Combinatorial species and calculus of functors both take the viewpoint that many interesting functors can be expanded in a kind of Taylor series. Many operations familiar from actual calculus can be imported into either setting. Is there anything deeper to this comparison? Might it be fruitful to try to transport questions, results, methods back and forth between the two areas?

Although they work with very similar analogies, besides the obvious differences, I should note that at present the goals of the two theories appear to be rather different (disclaimer: I am an expert in neither field). Typically, one will define a Combinatorial species in terms of its Taylor coefficients, and then reap the benefits of the perspective shift coming from the functorial viewpoint. For example the machinery allows one to construct new interesting species from old ones. On the other hand, in Goodwillie calculus one typically *starts off* with a functor, and the very *existence* of the Taylor coefficients is a perhaps surprising consequence of the machinery -- this is the information one wants to get at.
**EDIT**: Let me clarify that the derivatives of *any* functor exist in Goodwillie calculus -- the surprise is over once you learn the general theory (actually, this seems strange in terms of the analogy -- it seems to say that every functor is in some sense smooth... or perhaps it's just that the coefficients of the Taylor tower live in a category (the stabilization of the codomain) which has been constructed to accommodate very general notions of "direction" that ordinary notions of smoothness don't quite get at -- there's a degree of control which is more like the situation in algebraic geometry than in analysis). Of course, the Taylor tower doesn't converge for an arbitrary functor. I don't know of a way of talking about nonconvergent Taylor towers in the theory of species...

As an example of one kind of result in one area that one might want to try to mimic in the other, Joyal characterized (see Theorem 1 in the appendix here) the "analytic" functors in his sense (i.e. the ones with a power series expansion) as those functors $\mathsf{Set} \to \mathsf{Set}$ which preserve filtered colimits, cofiltered limits, and weak pullbacks. The only sufficient condition I've heard of for functors $\mathsf{Top} \to \mathsf{Top}$ to be analytic in Goodwillie's sense (i.e. so that the Goodwillie tower converges) involves explicit connectivity estimates. Is it possible that there exists a nice characterization of analytic functors in Goodwillie calculus, perhaps analogous to the one for species?

**EDIT**

A comparison that really deserves to mentioned is that a species (which is the same thing as an "analytic" functor in Joyal's sense) is the same thing as a functor $\mathsf{Set} \to \mathsf{Set}$ which is the left Kan extension of a functor $\mathrm{Core}(\mathsf{FinSet}) \to \mathsf{Set}$, where $\mathrm{Core}(\mathsf{FinSet})$ is the groupoid of finite sets and bijections. For certain codomain categories $\mathcal{D}$ (I think that $\mathcal{D}$ should stable?), an analytic functor $F: \mathsf{Top} \to \mathcal{D}$ in Goodwillie's sense is the same thing as a functor which is the inverse limit of a tower of functors $F_n: \mathsf{Top} \to \mathcal{D}$ such that $F_n$ is the left Kan extension of a functor $\mathsf{FinSet}_{\leq n} \to \mathcal{D}$, where $\mathsf{FinSet}_{\leq n}$ is the category of finite sets of cardinality $\leq n$ and *all* maps. Saul Glasman has a perspective which I think gives another version of this statement ~~ for more general codomains ~~.

exist.) $\endgroup$ – Timothy Chow Apr 4 '17 at 15:243more comments