Title: Building Cartesian Differential Categories as Kleisli Categories
Abstract: Cartesian differential categories (CDC) come equipped with a differential combinator, which provides a categorical axiomatization of the directional derivative from multivariable calculus, and so produces a derivative for every map. An important example of a CDC is the coKleisli category of a differential category. In 2018, BJORT constructed a CDC from abelian functor calculus. However, the BJORT example arises as a Kleisli category rather than a coKleisli category.
In this talk, I will generalize this story and explain how to construct a CDC as a Kleisli category. Since CDC are not self-dual, it is not as simple as taking the dual construction of the coKleisli category: in fact it is very different!
Title: Quillen-Barr-Beck cohomology for restricted Lie algebras
Abstract: The Hochschild cohomology for restricted Lie algebras classifies strongly abelian extensions of restricted Lie algebras. In this talk we define Quillen-Barr-Beck cohomology for the category of restricted Lie algebras and we prove that Quillen-Barr-Beck’s cohomology classifies general abelian extensions. Moreover, using Duskin-Glenn’s torsors cohomology theory, we prove a classification theorem for the second Quillen-Barr-Beck cohomology group in terms of 2-fold extensions of restricted Lie algebras. Finally, we give an interpretation of Cegarra-Aznar’s exact sequence for torsor cohomology. Thus,we obtain for a short exact sequence of restricted Lie algebras, an eight-term exact sequence for Quillen-Barr-Beck cohomology. This sequence replaces the five-term exact sequence proved by Eckmann-Stammbach in the context of Hochschild cohomology.
Title: Locally bounded enriched categories
Abstract: We define and study the notion of a locally bounded category enriched over a (locally bounded) symmetric monoidal closed category, generalizing the locally bounded ordinary categories of Freyd and Kelly. Along with providing many new examples of locally bounded (closed) categories, we demonstrate that locally bounded enriched categories admit fully enriched analogues of many of the convenient results enjoyed by locally bounded ordinary categories. In particular, we prove full enrichments of Freyd and Kelly’s reflectivity and local boundedness results for orthogonal subcategories and categories of models for sketches and theories. We also provide characterization results for locally bounded enriched categories in terms of enriched presheaf categories, and show that locally bounded enriched categories satisfy useful representability and adjoint functor theorems. We also define and study the notion of an alpha-bounded-small limit with respect to a locally alpha-bounded closed category, which parallels Kelly’s notion of alpha-small limit with respect to a locally alpha-presentable closed category. This is joint work with Rory Lucyshyn-Wright at Brandon University.
Title: TQFTs through Parallel Transport and Quantum Computing
Abstract: Quantum computing is captured in the formalism of the monoidal subcategory of Vect (category of complex vector spaces) generated by C^2 – in particular, quantum circuits are diagrams in Vect – while topological quantum field theories, in the sense of Atiyah, are diagrams in Vect indexed by cobordisms. We outline a program to formalize this connection. In doing so, we first equip cobordisms with machinery for producing linear maps by parallel transport along curves under a connection and then assemble these structures into a double category. Finite dimensional complex vector spaces and linear maps between them are given a suitable double categorical structure which we call FVect. Finally, we realize quantum circuits as images of cobordisms under monoidal double functors from these modified cobordisms to FVect, which are computed by taking parallel transports of vectors and then combining the results in a pattern encoded in the domain double category. This talk reports on joint work with Steven Rayan.
Title: Dagger linear logic and Categorical Quantum Mechanics
In this talk, I will present an overview of my PhD research on the foundations of Categorical quantum mechanics. Categorical quantum mechanics uses the graphical calculus of compact closed categories to provide a rigorous and diagrammatic language for describing and reasoning about quantum processes between finite dimensional systems. Compact closed categories provide a categorical semantics for compact multiplicative linear logic with negation. Compact refers to the idea that the multiplicative conjunction and disjunction coincide, and that the truth and the false connective coincide in the fragment of the linear logic. In our research, we have developed a diagrammatic framework for CQM using linearly distributive and *-autonomous categories, with the aim of addressing the dimensionality constraint. *-autonomous categories provide the categorical semantics for (non-compact) multiplicative linear logic with negation. We proved that one can always recover the framework for finite dimensional settings from our more general framework. We generalize the existing algebraic structures of CQM to our new framework and study its implications. Surprisingly, we arrived at a connection between complementary systems of quantum mechanics and exponential modalities (operators which allow non-linear resources) of linear logic in our new general framework.
Title: The Diamond Lemma through homotopical algebra.
The Diamond Lemma is a result indispensable to those studying associative (and other types of) algebras defined by generators and relations. In this talk, I will explain how to approach this celebrated result through the lens of homotopical algebra: we will see how every multigraded resolution of a monomial algebra leads to “its own” Diamond Lemma, which is hard-coded into the Maurer-Cartan equation of its tangent complex. For the reader familiar with homotopical algebra, we hope to provide a conceptual explanation of a very useful but perhaps technical result that guarantees uniqueness of normal forms through the analysis of “overlapping ambiguities”. For a reader familiar with Gröbner bases or term rewriting theory, we hope to offer some intuition behind the Diamond Lemma and at the same time a framework to generalize it to other algebraic structures and optimise it. This is joint work with Vladimir Dotsenko (arXiv:2010.14792).
