Speakers

Title: Systems of homotopy colimits

Abstract: Limits and colimit constructions are ubiquitous in category theory, and are one of the main tools used to understand how objects in a category relate to one another.  These are very concrete and easily stated in terms of universal properties: given a diagram in a category, the colimit is the initial cocone making the resulting diagram commute.  Homotopy colimits, on the other hand, have always been more difficult to define.  These so not satisfy a universal property in any category, and tend to be described in terms of a construction and properties.  In recent work with Brooks-Hess-Johnson-Rasmusen-Schreiner (BBHJRS for short) we attempted to enumerate the properties required for something to be a system of homotopy colimits.  Following a referee’s comments, we transformed the list into a more familiar categorical construction.  In this talk, I will offer an alternative definition of homotopy colimits using actegories.

Title: What can you buy with a bicategory?

Abstract: In this talk we’ll continue to get to know bicategories by learning about what you can do inside a bicategory. We’ll learn about pasting diagrams as well as adjoints and equivalences inside bicategories before presenting some examples of what these equivalences can look like in certain examples (and in doing so, we will recapture the notion of Morita equivalence of rings). Also Geoff had fun giving this talk.

Title: Saying Hello to Bicategories

Abstract: Bicategories are an important aspect of modern category theory and provide the first instance of “category theory up to coherent isomorphism” we see when hiking up Mount Higher Category Theory. In this talk I will introduce the notion of bicategories, what we can do with them, and explain some of their basic properties. There will also be many, many examples presented in order to both show people who are first becoming introduced to bicategories what flavours in which they can arise and also to keep the talk (somewhat) grounded.

This will be a 2 part talk on Friday Sept 13th and Friday Sept 20th.

Title: The scheaf of distributions

Abstract: Originally distributions were introduced in PDE-theory to generalize the notion of functions. However, this analytical concept also has some purely algebraic properties, which are worth considered.
In particular, they are an important example of sheaves in differential geometry. In this talk we are going to discuss two views on distributions and their connections. On one hand, distributions can be considered as generalized densities (a broader version of the determinant) and on the other hand as generalized functions. The latter, as we are going to show, form a sheaf, which gives us an insight on the sheaf-structure of distributions.

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.