Statement on Black Lives Matter
Title: String diagrams: by categories, for categories
Abstract: The formalization of string diagrams, in a 1991 paper by Andre Joyal and Ross Street, was a seminal event. They defined a basic diagrammatic language and proved its validity for rigorous mathematics in any monoidal category, and in doing so they laid the foundation for a substantial ongoing program of research on extensions and specializations of the original theory. This first half of this talk will address the question of how all this was possible – how it can be demonstrated that proofs by diagram are logically valid. We will introduce the foundations of string diagrams by studying the core argument of Joyal and Street’s 1991 paper. The second half will be devoted to a particular example: a lovely diagrammatic language for the 2-category of categories, as introduced in a recent monograph by Ralf Hinze and Dan Marsden. We will demonstrate how to use the language, and we will emphasize the benefits that it offers in the study of elementary category theory.
Title: 2-Segality and the S.-construction
Abstract: Waldhausen’s $S_\bullet$-construction gives a way to define the algebraic $K$-theory space of a category with cofibrations. Specifically, the $K$-theory space of a category with cofibrations $C$ can be defined as the loop space of the realization of the simplicial topological space $|iS_\bullet C |$. Dyckerhoff and Kapranov observed that if $C$ is chosen to be a proto-exact category, then this simplicial topological space is 2-Segal. A natural question is then what variants of this $S_\bullet$-construction give 2-Segal spaces. In this talk, we give the necessary background in this area and discuss work in progress that aims to address the preceding question.
Title: The Lie Algebra of a group object
Abstract: At a previous presentation Geoff Cruttwell asked me, if the vertical bundle of a principal bundle in a join restriction category is trivial. This was not just an interesting question on its own, it also pointed towards more going on with group objects in tangent categories.
In differential geometry, given a Lie-group G, its tangent space TuG is known as its Lie-Algebra. Generalizing this perspective, I will prove that the tangent bundle of a group object in a tangent category is trivial (isomorphic to a product). Then I will continue discussing how the addition from the tangent bundle structure interacts with the group multiplication and show that there is a negation, even if the tangent category was not required to have negatives. From this I will conclude that there is a Lie-Algebra structure on the elements of the tangent space.
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.
