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Apr 5, 2017 at 5:15 comment added Jim Conant 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 3:42 history edited Tim Campion CC BY-SA 3.0
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Apr 5, 2017 at 3:35 history edited Tim Campion CC BY-SA 3.0
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Apr 5, 2017 at 3:29 history edited Tim Campion CC BY-SA 3.0
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Apr 5, 2017 at 3:20 comment added Tim Campion Let's see... If I recall correctly, if $F$ is a species, then $\exp F$ is the species where the connected components come from $F$, right? So their Taylor series are related by the chain rule. The chain rule for Goodwillie calculus is an important result. Another thing that occurs to me is that I think in abelian functor calculus there is less complexity in how the layers of the Taylor tower fit together, so the analogy to species might be more direct.
Apr 5, 2017 at 3:09 comment added Timothy Chow A big thing that species give you is a way to estimate the asymptotic behavior of the Taylor coefficients, if you know how to build your object from simpler objects using certain standard constructions. For example if your combinatorial object of interest can be decomposed into a disjoint union of "connected components" then the theory tells you how to relate the Taylor coefficients of a connected object to the Taylor coefficients of a (possibly) disconnected object. If there's any cross-fertilization to be had, I'd think there'd need to be something analogous in Goodwillie calculus.
Apr 5, 2017 at 1:21 comment added Tim Campion In general, one of the important facts is that the derivatives of the identity are stable invariants -- they are associated to spectra, i.e. homology theories, which homotopy theorists actually have tools to analyze. The reason homology theories can be analyzed is that they satisfy excision -- which makes them "linear" in Goodwillie's sense.
Apr 5, 2017 at 1:15 comment added Tim Campion 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 1:12 comment added Tim Campion @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 4, 2017 at 15:24 comment added Timothy Chow 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 2:19 comment added Denis Nardin 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 1:22 history asked Tim Campion CC BY-SA 3.0