# Is there anything to the obvious analogy between Joyal's combinatorial species and Goodwillie calculus?

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 .

• Nice question. I am skeptical of the existence of an "easy" characterization of (weakly) analytic functors in Goodwillie calculus. To me it resembles characterizing the rational spaces: easy in the simply connected case (i.e. when you have connectivity estimates), very hard otherwise Apr 4, 2017 at 2:19
• I don't know anything about Goodwillie calculus. Can you describe a concrete calculation (as opposed to a theoretical construction) that uses it? In combinatorics, many concrete examples can be found in Flajolet and Sedgwick's "Analytic Combinatorics" or Stanley's "Enumerative Combinatorics 2" (using the language of exponential generating functions rather than species). There are many examples where one starts with a combinatorial construction, then derives a functional equation, then finally gets the Taylor coefficients. (But it's never a surprise that the Taylor coefficients exist.) Apr 4, 2017 at 15:24
• @TimothyChow I'm not an expert and would love if somebody more knowledgeable could chime in on this, but if you have an analytic functor $F$, then the Goowillie tower (i.e. the Taylor expansion) gives you a spectral sequence (the Goodwillie spectral sequence) which under favorable conditions will converge and thus give you a method to compute the homotopy groups of $F(X)$ from the homotopy groups of the Taylor approximations $F_n(X)$. Apr 5, 2017 at 1:12
• For example, when $F$ is the identity functor $\mathsf{Top} \to \mathsf{Top}$, one can thus compute unstable homotopy groups from stable data (cf. Behrens). An important fact in this case is that the derivatives of the identity have an operad structure (cf Ching). When $F(X)$ is the space of embeddings of $M \to X$, one can glean data about embedding spaces from data about immersion spaces (see Goodwillie and Weiss). Functors like algebraic K-theory can also be analyzed this way. Apr 5, 2017 at 1:15
• I'm guessing species come in when you compute homology or rational homotopy of Taylor towers. The part of the story I'm familiar with is spaces of long knots. Their rational homotopy is given by a spectral sequence which involves very combinatorial objects, and turn out to be a version of graph homology, at least on the first page. See my paper "Homotopy approximations to spaces of knots." Graph homology has a species interpretation (see a survey paper by Swapneel Mahajan.) Apr 5, 2017 at 5:15