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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.
Bio:
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.
Title: Differential bundles in the category of smooth manifolds
Abstract: In this talk we give a sketch of the proof that differential bundles in smooth manifolds are vector bundles. If we have time we describe how the addition map of a differential bundle may be recovered from the rest of the structure.
Title: Bertram’s Lie Calculus
Abstract: We will consider Bertram’s Lie Calculus, which develops the foundations of differential calculus and Lie theory concurrently. In particular, we shall consider his presentation of Connes’ Tangent Groupoid on the category of smooth manifolds and consider how this could relate to a categorical semantics for identity types.
Title: Generalized sketches with monad sorts
Abstract: I give a generalization of sketches which captures the theories of group presentations and certain notions of generalized multicategories. I conjecture that the categories of models of these sorts of sketches are locally finitely presentable, so that therefore this generalization stays within the realm of essentially algebraic theories.
Title: Kock-Lawvere Modules in a Tangent Category
Abstract: In the category of smooth manifolds, vector spaces have the property that T(V) is isomorphic VxV. When moving to abstract settings for differential geometry, such as synthetic differential geometry, this need not be the case, especially tangent categories without any sort of ring object. Cockett and Cruttwell introduced the notion of a differential object to axiomatize this structure in tangent categories.
In this talk we will introduce tangent categories with a scalar object – a differential object satisfying a universal property – and develop the theory of Kock-Lawvere R-modules, or KL modules. This will lead to the presentation of differential objects as models of an enriched sketch.
This is joint work with Jonathan Gallagher and Rory Lucyshyn-Wright.
Title: Involution algebroids and their homotopy theory II
Abstract: We define involution algebroids which generalise Lie algebroids to the abstract setting of tangent categories. We first sketch the relationship between our new definition and the classical one. Then as an illustrative example of how to work with the new definition we develop some of the homotopy theory of involution algebroids. This is joint work with Ben MacAdam.
Title: Involution algebroids and their homotopy theory
Abstract: We define involution algebroids which generalise Lie algebroids to the abstract setting of tangent categories. We first sketch the relationship between our new definition and the classical one. Then as an illustrative example of how to work with the new definition we develop some of the homotopy theory of involution algebroids. This is joint work with Ben MacAdam.
Title: Dagger linear logic for categorical quantum mechanics
Abstract: Categorical quantum mechanics is largely based on dagger compact closed categories. A well-known limitation of this approach is that, while it supports finite dimensional processes, it does not generalize well to infinite dimensional processes. A natural categorical generalization of compact closed categories, in which to seek a description of categorical quantum mechanics without this limitation, is in ∗-autonomous categories or, more generally, linearly distributive categories (LDCs).
Last winter, Robin, Cole and myself started development of this direction of generalization. An important first step is to establish the behavior of the dagger in this more general setting. Thus, we simultaneously developed the categorical semantics of dagger linear logic. The goal of my talk is to give the definition of Mixed Unitary Categories [1] which is a key structure in the development of the quantum mechanics in the setting of LDCs. I will also show a running example of this category during my talk.
Reference:
[1] Robin Cockett, Cole Comfort, and Priyaa Srinivasan “Dagger linear logic for categorical quantum mechanics”, arXiv:1809.00275 [math.CT], Sep. 2018.
This week we:-
– recap the definition of dagger-LDC, dagger mix and dagger isomix categories
– define unitary structure for dagger isomix categories
– show that compact LDCs are linearly equivalent to monoidal categories
– define a mixed unitary category
– (If time permits) Finiteness spaces, an example of a Mixed Unitary Category
Title: Dagger linear logic for categorical quantum mechanics
Abstract: Categorical quantum mechanics is largely based on dagger compact closed categories. A well-known limitation of this approach is that, while it supports finite dimensional processes, it does not generalize well to infinite dimensional processes. A natural categorical generalization of compact closed categories, in which to seek a description of categorical quantum mechanics without this limitation, is in ∗-autonomous categories or, more generally, linearly distributive categories (LDCs).
Last winter, Robin, Cole and myself started development of this direction of generalization. An important first step is to establish the behavior of the dagger in this more general setting. Thus, we simultaneously developed the categorical semantics of dagger linear logic. The goal of my talk is to give the definition of Mixed Unitary Categories [1] which is a key structure in the development of the quantum mechanics in the setting of LDCs. I will also show a running example of this category during my talk.
Reference:
[1] Robin Cockett, Cole Comfort, and Priyaa Srinivasan “Dagger linear logic for categorical quantum mechanics”, arXiv:1809.00275 [math.CT], Sep. 2018.
