Statement on Black Lives Matter
Title: Higher groupoids and higher generalized morphisms
Abstract: Higher groupoids play a crucial role in the active research area of interplay between higher categorical structures and other fields of mathematics. We give the notion of a Good Geometric category, where one can define and study these higher structures with applications to geometry, for example the category of smooth manifolds. We define the notion of higher groupoids in Good Geometric categories and organize them into an (∞, 1) categorical framework. The morphisms between the higher groupoids are given by bibundles which are Kan fibrations over the interval. Higher morphisms will be modelled by Kan fibrations over the higher simplices. This approach gives a more combinatorial and geometric way of approaching anafunctors and higher generalised morphisms between groupoids. This is of particular interest in higher gauge theory and string theory, where the higher connection on higher bundles will give the notion of parallel transport of strings and surfaces.
Title: All Concepts are Kan Extensions… and Kan Extensions are Concepts: A Categorical Theory of Vector Symbolic Architectures
Abstract: It is widely understood by category theorists that Kan extensions subsume all other notions in category theory (limits, ends, initial objects, adjoints, etc.). In this talk, I introduce vector symbolic architectures (VSAs)—a distributed model of data storage and manipulation that have gained popularity alongside the neuromorphic hardware upon which engineers hope to implement VSAs. Broadly, VSAs populate a vector space with a collection of “primitive” vectors, then use two operations, ‘binding’ and ‘bundling’, to create new vectors (ex. “red” and “car” bind as “red car”). Despite their many implementations, VSAs have little unifying theory. I discuss my recent work generalising and unifying VSAs. I focus on extending from vectors to co-presheaves and demonstrating that bind and bundle arise as right Kan extensions of the external tensor product and direct sum.
Title: Introduction to equivalences between bicategories and 2-categories
Abstract: This is part four of our series of introductory talks on bicategories. This time we will be considering different ways in which bicategories can be considered to be equivalent. While sets are considered “the same” when they are isomorphic, an equivalence of categories (the right way of “same-ness” for categories) can be seen as being isomorphic up to isomorphisms. Analogously we will see that two bicategories are biequivalent if they are isomorphic up to isomorphisms up to isomorphisms. In practice it is often easier to check biequivalence using the Whitehead theorem, which I will present and apply to examples.
Title: An Introduction to Embedding Calculus and the Role of Automorphisms of the (Framed) Little Disk Operad
Abstract: Embedding calculus is a powerful tool which is useful in making quantitative and qualitative conclusions about the topology of embedding spaces. In this talk, we will give a geometric description of the fiber of $\operatorname{Emb}(M,M) \rightarrow T_\infty \operatorname{Emb}(M,M)$ using smooth structures as well as an algebraic description of its delooping using automorphisms of the little disk operad. In the second part of the talk, we will describe some known properties of that automorphism space and will then explain some ideas that go into the proof of showing that the endomorphisms of the framed little disk operad agree with the automorphisms.
Weighted pullbacks in V-graded categories
As a replacement we have the talk on morphisms between bicategories
Title: The Three Flavours of Morphisms Between Bicategories
Abstract: In everybody’s first course in category theory, we learn that categories have two different flavours of morphism: 0-morphisms (which are categories themselves), 1-morphisms (which are functors), and 2-morphisms (which are natural transformations). In this talk I’ll continue with our theme of “Geoff teaches everyone the magic of bicategories” by showing that this pattern of having multiple flavours of arrows continues with bicategories by introducing us to the 0-morphisms (which are bicategories), the 1-morphisms (which are pseudofunctors), the 2-morphisms (which are pseudonatural transformations), and the 3-morphisms (which are called modifications). Depending on time, we’ll also see various examples of these gadgets and how they can arise “in nature.”
Title: The Intersection between Number Theory, Abstract Algebra, and Cryptography
Abstract: In this talk we will cover the mathematics behind a security system that is thought to be secure against quantum computer hacking; the Supersingular Ell Isogeny Cryptosystem. Specifically the Supersingular Ell Isogeny Graph. This abstract object is made of isomorphic elliptic curves and equivalent homomorphisms between them. The Supersingular Ell Isogeny graph is an extremely complicated graph that is used for cryptosystems (for the future), but we will use a simpler approach to find patterns within this convoluted graph.
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.
