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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.

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    $\begingroup$ 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. $\endgroup$ – Phil Tosteson Sep 18 '19 at 18:34
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    $\begingroup$ @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. $\endgroup$ – Valery Isaev Sep 18 '19 at 18:53
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I contacted Geoffrey Cruttwell with regards to this question. Here is his reply:

There hasn’t been any published paper on a tangent category of combinatorial species. However, the ideas can be found in a talk by my co-author, Robin Cockett here: Can you Differentiate a Polynomial?. The idea there is to show that polynomial functors form a Cartesian differential category (and thus a tangent category, since any Cartesian differential category is a tangent category). A particular example of this differentiation is then differentiation of combinatorial species (eg., as in wikipedia). We haven’t yet gone any further with these ideas, but there may be interesting developments if one attempts to apply some of the tangent category theory to this example (though it may be a bit tricky, as this example is perhaps more naturally a differential/tangent bicategory).

So, there is no differential category of species. Instead, there is a (putative) differential bicategory in which 1-morphisms (between particular objects) are species and the operation of differentiation on them is the usual differentiation of species. To see this, species should be represented as analytic functors. Then the construction of this bicategory should be similar to the bicategory of polynomial functors (which are a special case of analytic functors) which is described in the slides linked above.

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