Speakers

Title: String Diagrams for Regular Restriction Categories

Abstract: I will discuss how cartesian restriction categories can be reasoned about using string diagrams for monoidal categories, including how this extends to regular restriction categories. Specifically, we will see that every cartesian bicategory of relations (in the sense of Carboni and Walters) has a regular restriction category of partial maps as a subcategory, and that from a regular restriction category we can construct the category of relations of which it is the partial map subcategory. This all works whether or not our restriction categories are split.

Title:          Dagger Frobenius relations for dagger linearly distributive categories

Abstract:   Commutative dagger Frobenius algebras play a central role in categorical quantum mechanics due to their correspondence to orthogonal basis in the category of finite-dimensional Hilbert spaces. Subsequently, such algebras represent quantum observables. Measurement in a dagger monoidal category is an isometry, m: A \to X (i.e, m^\dagger m = 1_X) where A is any object, and X is a special commutative dagger Frobenius Algebra.

In this talk, I will generalize dagger Frobenius Algebras from dagger monoidal categories to dagger linearly distributive categories. We refer to the generalization as dagger linear monoids. I will provide the conditions under which a dagger linear monoid gives a dagger Frobenius algebra in a unitary category. We show the correspondence between dagger linear duals and dagger linear monoids. We find that in a complete and cocomplete category, limit of dagger linear monoids is a dagger linear monoid. Thus one can represent, possibly, infinite dimensional quantum observables using dagger linear monoids.

If time permits, I will discuss measurements for dagger linear monoids. A measurement for a dagger linear monoid is a retract from the linear monoid to a special commutative dagger Frobenius Algebra within the unitary core.

Title: The ZX& calculus

Abstract: Consider ZX&, the fragment of the ZX calculus generated by the copying/addition Frobenius algebras, the not gate and the and gate.  I prove that this fragment is complete and universal for a prop of spans of sets, by freely adding units and counits to the inverse products of TOF (the category generated by the Toffoli gate and computational basis states/effects).  To prove completeness, I first show show that adding a counit to TOF is the same as computing the classical channels, which is the the same as the discrete Cartesian completion.  The completeness of ZX& is obtained via a two way translation between this extension of TOF with units and counits, and ZX&.

Title:Enriched limit doctrines, Lawvere theories.

Abstract: Following Lack and Rosicky’s “Notions of Lawvere Theories”, we will look at the theory of strongly finitely presentable objects. This leads to a notion of enriched Lawvere theory that mirrors the classical case, particularly commutative theories and morphisms of theories.

Title: The fast Weil to go Faa (with Ben MacAdam)

Abstract:
This talk will introduce the notion of a tangent complex in an arbitrary category.  We will then use tangent complexes to show that tangent categories are precisely coalgebras of a comonad. We will then show how to reconstruct the Faa di Bruno construction as a subconstruction of the cofree tangent category, and we will not require any combinatorics to do this.

Title: Persisten Homology and it Generalization

Abstract: Persistent homology has seen more widespread use as a tool for analyzing data over the last decade. Generalizations of the theory to multiple parameter filtrations would have even broader applications. In this talk we’ll quickly see how (single parameter) persistent homology works and look at McCleary and Patel’s generalization of the notion of a persistence diagram by exploring a counter-example to their first attempt.

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

Abstract:

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.