# Combinatorial species and differential categories

It is mentioned in the introduction to [1] that (Cartesian) differential categories might be the unifying framework for differentiation in various branches of mathematics including combinatorics. It is also mentioned in [2] and other papers on tangent categories that there are tangent categories of combinatorial species. I don't see how the obvious definition of the category of species can be made into a tangent category, but maybe this is true for some other category of species.

Questions: What is the relationship between combinatorial species and differential/tangent categories? Is there a differential or tangent category of species? Is the operation of differentiation of species related to these structures?

[1] Blute, R.; Cockett, J. R. B.; Seely, R. A. G., Cartesian differential categories, Theory Appl. Categ. 22, 622-672 (2009).

[2] Cockett, J. R. B.; Cruttwell, G. S. H., Connections in tangent categories, Theory Appl. Categ. 32, 835-888 (2017). ZBL1374.18016.

• I don't know about cartesian differential categories, but you might be interested in a structure on species that nlab does not mention: the derivative. The derivative of a species $M$ is $(DM)_X = M_{X \sqcup *}$. On exponential generating functions this operation really is a derivative, it satisfies the Leibniz rule with respect to the Day product, and so on. – Phil Tosteson Sep 18 '19 at 18:34
• @PhilTosteson Yes, I mentioned it in the question. I don't see how it can be related to the structure of a differential category since it is an operation on objects, but the structure of a differential category consists of a function on $\mathrm{Hom}$-sets. Nevertheless, at some very informal level, the derivative of species looks similar to the derivative in other differential categories (just adding an element), but I don't know if there is something behind this intuition. – Valery Isaev Sep 18 '19 at 18:53