Speakers

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.

Title: A Topology on Categories of Locally Ringed Space and Applications to Arithmetic Geometry

Abstract: In this talk we present some relevant background on locally ringed spaces and consider a problem that appears in arithmetic geometry. We then use this problem to motivate a class of functors that arise in arithmetic geometry, provide examples of these functors, and show how these functors can induce a Grothendieck topology on their codomains. We then use this topology and its topos of sheaves to make progress towards our motivating problem.

Title: Cubical Sets and A-Homotopy Theory

Abstract: In this talk, I will describe the cubical category and cubical sets. Then I will show how A-homotopy theory, a discrete homotopy theory for graphs, gives us an example of a cubical set. Time permitting, I will discuss how this cubical set might lead us to a chain complex.

Title: Extending CNOT to real stabilizer quantum mechanics

Abstract: The stabilizer formalism for quantum mechanics is an important tool for implementing fault tolerant quantum circuits.  In this talk we first give a brief overview of the stabilizer formalism.  We also will discuss the angle-free fragment of the ZX calculus, which is complete for the real fragment of stabilizer quantum mechanics.  We use this fact to extend the category CNOT to be complete for this fragment of quantum mechanics.