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Added a link to Todd’s nice page on free cartesian category.
(Only seeing this now.)
That sounds right, mattecapu. But I think a nice way of viewing it is by using distributive laws between spans. The rough idea is that morphisms in the free cartesian category can be presented in a “standard” form, as a “purely cartesian morphism” (a morphism in the free cartesian monoidal category on the underlying set of objects) followed by a morphism in the monoidal category. I’ll denote this by . When it comes time to compose two such morphisms, say followed by , you rewrite into a new form by using naturality of copy and delete, and then the composite will be . The distributive law is this rewriting process.
Slightly more formally, I think it goes like this. Suppose given a monoidal category whose underlying span is
I think technically it might be easier dealing with strict monoidal categories here, so I’ll just assume that for now. By strictness, the monoidal structure on the object level gives a map where the domain is the free monoid on . Now, the free cartesian monoidal category on (the discrete category) has as its objects elements of , and so its underlying span will look like
and now I think the underlying span of the free cartesian monoidal category on the monoidal category is the span composite of
followed by . Abusing notation, let denote this span composite.
The distributive law will be a morphism of spans of the form . I’ll only give an example for how this goes. Suppose is a morphism in , and let be the copy morphism. Then is formally rewritten as . Hopefully this makes the general idea clear.
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