#### Statement on Black Lives Matter

**Title:** Adjointness in Modal Semantics

**Abstract:** Coalgebraic modal logic is a set of approached to modal semantics that defines frames and models as coalgebras for a functor. This approach has captured the traditional notions of Kripke and Neighbourhood frames, as well as many other types of semantic structures unfamiliar in traditional philosophical logic. In this talk I will present the coalgebraic perspective on Kripke and Neighbourhood frames, and show how this perspective can lead us in helpful new directions. Once we adopt a coalgebraic perspective, we can see that the central invariance result about modal definability—that all formulas of the basic modal language are invariant under bounded morphisms—follows from a single, fundamental fact about functions: that direct images are left adjoint to inverse images. However, there is also a right adjoint to inverse image: the “codirect image.” Using the right adjoint to inverse image, instead of the left, leads to another, non-traditional, modal language and another invariance result. This talk will explain how this adjoint sequence arises from a coalgebraic perspective on Kripke frames, and discuss its implications for defining modalities in the Neighbourhood frames.

**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:** 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.