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 fully-faithful, 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 involution algebroids are a coreflective subcategory of smooth groupoids.
This talk is based on joint work with Matthew Burke.
Title: Props and Distributive Laws
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?
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.
Title: What is the category associated to a programming language with recursion and partiality, and can we make one from a differential language?
Abstract: This talk will introduce a famous concept in the functional programming community known as “immoral reasoning is acceptable.” This is the first talk in a 2 talk series to develop these ideas for a functional differential programming language. We will develop the category of partial equivalence relations for a functional programming language to give a way to move between denotations of total and partial functions.
The dual fibration, part two: partial case
Last week we reviewed how to construct the dual fibration to a given fibration, and saw that this construction gives some interesting examples. In this second part we’ll see how to work with this idea in the setting of restriction categories. We’ll begin by defining and working with latent fibrations (a version of the fibration notion for restriction categories), then show that certain kinds of latent fibrations have a dual.
This is joint work with Robin Cockett, Jonathan Gallagher, and Dorette Pronk.