I will review the theory of monads and their Kleisli categories. Then I'll argue that taking a double category point of view sheds some light on their morphisms and 2-cells.
Details about the Atlantic Category Theory (ATCAT) Seminar for the 2019 fall term can be found below. Information about past seminars can be found here.
Robert Paré (Dalhousie University)
Kleisli Double Categories I
I will review the theory of monads and their Kleisli categories. Then I'll argue that taking a double category point of view sheds some light on their morphisms and 2-cells.
Robert Paré (Dalhousie University)
Kleisli Double Categories II
I will continue my case study of double categories arising from monads and comonads. I will look at companions and conjoints, double functors, and horizontal and vertical transformations, all motivated by the Kleisli construction.
Geoffrey Cruttwell (Mount Allison University)
Vector Fields and Flows, Categorically I
In this series of three talks, we'll investigate how to define and work with the idea of vector fields and their associated flows in the setting of a tangent category. Tangent categories are categories equipped with an abstract version of the tangent bundle for smooth manifolds; as such, vector fields are easy to define in this setting. However, to talk about their associated flows requires more care: we need an object "which uniquely solves ordinary differential equations" in the tangent category. We'll investigate this idea, and then focus in particular on how considering tangent categories of vector fields and flows gives us a new perspective on commutation of vector fields and flows. If time permits, we will also consider flows for linear vector fields in this abstract setting. This is based on joint work with Robin Cockett, JS Lemay, and Rory Lucyshyn-Wright.
Geoffrey Cruttwell (Mount Allison University)
Vector Fields and Flows, Categorically II
This is a continuation of the September 24th talk.
Geoffrey Cruttwell (Mount Allison University)
Vector Fields and Flows, Categorically III
This is a continuation of the September 24th and October 1st talks.
Theo Johnson-Freyd (Perimeter Institute)
Condensation and Idempotent Completion
Idempotent (aka Karoubi, aka Cauchy) completion appears throughout mathematics: for instance, it converts the category of free modules to the category of projective modules. I will explain the higher-categorical generalization of idempotent completion. I call it "condensation", because, as I will explain, if you start with a category of gapped phases of matter, then its idempotent completion consists of those phases that can be condensed from the phases you already have. In particular, if you start just with the vacuum phase, and idempotent complete, you recover a very large class of gapped phases, including the Turaev--Viro--Barrett--Westbury models. Moreover, every condensable-from-the-vacuum phase of matter is fully dualizable (i.e. determines a fully-extended TQFT), and conversely every condensable-from-the-vacuum TQFT has a commuting projector Hamiltonian model, and so one finds an equivalence between large classes of TQFTs and condensed phases. Based on joint work with Davide Gaiotto.
Frank Fu (Dalhousie University)
Interpreting a Simple Dependent Type System in a
Locally Cartesian Closed Category
It is well known that locally cartesian closed category (LCCC) can be used to understand dependent types (Seely 1984). In this talk, I will first explain how to define a notion of dependent function space in LCCCs. Then we will see how to interpret types as objects, terms as morphisms in LCCCs for a very simple dependent type system. Finally, I will describe how LCCCs can naturally model substitution for types and terms. This talk is based on an ongoing project with Kohei Kishida, Neil Julien Ross, Peter Selinger.
Jeff Egger
On the Naturality of Fourier-Stieltjes Transforms
I
Some years ago, I gave a well received talk on the naturality of Fourier transforms (which can be defined on any abelian locally compact group). In writing up my notes on the topic, I found myself increasingly drawn to Fourier-Stieltjes transforms, which seem to be better behaved (categorically speaking). This will be a relatively basic talk: I will not assume that you remember much undergraduate analysis.
Jeff Egger
On the Naturality of Fourier-Stieltjes Transforms
II
I will wrap up some loose ends from my previous talk.