Speakers

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.

Title: A Two Dimensional Setting for the Calculus of Infinity Functors
Abstract: In this talk we combine two related approaches to the theory of infinity categories. On the one hand we use derivators to work with (homotopy) (co)limits within small infinity categories. Using this theory we define the excisive functors, suspension functors etc.. that are commonly used in the Goodwillie calculus. On the other hand we use the homotopy 2-category of quasi-categories developed by Riehl and
Verity to describe relationships between the small infinity categories themselves. Using this theory we work out how to form colimits in an infinity category of excisive functors.

Title: Quantum Channels for Mixed Unitary Categories
Abstract: Categorically, quantum processes are modelled as completely positive maps in dagger compact closed categories [1,2,3]. The limitation however is that the quantum processes modelled are for finite-dimensional systems. A natural setting for quantum processes between infinite dimensional systems is a *-autonomous category or more generally a linearly distributive category. The goal of my talk is to introduce dagger linearly distributive categories and mixed unitary categories in which one can discuss about quantum processes possibly in infinite dimensions. CP-infinity construction [4] on dagger symmetric monoidal categories generalizes CPM construction [3] to arbitrary dimensions. I will show a generalization of CP-infinity construction to mixed unitary categories.
References:
[1] Bob Coecke, Chris Heunen, and Aleks Kissinger. Categories of quantum and classical channels. Quantum Information Processing, 15(12):5179–5209, December 2016.
[2] Bob Coecke and Aleks Kissinger. Picturing Quantum Processes. Cambridge University Press, Cambridge, England, 2017.
[3] Peter Selinger. Dagger compact closed categories and completely positive maps. Electronic Notes in Theoretical Computer Science, 170:139–163, 2007.
[4] Coecke, Bob, and Chris Heunen. Pictures of complete positivity in arbitrary dimension. Information and Computation 250 (2016): 50-58.