Statement on Black Lives Matter
Title: Reinventing the Wheel; An introduction to fibered categories through the eyes of a representation theorist.
Abstract: Given two isomorphic groups G and H, one may identify representations of H with representations of G along some fixed isomorphism from G to H. However, this process of identifying representations depends crucially on the isomorphism of groups one begins with, and thus we are left asking ourselves how to privilege one isomorphism over the other, or if we should. In this talk, we will discuss how the above question naturally leads one to the concept of a fibered category. We will also briefly discuss other examples of fibered categories, and speculate about generalizations of the original question.
Title: The Many Faces of the Eilenberg-Zilber Theorem
Abstract: The Eilenberg-Zilber (EZ) theorem is a powerful tool in homological algebra and algebraic topology, being a key ingredient in the Kunneth theorem. It also serves as the basis for defining the cup product, which in turn establishes cohomology as a graded ring. Since its initial proof in 1953, the main generalization beyond simplicial R-modules has been to bisimplicial objects in abelian categories.
The method of proof, acyclic models, was introduced together with the first EZ theorem in 1953. Acyclic models is actually a theorem of Eilenberg and Maclane’s, and was originally used to prove the equivalence of singular homology based, respectively, on simplices and on cubes. In his 2002 monograph “Acyclic models”, Barr collects various versions of the acyclic models theorem, then shows its applications in Cartan-Eilenberg cohomology and homology on manifolds (among other things).
In this talk I will introduce all of the concepts required to understand EZ theorems, as well as acyclic models. We will examine recent developments in these concepts to motivate a conjecture for an EZ theorem with weaker hypotheses. Finally, we will also examine how acyclic models is used to prove the most recent EZ theorem, revealing potential obstacles to generalizing it further.
Title: Comonoids in Poly are categories
Abstract: The goal of my talk is to sketch out an easy to follow, diagrammatic proof of the above statement.
Title: Tutorial on Curry-Howard
Abstract: I will provide a (very basic) tutorial on the Curry-Howard correspondence between proofs (in natural deduction) and terms in the typed $\lambda$-calculus. No prior knowledge of either will be presupposed.
Title: The pebbling comonad in finite model theory (an exposition)
Abstract: In this talk, I will exposit some of the main ideas and results of the seminal 2017 paper “The pebbling comonad in finite model theory” by Samson Abramsky, Anuj Dawar, and Pengming Wang. In this paper, the authors demonstrate that pebble games, which are a powerful combinatorial tool in the study of finite model theory, constraint satisfaction, and database theory, admit a natural formulation in terms of comonads, which leads to comonadic characterizations of many central concepts in finite model theory. Specifically, given a relational signature S and S-structures A and B, the authors establish that winning strategies for the Duplicator in existential pebble games from A to B are equivalently given by morphisms from A to B in the coKleisli categories of certain comonads on the category of S-structures. The paper thereby provides a connection between two broad topics in logic and computer science that have previously been largely disjoint: the interaction of logic with computational complexity, and the study of the semantics of programs and processes. Time permitting, I will also discuss some extensions of their results that I have recently proved.
Title: Grandad of all Computation
Abstract: Moses Schönfinkel invented “combinatory logic”—aka combinatory algebra (CA )—in 1920. It consisted of a binary operation and two constants $S$ and $K$ (called combinators by Haskell Curry who further investigated CAs in the 1950s) which satisfy just two identities. That such a simple gadget can generate all computable functions is—well—amazing.
However, that was not the end of the story.
In 1975 Solomon Feferman introduced the notion of a partial combinatory algebra (PCA) and showed that it too could generate all computable functions … but furthermore had as a prime example the usual notion of computability which the average CS student meets in theory courses via Turing machines. Again, this is an amazingly simple structure which can express all computation: it consists of a partial binary operation and combinators $S$ and $K$ which satisfy just four identities.
In 2008 Pieter Hofstra and I introduced Turing Categories: we argued that this notion subsumed all the previous notions of computability. Furthermore, it turned out that the initial Turing category was generated by a generic PCA … and, thus, this gadget was, therefore, the grandad of all computation.
Now CAs are well known to have a confluent rewriting system, which this is very important as it is the rewriting system which generates computation. By analogy to a CA, this generic PCA should have a nice rewriting system. However, it is still has not been proven that PCAs do have a confluent rewriting system!
