Calculus of Species
Arithmetic on generating functions corresponds to certain "natural" operations on species. The basic operations are addition, multiplication, composition, and differentiation; it is also necessary to define equality on species. Category theory already has a way of describing when two functors are equivalent: a natural isomorphism. In this context, it just means that for each A there is a bijection between F-structures on A and G-structures on A, which is "well-behaved" in its interaction with transport. Species with the same generating function might not be isomorphic, but isomorphic species do always have the same generating function.
Read more about this topic: Combinatorial Species
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