Speakers

Title: Frolicher spaces, Weil spaces, Diffeological spaces and abstract differential geometry Abstract: In differential geometry, obtaining a smooth structure on spaces of smooth maps is a motivation for the introduction of various kinds of generalized smooth spaces. Frechet manifolds do allow a manifold structure on the space of smooth maps between *finite* dimensional smooth manifolds, but the category is not cartesian closed. Frolicher spaces, Weil spaces, and diffeological spaces all have the advantage that they generalize smooth manifolds, and are cartesian closed categories. We will describe these categories as Tangent categories. They are *not * tangent categories, but they should be. We will introduce a notion of generalized microlinearity, based on the work of Nishimura, whose notion was based on the work of Wraith and Kock, to extract a tangent full subcategory.

Title: Using Postulated Colimits in Coq
Abstract: In this talk we define and construct finite colimits in the Coq proof assistant in a context that is similar to the category of sets. First we review without proof the key mathematical ideas involved in the theory of postulated colimits as described in a note of Anders Kock. This theory gives us a way to prove results about colimits in an arbitrary sheaf topos. Then we give an inductive definition in Coq of the fundamental notion of zigzag in this theory. We finish by proving the result analogous to the (mathematically easy) result that in the category of sets pushouts of monomorphisms are monomorphisms.

Title: Localisation of Simplicial Presheaf Categories
Abstract: In this talk we describe a special case of left Bousfield localisation that is of interest in the calculus of functors. In particular we work in the category of simplicially enriched presheaves of a small  category. First we sketch how the classical small object argument constructs factorisations from a set of presheaf morphisms. Then we describe how to generate a model category from these maps (and an appropriate specification of weak equivalences) by augmenting the small object argument with some results due to J. Smith. Finally if time permits we will work out the special case of interest in the calculus of functors.

Title:  Linearly distributive categories and daggers do mix!
Abstract:  We shall explain the basic structure of a dagger *-autonomous category and exhibit a basic example using finiteness spaces.  Time permitting we will discuss how the CPM construction can be generalized to this setting.

Title: The Calculus of Functors using Sheafification
Abstract:
In classical calculus we approximate an appropriately differentiable function using a sequence of simpler functions called the Taylor polynomials. In an analogous way we can approximate a functor whose domain and codomain are appropriately topological by using a sequence of simpler functors. These simpler functors can be described using a universal property and a condition asserting that certain pullbacks are taken to certain homotopy pushouts. In this talk we present an alternative perspective based on a paper of de Brito and Weiss in the case that the domain of the functor is the category of smooth manifolds. First we describe the approximating ‘polynomial’ functors as generalised sheafifications with respect to a sequence of Grothendieck coverages. Then we explore how this approach generalises when we replace the category of smooth manifolds with more general categories.

Title: Proving Teleportation protocol using ZX-calculus
Abstract: In my previous talks, I introduced environment structures and discarding maps. In this talk I will use discarding maps and ZX- calculus to prove the correctness of teleportation protocol. ZX- calculus is a universal graphical calculus for reasoning about quantum processes. With discarding maps, one can graphically represent classical control and measurements. ZX-calculus along with the discarding maps provides a simple language for deriving the correctness of quantum information theoretic protocols. We will prove one such protocol in this talk namely quantum teleportation.

Abstract:
The first chain rules in functor calculus were established by Arone-Ching and Klein-Rognes, who considered functors of spaces or spectra.  The Arone-Ching chain rule stemmed from earlier work of M. Ching, in which he established that the derivatives of the identity functor form an operad, whose homology is the classical Lie operad.  The chain rule, in conjunction with Ching’s earlier work, lead to a classification theorem for all functors of spaces or spectra.  That is, it explained how you could produce a functor from its derivatives.  However, Arone-Ching’s work was quite complicated and difficult.  One of the motivations for establishing a chain rule for abelian functor calculus was to try to look for a simplification of their work.  Indeed, in BJORT, we established the chain rule by first establishing that the category of abelian categories is a cartesian differential category.  This, together with the associated tangent structure, lead to a much simpler proof of the chain rule.  Following the program laid out by Arone and Ching, we are now looking for the expected operad structure and classification theorems.  In this talk, I will explain the derivative (as opposed to the directional derivative) for abelian functor calculus, and the candidate for our operad.