University of Regina Topology Seminar

A long standing slogan in Goodwillie's functor calculus is that the derivatives of the identity functor on a suitable model category should come equipped with a natural operad structure. A result of this type was first shown by Ching for the category of based topological spaces. It has long been expected that in the category of algebras over a reduced operad $\mathcal{O}$ of spectra that the derivatives of the identity should be equivalent to $\mathcal{O}$ as operads.

In this talk I will discuss my recent work which gives a positive answer to the above conjecture. My method is to induce a "highly homotopy coherent" operad structure on the derivatives of the identity via a pairing of underlying cosimplicial objects with respect to the box product. This cosimplicial object naturally arises by analyzing the derivatives of the Bousfield-Kan cosimplicial resolution of the identity via the stabilization adjunction for $\mathcal{O}$-algebras. Time permitting, I will describe some additional applications of these box product pairings including a new description of an operad structure on the derivatives of the identity in spaces.

By considering algebras over an operad $\mathcal{O}$ in one's preferred category of spectra, we can encode various flavors of algebraic structure (e.g. commutative ring spectra). Topological Quillen (TQ) homology is a naturally occurring notion of homology for these objects, with analogies to both singular homology and stabilization of spaces. In this talk, we will begin by discussing a fibration theorem for TQ-completion, showing that TQ-completion preserves fibration sequences in which the base and total $\mathcal{O}$-algebra are connected. We will then describe a few results that hint towards an intrinsic connection between TQ-completion and the convergence of the Taylor tower of the identity functor in the category of $\mathcal{O}$-algebras. Lastly, time permitting, we will discuss recent joint work with Crichton Ogle on the chromatic localization of the homotopy completion tower of $\mathcal{O}$-algebras and connections to functor calculus.

We will discuss the equivalence between differential graded (DG) algebras and algebras over the Eilenberg-MacLane spectrum HZ.

We will describe the derived category of a DG-algebra as a triangulated category. It arises as the homotopy category of a model structure on DG-modules. We will tackle the question: When do two DG-algebras have equivalent derived categories? In stark contrast to the case of rings, for DG-algebras, the derived category does not determine the homotopy theory of DG-modules.

I will talk about topological equivalence between DG-algebras and look at the difference between topological equivalence and quasi-isomorphism.

$A_{\infty}$ algebras are introduced in the context of cochain complexes. Motivation for the defining equations of $A_{\infty}$ algebras is provided before it is demonstrated that $A_{\infty}$ algebras can be viewed as coderivations on a tensor coalgebra. Finally, a sketch of Kadeishvili's theorem is provided, showing that the cohomology of differential graded algebras admits an $A_{\infty}$ algebra structure quasi-isomorphic to the DG-algebra.

Rational homotopy theory is modeled by the homotopy theory of (graded) commutative DG-algebras (CDGAs). We introduce the notion of rational equivalence of spaces, and then develop the theory of CDGAs. We start with some basic standard notation for CDGAs and then discuss equivalences and formality. We describe a number of examples of CDGAs and maps between them. Looking at the role of cofibrancy, we describe the difference between CDGAs and DGAs and finish with an example of a non-formal CDGA.

I will explain how (graded) commutative DG-algebras (CDGAs) model rational homotopy theory. The functors in both directions are: the PL de Rham complex of a simplicial set, and the spatial realization of a CDGA. I will describe the model structure on rational CDGAs, which is not available in positive characteristic. Then I will sketch why the two functors induce an equivalence of homotopy categories when restricted to simply-connected objects.

In this seminar talk, we give some examples of minimal models together with some computations using them. These computations include the rational homotopy groups of spheres and the rational cohomology of Eilenberg-MacLane spaces.

I will talk about the Sullivan models for homotopy fibers and pullbacks, and look at the rational cohomology of loop spaces of simply connected spaces.

We introduce differential graded Lie algebras and the Quillen model for rational homotopy theory, and do a few examples. We will also describe a connection with the Adams-Hilton model and Milnor-Moore's work on the homology of loop spaces and rational homotopy groups.

I will explain how a Sullivan minimal model of a space encodes the Whitehead product on rational homotopy groups, namely as the dual of the quadratic part of the differential. I will look at a few examples and computations.