Speakers

Title: Introduction to univalence in Coq
Abstract: This week we begin a sequence of talks on homotopy type theory (HoTT). In this first talk we introduce the elementary definitions and tactics that are required to get started with HoTT using the Coq proof assistant. We endeavour to introduce as little programming syntax as possible and develop the theory using only a very few syntactic constructs that have clear mathematical interpretations.
The audience is encouraged to follow the development of the theory on their own computers so please bring a laptop if you want to do this! (You will need about 300mb of space to install the Coq proof assistant.) There will be short exercises that the audience can complete at their own pace which will not be vital to the theory but rather are intended to familiarise the audience with the proof assistant and tactics.

Title: Every CDC embeds into the coKleisli category of a monoidal differential category
Abstract: The coKleisli category of a monoidal differential category is always a Cartesian differential category. However, it seems that not every CDC arises this way. In the category of smooth maps between finite dimensional real vector spaces, there does not appear to be a “bang” on the subcategory of linear maps, as the “bang” should give rise to an infinite dimensional space. However, the question of whether any CDC embeds into a coKleisli category of some monoidal differential category has been floating around for a while. This talk will address this question directly.

Title: A structural definition of symmetric multicategories
Abstract: Symmetric multicategories are a basic structure in the categorical semantics of linear logic. One can define them elementarily, but already the coherence are difficult to track. The problem compounds when one tries to define functors and natural transformations to get a 2-category of symmetric multicategories, which is necessary for stating properly 2-categorical theorems like coherence. I will give a structural definition of symmetric multicategories based on profunctors which will provide the basic definition upon which the 2-category of symmetric multicategories can be built.

Title: Linearisation of infinity categories
Abstract: In a paper on the Goodwillie calculus Lurie defines a linearisation procedure that forms a map of infinity bi-categories. In this talk we show how this result transfers into the 2-categorical setting for quasi-categories developed by Riehl and Verity. If we have time we sketch how to express a tangent bundle pseudo-functor in terms of the linearisation pseudo-functor.

Title: The differential lambda-calculus: syntax and semantics for differential geometry
Abstract: This talk will introduce semantics for the differential lambda-calculus using tangent categories. We will show how to obtain models of the differential lambda-calculus that stem from differential geometry. We will also explore the coherence required for tangent categories to model the differential lambda-calculus from different points of view.

Title: Operad structures in abelian functor calculus
Abstract: Abelian categories are a cartesian differential category, and the derivative corresponds to the same derivative which is used in functor calculus (a branch of homotopy theory). In 2011, Cockett and Seely showed that any Cartesian differential category has a higher-order chain rule for the derivative.
B. Johnson, S. Yeakel and I have identified this higher order chain rule in the abelian functor calculus example. Furthermore, we have shown that a consequence of the higher-order chain rule is that higher order derivatives of a functor of R-modules form an operad (a monoid in the category of symmetric sequences). The existence of this operad was predicted by a similar result for functors of topological spaces (discovered by G. Arone and M. Ching). In the case of abelian calculus, we have identified this operad as a consequence of the existence of a (lax) functor from abelian categories to the category Faa(AbCat), as defined by Cockett and Seely. We see our result as a kind of translation between the homotopy theoretic and category theoretic results.
In this talk, I will define the abelian functor calculus derivative, explain the higher order chain rule and produce the resulting operad.

Title: A Tangent Category of Fibrant Objects
Abstract: We shall consider Getzler and Behrend’s construction of a category of fibrant objects from a descent category in the setting of tangent categories. This will generate a category of fibrant objects with a well defined tangent structure. A particularly important class of objects will be those equipped with an infinite family of higher order horizontal connections.

Title: Elements of the Theory of Quasi-categories
Abstract: We outline some of the theory of quasi-categories that is required to set up the functor calculus. First we review the definition of left, right and inner factorisation systems and describe an alternative characterisation of these factorisation systems that makes certain computations more straightforward. Then we define quasi-categories and prove that the internal hom of quasi-categories is a quasi-category. In order to define a tangent bundle functor we first need to define the (large) quasi-category of quasi-categories and recall how the representable (infinity) functors are defined in this setting. If we have time we describe how to use this work to define the functor of excisive functors that conjecturally constitutes the tangent bundle functor.

Title: A Two Dimensional Setting for the Calculus of Infinity Functors: Part II
Abstract:
In this talk we continue describing the calculus of infinity functors in terms of derivators. First we recall what a derivator is, the basic examples of derivators, the definition of Cartesian square and what it means to be an excisive morphism of derivators. Then we develop the basic theory of pointed and stable derivators and prove that the derivator of reduced excisive functors between two derivators is stable. If we have time we describe the zeroth and first order approximations of a derivator and define what a pre-stable derivator is.

Title: Cartan Calculus for Tangent Categories 2
Abstract: We develop the string calculus for Cartesian Tangent categories, and consider the shuffle operation and Noether’s theorem in a tangent category.