Title: Latent fibrations: some theory, some examples
Abstract: Latent fibrations are to restriction categories what fibrations are to ordinary categories.
I shall introduce their basic theory and explore some basic examples including the “standard” latent fibration and the latent fibration of “propositions”.
Title: Taking the derivative of computations, backwards
This talk will introduce reverse differential restriction categories. The reverse derivative is a fundamental operation in machine learning and differential programming. Reverse differential categories provide an axiomatization of the reverse derivative. In this talk, we will expand the axiomatic framework for reverse differentiation by combining it with restriction structure; this allows for reverse differentiation functions that may be partial (such as those defined by while-loops).
Title: Completely positive maps and the Cartesian completion of a discrete inverse category
Abstract: In this talk, I relate Giles’ Cartesian completion of a discrete inverse category to Coecke and Heunen’s CP^∞ construction of quantum channels for well behaved symmetric †-monoidal categories. In particular, by taking the subcategory of classical channels of CP^∞(C), CP*(C) for a well-behaved discrete inverse category C, one obtains precisely the Cartesian completion of C.
Title: Exponential Functions for Cartesian Differential Categories.
Abstract: We introduce differential exponential maps in Cartesian differential categories, which generalizes the exponential function from classical differential calculus. In particular, differential exponential maps can be defined without the need for limits, converging power series, multiplication, or unique solutions of certain differential equations — which most Cartesian differential categories do not necessarily have. Every differential exponential map induces a commutative rig, called a differential exponential rig, and conversely, every differential exponential rig induces a differential exponential map. Examples of differential exponential maps in the Cartesian differential category of real smooth functions include the exponential function, the complex exponential function, and the dual numbers exponential.
Title: Lie algebroids are the same as involution algebroids in the category of smooth manifolds.
Abstract: Involution algebroids are a generalisation of Lie algebroids that make sense in any tangent category. The aim of this talk is to sketch a proof that the category of Lie algebroids is isomorphic to the category of involution algebroids in the category of smooth manifolds. Our method is to use the structure equations of the Lie algebroid to mediate between the two definitions. The advantage of this approach is that it reveals that the Tulczyjew involution in Lagrangian mechanics satisfies the involution algebroid axioms. This is joint work with Ben MacAdam and is based on an idea of Richard Garner.
Title: Graphs and Anchored Bundles
Abstract: Lie’s second theorem for groupoids states that there is a full and faithful functor from the category of Lie groupoids to Lie algebroids. We will consider a simplification of this theorem by looking at reflexive digraphs rather than groupoids and the corresponding functor to anchored bundles.
Title: Dagger linear monoids in dagger LDCs
Categorical quantum mechanics describes quantum observable as dagger Frobenius Algebra with extra properties in the category of finite-dimensional Hilbert Spaces. In this talk, I will generalize the notion of dagger Frobenius Algebras to dagger linearly distributive categories (dagger-LDCs). We call the dagger Frobenius Algebras in this setting as dagger linear monoids. We observe that in the setting symmetric dagger LDCs, a dagger linear dual gives rise to endomorphism monoids (usually referred to as the pants monoid) for which the multiplication is anti-isomorphic to the dagger of the comultiplication. Finally, we examine the conditions under which the anti-isomorphic pants monoid coincides with the usual pants monoid in a unitary category.
Title: Generalized Algebraic Theories and Differential Objects
Abstract: We introduce enriched algebraic theories with generalized arities, and see how this can be applied to tangent categories. In particular, we will show that every tangent category embeds into a so-called linear/nonlinear system between a monoidal differential category and cartesian tangent category.
This is part of an ongoing collaboration with Jonathan Gallagher and others.
Abstract: In this talk we’ll investigate how to define vector fields and their flows in a tangent category, and how to prove a result about commutation of vector fields and flows in this setting. In the first half of the talk, we’ll introduce the notion of a “curve object” in a tangent category: an object which “uniquely solves ordinary differential equations in the tangent category”. In the second half of the talk, we’ll see how considering “vector fields and flows in the tangent categories of vector fields and flows” leads to a new proof of the commutation theorem for vector fields and flows (Proposition 18.5 in Lee’s “Introduction to Smooth Manifolds”).
Bio: Geoff is an Associate Professor at Mount Allison University in Sackville, NB. His general interest is category theory. He is currently investigating how to generalize as much of differential geometry as possible to the setting of tangent categories. Geoff received his PhD at Dalhousie University in 2009 under the supervision of Richard Wood, and subsequently did postdoctoral research at the University of Calgary (supervised by Robin Cockett) and the University of Ottawa (supervised by Rick Blute) before taking up his current position at Mount Allison.
We review recent results using Cartesian differential categories to model backpropagation through time, a training technique from machine learning used with recurrent neural networks. We show that the property of being a Cartesian differential category is preserved by a variant of a stateful construction commonly used in signal flow graphs. Using an abstracted version of backpropagation through time, we lift the lift the differential operator from the starting differential category to the stateful one.
David is a project research at the ERATO MMSD project in Tokyo. This project aims to extend formal methods and software verification techniques to cyber-physical systems, with particular emphasis on applications to automotive control and manufacturing.
David received a PhD in mathematics at Indiana University in August 2017 as a student of Larry Moss. His academic research interests are primarily in coalgebra, logic, and category theory. Since moving to Tokyo, he has been developing an interest in quantitative refinements of bisimulation and other coalgebraically defined structures. He has also been looking into deep learning and neural networks.