Title: Differential bundles as functors
Abstract: Differential bundles are the generalisation of vector bundles in tangent categories. Following an idea by Michael Ching, we will consider lax morphisms of tangent categories, functors between tangent categories preserving the tangent structure, and show that they induce differential bundles under certain conditions. Then we will continue to show that the categories of differential bundles in a tangent category X with additive/linear morphisms are equivalent to the category of tangent functors from the category of free commutative monoids into X.
Title: A Ternary Notion of Logical Consequence
Abstract: Slightly altering and extending McGee’s semantics for conditionals, we define a ternary notion of logical consequence for the validity of natural language arguments. The ternary logical consequence can be regarded as a unification of two kinds of validity in the literature. By the new notion of logical consequence, an inference is not just valid or invalid, but valid or invalid under a set of assumptions. Based on this notion, we give a unified solution to some typical puzzles concerning conditionals and epistemic modals, including the (in)validity of modus ponens, modus tollens, Import-Export, conditional excluded middle, Or-to-If, and fatalism arguments, as well as the puzzle of Moore sentences and the scope ambiguity problem in modal conditionals.
Title: Tutorial on Curry-Howard Part II
Abstract: I will provide a (very basic) tutorial on the Curry-Howard correspondence between proofs (in natural deduction) and terms in the typed -calculus. No prior knowledge of either will be presupposed.
Title: A categorical introduction to Drazin inverses
Abstract: Michael Drazin introduced the idea of a “pseudoinverse” for rings and semigroups in 1961. These inverses categorically are rather special as (like ordinary inverses) they are preserved by all functors when they exist. A category is Drazin when all maps have a Drazin inverse. The aim of the talk is prove that when a category has “expressive” rank it must be Drazin. This provides some examples of Drazin categories … time permitting I will explain how Drazin inverses relate to the Fitting decomposition and the Jordan -Chevalley decomposition.
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.