[Alta-Logic] Alberta Mathematics Dialogue 2018 Abstracts

[Matthew Burke]

Please find below the times, speakers and abstracts for the Alberta mathematics dialogue.

As a reminder about arrangements, we will be in ST143 on Friday March 4. The online registration form is now closed but on-site registrations will be available in the Science Theatre Atrium at the workshop.

(2:30-3:00)
Rory Lucyshyn-Wright, Mount Allison University
Distribution monads, algebraic dualities, and the relation between probability and convexity

Abstract: The concept of dualization of linear spaces admits a far-reaching generalization through the notion of an algebraic duality [3], i.e., a contravariant adjunction between (enriched) algebraic categories. Every algebraic duality is induced by a dualizing algebra and determines an induced notion of distribution that specializes to yield various kinds of measures, Schwartz distributions, filters, closed subsets, compacta, and so forth [2].

In this talk, we will treat a specific example, beginning with the algebraic category of convergence convex spaces, with the unit interval as the dualizing algebra, and we will discuss the associated algebraic duality. We will show that the induced notion of distribution is a generalization of the notion of Radon probability measure. The proof employs R. C. Buck's representation theorem [1] involving the strict topology on the space of bounded continuous functions. Along the way, we also prove a representation theorem for bounded Radon measures that is formulated in terms of the unit interval and certain simple categorical constructions.

[1] R. C. Buck, Bounded continuous functions on a locally compact space. The Michigan Mathematical Journal 5 (1958) 95–-104.

[2] R. B. B. Lucyshyn-Wright, Functional distribution monads in functional-analytic contexts. Advances in Mathematics 322 (2017), 806-860.

[3] R. B. B. Lucyshyn-Wright, Algebraic duality and the abstract functional analysis of distribution monads. Talk at CT 2017: International Category Theory Conference, Vancouver, July 2017.

(3:00-3:30)
Jonathan Gallagher, University of Calgary
Smootheology, Weil algebras, and tangent structure

Abstract: In this talk we will develop categories of generalized smooth spaces, or smootheologies, from the tangent categories perspective. We will make use of actions by the category of Weil algebras to extract tangent categories from Weil, Sikorski, Diffeological, and Frölicher spaces.

(3:30-4:00)
Matthew Burke, University of Calgary
A Tangent Category of Infinity Categories

Abstract: In classical calculus we approximate an appropriately differentiable function using a sequence of simpler functions called the Taylor polynomials. In an analogous way the Goodwillie calculus describes how to approximate a functor whose domain and codomain are appropriately topological by using a sequence of simpler functors. In this talk we take the first steps towards describing some fundamental parts of the Goodwillie calculus in terms of the theory of tangent categories. A tangent category is an axiomatisation of the tangent bundle functor on the category of smooth manifolds that was introduced by Rosicky in 1984 and then extended by Cockett and Cruttwell in 2013.

First we give a brief introduction to the Goodwillie calculus using infinity categories which follows previous work by Lurie. Next we identify the main components of a tangent category on the category of presentable infinity categories. We will then describe the connection of this work to a paper of Bauer et al. that constructs a directional derivative in the calculus of functors that, along with a suitably chosen ambient category, satisfies all the axioms for being a Cartesian differential category. This work is part of a joint project with Kristine Bauer and Michael Ching.