Speakers

Title: Strict Monoidal Categories as Internal Monoids.

Abstract: Internalization is a paradigm for the generalization of classical mathematical definitions, e.g. monoid generalizes to internal monoid in a monoidal category, or group generalizes to internal group in a category with finite products. In this talk, we will demonstrate that strict monoidal categories may profitably be studied as internal monoids in the 1-category of 1-categories. We present detailed calculations pertaining to limits and colimits of internal monoid objects, while relying on little more than the material of an introductory course in category theory.

Title: Vogan’s conjecture for p-adic GL_n

Abstract: Local Arthur packets are sets of representations of p-adic groups that help us realize important classes of automorphic forms. They have geometric analogues, called ABV-packets. This was first proposed for p-adic groups by David Vogan following his joint work with Adams and Barbasch for real groups. This theory was then adapted by Cunningham et al. for the non-archimedean case. They defined ABV-packets and formulated the conjecture that ABV-packets generalize local Arthur packets. They called it “Vogan’s conjecture” to honour the work that led to it, in addition to providing a wealth of examples as evidence. In this talk, I will introduce ABV-packets and present a proof of Vogan’s conjecture for p-adic GL_n .

Title: Type Classes in CaMPL

Abstract: Overloading, which allows functions to exhibit different behaviors based on the types involved, is one of the most useful facilities of a typed programming language. In the Haskell programming language, this facility is provided by type classes. Type classes offer a systematic solution for overloading, providing uniform operations for arithmetic, equality, and displaying values, and so on. In addition, higher-order type classes are used to implement important more advanced structures such as functors and monads.

Categorical Message Passing Language (CaMPL) is a concurrent programming language based on a categorical semantic given by a linear actegory. CaMPL has with type inference for both sequential and concurrent programs. The sequential side of CaMPL is a functional-style programming language, while the concurrent side supports message passing between processes along channels with concurrent types called protocols.

In this presentation, first the importance of type classes will be discussed, then CaMPL will be introduced, and its different features will be explained. Finally, the process of integrating type classes into both sequential and concurrent tiers of CaMPL will be investigated.

Title: Awesome Reversible Knitting
Abstract: Reversible Knitting is a specialized branch of knitting focussed on creating fabric that looks beautiful on both sides, lies flat and doesn’t curl, and has a wonderful hand feel and texture. It is technically intriguing, creatively satisfying, and practically flexible!
This talk will provide applications of math in reversible knitting, touching on a variety of concepts including topology, notation, sets, balance and duality, tiling and geometry, rotational symmetry and translation, flat vs circular vs möbius knitting, gauging and ratios, and even physics!
Starting with the two fundamental knitting stitches, Knits and Purls, we will explore reversible knitting definitions, the fascinating Möbius knitting and end with the “reversiblest” stitch pattern known.
Although no physical knitting experience is required or provided, I will supplement the talk with a selection of knitted fabrics as examples of each concept.

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