Speakers

Title: Ehrhard’s exponential modalities are free!

Abstract: Ehrhard introduced models of linear logic based on “Finiteness spaces” in 2005.   Priyaa Srinivasan, Cole Comfort and I used one of these models (finiteness matrices) as a model for our version of infinite dimensional quantum mechanics.  To model certain quantum phenomena, we needed free exponential modalities and set out to prove that Ehrhard’s are free … to discover (belatedly) that they are, indeed, free thanks to JS pointing us at Christine Tasson’s PhD. thesis!

The talk introduces the linear logic models based on finiteness spaces and develops what we expected to be the free exponential modalities therein (which is the E_\inft of Tasson, Tabareau, and Mellies) … but which we belatedly discovered fails to be such.

Title: Complementarity in dagger linearly distributive categories

Abstract: Complementarity is key feature that distinguishes quantum from classical mechanics. Two physical variables are complementary if measurement of one variable leads to maximum uncertainty about the value of the other, and vice versa. Algebraically, complementarity is described as two commutative dagger Frobenius Algebras interacting by the Hopf Law in a dagger symmetric monoidal category. The goal of this talk to set up complementarity within the framework of dagger linearly distributive categories. As an example of our algebraic description of complementarity in this setting, I will show that splitting certain kind of idempotents on exponential modalities (! and ?) gives rise to complementary observables.

Title: The Grothendieck Construction for Double Categories

Abstract: I will describe what a double index functor is and describe a Grothendieck construction of a double category of elements and show that it forms a lax double colimit for the diagram. I will discuss how these colimits are related to other notions of 2-categorical colimits and discuss the relationship with other double categorical Grothendieck constructions.

Title: Lie integration as a left Kan extension

Abstract: Charles Ehresmann first introduced Lie groupoids as groupoids in the category of smooth manifolds through his initial work in sketch theory. His student Pradines developed Lie algebroids as the “infinitesimal approximation” of a Lie groupoid (extending the Lie group-Lie algebra correspondence). The question of what Lie algebroids could be “integrated” into a source-simply-connected Lie groupoid was one of the significant problems in differential geometry and Lie theory, and was resolved by Crainic and Fernandes in their paper “On the integrability of Lie brackets.”

In this talk, we will use Kelly’s enriched sketches to show that Lie integration is a left Kan extension. The construction follows in three steps: we first show that Lie algebroids are equivalent to involution algebroids, we next show that involution algebroids are sketchable, and finally we show that the theory of involution algebroids has a left-exact inclusion into the theory of smooth groupoids. A useful consequence of this result is that in presentable tangent categories (such as models of synthetic differential geometry) we can see that there is an adjunction between involution algebroids and smooth groupoids.

This talk is based on joint work with Matthew Burke.

Title: Props and Distributive Laws

Abstract:
In this talk, I will review Lack’s technique of composing props, and give examples thereof.
Many well-known concrete structures are presented by props; for example, (FinSet,+) is presented by the prop for the free commutative monoid.  And by composing this prop with its opposite category using Lack’s technique, we obtain either bicommutative bialgebras or bicommutative Frobenius algebras, which are presentations for spans or cospans of finite sets, respectively—depending on the direction we compose things.
This modular process of building up props continues, eventually yielding much more complex structures.  I intend to continue this process of building up props to the point where a fragment of the ZH calculus is obtained.

Title: Approaches to (∞,n)-categories
Abstract:  The structure of an (∞,n)-category, or homotopical higher category, has become important not only in category theory and abstract homotopy theory, but is also arising in a number of other areas of mathematics, including topology, mathematical physics, and algebraic geometry.  In this talk, we’ll start with the idea of what an (∞,n)-category should be, and why we might want to consider such structures.  Then we will consider ways to model them by explicit mathematical objects, and why it is good to do so in different ways.  We will focus on small values of n but give an indication of how these models can be generalized to higher n.

Title: A Complete Axiomatisation of Partial Differentiation

Abstract. Looking at recent work on categories equipped with differential structure (e.g., Blute, Cockett, and Seely’s cartesian differential categories)  one naturally asks if the categorical axioms are not only sound but also complete for natural examples, (e.g., smooth functions). Here we look at a related question: whether the well-known rules of partial differentiation are complete for smooth functions.
To do so, we first formalise these rules in second-order equational logic,  a version of equational logic with function variables and binding constructs. We prove the resulting theory is complete with respect to smooth functions, indeed even with respect to polynomial interpretations. The proof makes use of Severi’s interpolation theorem that all multivariate Hermite problems are solvable.

Title: Representable tangent ∞-categories and functor calculi

Kristine Bauer, Matthew Burke and I have constructed a tangent structure T on a certain ∞-category of ∞-categories which is related to Goodwillie’s calculus of functors. This tangent structure is not representable, though it looks kind of like it almost is. I will try to describe a way in which we might view T as (co?)represented by a certain ∞-topos (the ∞-topos of parameterized spectra).

One of the reasons for taking this perspective is to try to fit other versions of functor calculus from homotopy theory into the tangent category framework. In particular, I will describe the “manifold calculus” (of Goodwillie and Michael Weiss) and “orthogonal calculus” (of Weiss), and ask to what extent they also can be viewed in terms of tangent categories or generalizations thereof.

Title: When is a Hopf Monad a Trace Monad?

Abstract:
Trace monads are monads on trace monoidal categories which lift the trace to the Eilenberg-Moore category. Hopf Monads are comonoidal monads whose fusion operators are invertible, and it has been shown that a Hopf monad on a compact closed category lifts the compact closed structure to the Eilenberg-Moore category. Since every compact closed category is also a trace monoidal category, a natural question to ask is what is the relationship between Hopf monads and trace monads. In this talk, I will give introductions to trace monads and Hopf monads, and give a necessary condition for when a Hopf monad is a trace monad, and also give examples of trace monads that are not Hopf monads. We conjecture that not all Hopf monads are a trace monad, but unfortunately, we do not yet have an example of a Hopf monad which is not a trace monad! This is joint work with Masahito Hasegawa.

Title: Embedding theorem for affine tangent categories.

Abstract: Affine tangent categories were introduced by Blute, Cruttwell and Lucyshyn-Wright to model the category of affine manifolds and affine mappings. In this talk, we show that every affine tangent category embeds into an affine tangent category with a tensor representable tangent structure.

Joint work with Jonathan Gallagher and Rory Lucyshyn-Wright.