The talk is aimed to introduce Turing categories, PCAs, and a suggestion for what the rewriting system should be.
Title: Enriched algebraic theories, monads, and varieties
Abstract: In this talk, I will summarize the research on enriched algebraic theories, monads, and varieties that I produced with Rory Lucyshyn-Wright during my previous postdoctoral fellowship at Brandon University. I will start by providing a historical overview of the subject, which originated with Birkhoff, Lawvere, and Linton in the 1930s and 1960s. Birkhoff initiated the study of universal algebra by defining the notion of an (equational) variety of algebras, which is a class of algebraic structures axiomatized by certain equations. Lawvere and Linton then established purely categorical formulations of Birkhoff’s varieties, in terms of Lawvere theories and finitary monads on Set. I will then describe the ways in which, over the next 50 years, various researchers generalized Lawvere theories and finitary monads to certain enriched settings, by developing enriched notions of Lawvere theory and monad. None of these frameworks developed an enriched notion of variety, and moreover they were largely formulated for locally presentable bases of enrichment, which exclude many important categories of a topological flavour. Moreover, there was no overarching framework that encompassed all of these works. To address these issues, Rory and I developed notions of enriched Lawvere theory, enriched monad, and enriched variety that encompassed all of these prior works and also extended their applicability to many new mathematical settings (especially in topology and analysis). I will conclude by mentioning some of my current and planned research on this topic, which I will talk about at future seminars.
Title: CoCalculus from Monads
Abstract: This is a report from team Functor Calculus at the 4th Women in Topology conference. In earlier work, K. Hess and B. Johnson invented a way of producing what they call “calculus towers” from comonads. These calculus towers generalize functor calculus, and recover other important examples of towers in topology such as certain towers of localizations. In joint work with R. Brooks, K. Hess, B. Johnson, J. Rasmusen and B. Schreiner we produced a dualization of the Hess-Johnson machinery. The resulting “cocalculus towers” recover dual functor calculus, an important co-tower approximating homotopy functors to spectra. Both types of functor calculus are used to approximate important functors in homotopy theory in a manner similair to the Taylor series approximations of functions, but an important distinction for functors is that it is not always possible to recover the entire series from it’s homogeneous parts. Using dual calculus, R. McCarthy and several of his students (including me) produced a variety of results which indicated when one can recover the series from the homogeneous pieces. In this talk, I will explain the generalization of dual calculus to “cocalculus” and explain what kind of results we hope this may lead to in the future.
Title: Counterfactual Logics via Comparative Possibility
Abstract:
In 1973, C. I. Lewis published a landmark text developing a theory of counterfactual logics. The primary goal of the text was in providing a logical framework where one could reason about sentences of the form: “if P were true, then Q would be true”. Such sentences were handled with a new connective and the majority of his text explores the semantics and behavior of this new connective. More general semantics and other connectives were introduced as well and briefly discussed, including a comparative possibility relation. It isn’t until the last chapter of his book that Lewis actually described the syntax for the logics, a reformulation of the semantics, soundness and completeness theorems, and a list of axioms, among other things. This final chapter is unusual for many reasons, the logic and semantics rely on the comparative possibility relation instead of the counterfactual relation and all of the proofs and definitions are written in extremely terse prose. Moreover, this chapter discusses a large family of logics as well as their related modal logics.
In this talk we’ll first go over the sort of counterfactual logic and semantics that Lewis is primarily interested in. Then we’ll discuss the semantics and logic in terms of the comparative possibility relation. Finally, we’ll give an overview of how the soundness and completeness proofs work and where the difficulties lie.
Title: Categorically generalizing bundles
Abstract: Vector bundles and principal bundles are important objects in differential geometry. Vector bundles over a manifold M are manifolds that are locally isomorphic to the Cartesian product M times V for some vector space V, while principal bundles over M are manifolds that are locally isomorphic to M times G for some Lie group G. I will explain some of the many ways to formulate these ideas categorically which generalize different aspects.
Tangent categories are the categorical generalization of the smooth structure giving rise to the tangent bundle. In them differential bundles generalize vector bundles.
On the other hand, join-restriction categories generalize the concept of locality from differential geometry and can be used to define principal bundles.
In the end I will give an outlook about how to generalize these concepts to infinity categories.
