University of Regina Topology Seminar

For F a homotopy functor from spectra to spectra, the Goodwillie derivatives $\partial_* F(X)$ based at an object X form a symmetric sequence of spectra. Given two such functors F and G, how are the derivatives of the composite FG related to the derivatives of F and of G? In this talk, I will present a chain rule due to Ching. Namely, there is an equivalence of symmetric sequences $\partial_*(FG)(X) \simeq \partial_*F(GX) \circ \partial_*G(X)$, where $\circ$ denotes the composition product.

I will compare Ching's chain rule to Faà di Bruno's formula for higher derivatives of a composite of two functions, along with the combinatorics involved. I will also sketch the proof of the chain rule.

This talk will go over the paper of Arone and Ching entitled "Operads and chain rules for the calculus of functors". I will explain the central ideas of the proofs of the two main results stated as follows. Theorem 1: If $F: C \to D$ is a homotopy functor with each C or D either equal to Top_* (the category of pointed spaces) or Spec (the category of spectra), then the derivatives of F can be given the structure of a bimodule (over two operads) in a natural way. Theorem 2: If $G: C \to D$ and $F: D \to E$ are two "nice functors", there is a formula (called "chain rule") describing the derivatives of the composite functor FG in terms of those of F and G.

Classical stack theory concerns itself with glueing together objects along isomorphisms. However, I want to present a general framework which allows us to explicitly glue together objects along some kind of notion of weak equivalence (e.g. quasi-isomorphisms of complexes of sheaves, stable equivalences of spectra, etc.). The case we focus on is when our weak equivalences are those of a simplicial model category, although this can be generalized.

In particular, we show how various glueing conditions involving weak equivalences hold for presheaves of simplicial model categories satisfying an analogue of hyperdescent. The proof, which will be outlined briefly, uses the homotopy coherent nerve functor and a homotopy-theoretic description of classical stack theory due to Jardine.

I will go over the second main result of the Arone-Ching paper which states that if $G : C \to D$ and $F : D \to E$ are two "nice functors", there is a formula (called "chain rule") describing the derivatives of the composite functor FG in terms of those of F and G.

We present the first two sections of the Johnson-McCarthy paper which introduces their Taylor tower and studies the cross effects functor.

We present the second and third sections of the Johnson-McCarthy paper which studies the cross effects functor and the degree of the functors in their Taylor tower.

We present the last two sections of the Johnson-McCarthy paper which studies convergence and the relationship with the Goodwillie Taylor tower.

We approach manifold topology by examining configurations of finite subsets of manifolds. The homotopy types of such configurations organize as an ∞-category, the construction of which makes use of stratified spaces and exit-path ∞-categories thereof. The goal of this talk is to supply the constructions of these ∞-categories and to identify these ∞-categories in the case of $S^1$ and $\mathbb{R}^n$ in terms of the combinatorially defined parasimplex category and Joyal’s category $\Theta_n$, respectively.

Let M be a manifold, and let O(M) be the poset of open subsets of M ordered by inclusion. Manifold calculus, due to Goodwillie and Weiss, can be defined as the study of contravariant functors F from O(M) to spaces. The philosophy is to take a functor F and replace it by its Taylor tower which converges to the original functor in good cases, much like the approximation of a function $f: \mathbb{R} \to \mathbb{R}$ by its Taylor series. This talk will focus on the role the derivatives of a functor F play in this theory, and the analogies with ordinary calculus.