Statement on Black Lives Matter
Title: The OTHER polynomial functors
Abstract: Last week, we learned about polynomial functors from David Spivak. This week, I will talk about a different kind of polynomial functors – those arising in homotopy theory and algebra. In this talk, I will give three different definitions of what it means to be a polynomial degree n functor in the homotopy theoretic sense. I will then open the floor to questions, with an eye towards comparing to the polynomial functors that David Spivak presented.
Title: Ask me anything (AMA) on polynomial functors
Abstract: The category of polynomial functors in one variable and natural transformations between them is incredibly rich, e.g. it has infinitely many monoidal closed structures, including cartesian closure. Its substitution comonoids are categories and the associated double category includes multivariate polynomials as a full sub-doublecategory. Applications of polynomial functors include Moore and Mealy machines, data migration, algebraic datatypes, and much more. In this talk, I’ll explain whatever I know that is of interest in this circle of ideas.
Unfortunately this talk does not take place.
Title: The category Lex as a tangent (2-)category
Abstract: In this talk, I will report on joint work in progress (with Robin Cockett and Ben MacAdam) on how the additive bundle construction (almost) equips the category Lex of lex categories with a tangent category structure. Let Lex be the category of (small) lex categories (i.e. categories with finite limits) and finite-limit-preserving functors. Given a lex category C, an additive bundle in C consists of an object X of C equipped with a commutative monoid object in the slice category C/X over X. The additive bundles in C form a lex category AddBun(C), and the assignment of the lex category AddBun(C) to a lex category C extends to an endofunctor T : Lex —> Lex. This additive bundle endofunctor almost equips the category Lex with the structure of a tangent category, except that a few of the required coherences only hold up to isomorphism rather than equality. I will give an overview of this result and, noting that Lex is in fact a 2-category and that T is a 2-functor, I will also describe our current work in progress towards defining a notion of tangent 2-category and showing that the 2-functor T equips the 2-category Lex with the structure of a tangent 2-category.
Title: Acyclic models
Abstract: Acyclic models, as its commonly seen today, is a proof technique used to show when two chain complexes are chain equivalent or have isomorphic homology. It originated as a theorem by Eilenberg and MacLane (1953), where it was immediately used to show the Eilenberg-Zilber theorem (1953). This theorem, proven directly via acyclic models, gives us a Künneth theorem and defines the cup product, which turns cohomology into a graded ring.
This talk will be an exposition on (one version of) the acyclic models theorem, as given by Michael Barr in 2002. I will give the necessary definitions to understand Barr’s modern formulation of acyclic models, and then prove it. I will assume basic knowledge of chain complexes, but that will be briefly reviewed. Time permitting, I will also discuss how the Eilenberg-Zilber theorem follows directly from it and potential avenues to generalizing acyclic models.
Zoom link: https://ucalgary.zoom.us/j/97124679740
Title: String diagrams for categories
Abstract: A colorful diagrammatic language for the 2-category of categories has been exposited in a recent monograph by Dan Marsden and Ralf Hinze. In this tutorial, we will demonstrate how to use the language, and we will emphasize its benefits for the study of elementary category theory.
Title: Localization in restriction categories
Abstract: The aim of this talk about work in progress is to describe the local/separable factorization for restriction functors and for join restriction functors. I realized something new and quite important
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.
