how I am defining the morphisms. Take a multiset $$S_A$$ and let $$a_i$$ be one of the terms, equally for $$S_B$$ Y $$b_j$$. We have an indexing set. $$C$$ and let the $$f, g$$ be arms of such a span $$f: C rightarrow A$$ Y $$g: C rightarrow B$$. Leave $$c_i in C$$ and let $$f (c_i)$$ be the established element of the term $$a_i$$equally for $$g (c_j)$$. This makes sense to me in terms of three columns in a database, $$A, B, C$$ where $$C$$ It's like a primary key.

I'm not sure how to define the composition of span. I'm thinking a lot about David Spivak's work these days, so I'll suggest a definition of his text, see section 2.5.2.3, which I can not reproduce here. He uses fiber products.

Does my definition of objects and morphs define a category?