Title: PD operads and partition Lie algebras.
Partition Lie algebras, introduced by Brantner and Mathew, are homotopy theoretic refinements of Lie algebras that appear in deformation theory in positive characteristic. However, since partition Lie algebras are defined in a purely ∞-categorical way, it is hard to get one’s hands on them. For instance, it is not so clear how to describe them explicitly in terms of point-set models, because there is no operad whose algebras are partition Lie algebras.
In this talk, I will discuss a homotopy-theoretic generalization of the notion of an operad, called a “PD operad”, whose algebras can also carry certain kinds of divided power operations. The homotopy theory of such PD operads and their algebras can be understood quite explicitly in terms of chain complexes, using some homological algebra for representations of the symmetric groups. As a particular example, I will describe the PD operad that controls partition Lie algebras. Based on joint work with Lukas Brantner and Ricardo Campos.
Title: Equivariant Categories on Varieties
The derived category of sheaves on a variety is a central object of study in arithmetic geometry and the representation theory of algebraic groups. However, while the derived category of a variety is quite important, people who study groups acting on spaces have needed an equivariant version of the derived category so as to give a natural place from which to derive equivariant cohomology. It wasn’t until the nineties that a good definition of the equivariant derived category was discovered in a usable fashion by taking descent data through the derived categories of resolutions of an action on the base space. However, the constructions of equivariant derived categories differ greatly from the equivariant categories of sheaves that precede them and consequently make it difficult to see how to build categories of equivariant objects over a variety in a uniform way.
In this talk I will introduce a formalism that shows how to give a uniform and more general construction of equivariant categories overtop a variety by using certain pseudofunctors to construct equivariant categories. After introducing these categories, I will also show how this construction allows us natural ways to deduce properties of the equivariant category simply by looking at properties of the pseudofunctor. Finally, I’ll show that the functors and natural transformations that arise between equivariant categories that come from equivariant data are induced by pseudonatural transformations and modifications which give us techniques for lifting adjoints in a 2-category of pseudofunctors to what I call equivariant adjoints between equivariant categories. If there is any time left, I’ll also give some comments as to how these techniques allow us to produce triangulations on certain families of equivariant categories that contain the equivariant derived category.
Title: Bifold algebras
Given V-enriched algebraic theories T and U for a system of arities J in the sense of , commuting pairs of T- and U-algebra structures on the same object may be described equivalently as bifunctors that preserve J-cotensors in each variable separately, and these we call bifold algebras. Bifold algebras may be described equivalently as algebras for a theory called the tensor product of T and U, provided that J and V satisfy certain conditions that we do not assume in this talk. Every bifold algebra has two underlying algebras, which we call its left and right faces. (By an algebra, here we mean a pair consisting of a theory T and a T-algebra A.)
In this talk, we construct a two-sided fibration  of bifold algebras over various theories, and we show that the notion of commutant for algebras [3, 4] arises via universal constructions in this two-sided fibration. Using this method, we develop a functorial treatment of commutants for algebras over various theories. On this basis, we study bifold algebras in which one face is the commutant of the other, and vice versa, and we discuss examples in algebra, order theory, and topology.
 R. B. B. Lucyshyn-Wright, Enriched algebraic theories and monads for a system of arities. Theory and Applications of Categories 31 (2016), 101-137.
 R. Street, Fibrations and Yoneda’s lemma in a 2-category. Lecture Notes in Mathematics 420 (1974), Springer.
 R. B. B. Lucyshyn-Wright, Commutants for enriched algebraic theories and monads. Applied Categorical Structures 26 (2018), 559-596.
 R. B. B. Lucyshyn-Wright, Functional distribution monads in functional-analytic contexts. Advances in Mathematics 322 (2017), 806-860.
Title: Homotopy theory for Kan simplicial manifolds and a smooth analog of Sullivan’s realization functor
Abstract: Kan simplicial manifolds, also known as “Lie infinity-groupoids”, are simplicial Banach manifolds which satisfy conditions similar to
the horn filling conditions for Kan simplicial sets. Group-like Lie infinity-groupoids (a.k.a “Lie infinity-groups”) have been used to construct geometric models for the higher stages of the Whitehead tower of the orthogonal group. With this goal in mind, Andre Henriques developed a smooth analog of Sullivan’s realization functor from rational homotopy theory which produces a Lie infinity-group from certain commutative dg-algebras (i.e. L-infinity algebras).
In this talk, I will describe a homotopy theory for both these commutative dg-algebras and for Lie infinity-groups, and discuss some examples that demonstrate the compatibility between the two. Conceptually, this work can be interpreted either as a smooth analog of classical results of Bousfield and Gugenheim in rational homotopy theory, or as a homotopy-theoretic analog of classical theorems from Lie theory.