Statement on Black Lives Matter
Title: An Introduction to A-Homotopy Theory: A Discrete Homotopy Theory for Graphs
Abstract:
A-homotopy Theory was invented by Ron Aktin in the 1970s and further developed by Helene Barcelo in the early 2000s as a combinatorial version of homotopy theory. This theory respects the structure of a graph, distinguishing between vertices and edges. While in classical homotopy theory all cycles are equivalent to the circle, in A-homotopy theory the 3 and 4-cycles are contractible and all larger cycles are equivalent to the circle.
In this talk, we will examine the fundamental group in A-homotopy from the perspective of covering spaces. We will also establish explicit lifting criteria and examine the role of the 3 and 4-cycles in these criteria.
Title: Free co-completion, presheaves and sheaves
Abstract: In response to a special request this talk describes some fundamental aspects of sheaf theory. First we introduce the algebra of ends and co-ends for the purpose of making our subsequent calculations more concrete. Then we describe how the construction of a presheaf category of a small category corresponds to the free co-completion of that category. Finally we describe how the sheaf construction allows us to construct a non-free co-completion and choose which co-cones in the original category become colimit co-cones in the co-completion category. If we have time we sketch how to interpret various logical formulae in the internal logic of a sheaf category.
Title: Day convolution, infinity-operads and Goodwillie calculus (slides)
Abstract: Goodwillie calculus is a branch of homotopy theory that provides systematic approximations to a suitable functor (say from the category of topological spaces to itself) in the form of a “Taylor tower”, analogous to the Taylor series from ordinary calculus. In this talk, I will describe how some aspects of the Taylor tower construction are related via Day convolution. The slogan will be that “the nth derivative is an n-fold Day convolution of the first derivative”.
An important consequence of this observation is that the derivatives of an identity functor on a category C are a coloured operad (or symmetric multicategory), with the derivatives of a functor from C to D forming a bimodule over the operads corresponding to C and D. The context for all of this work is Lurie’s theory of infinity-categories though no technical background from that theory will be required in this talk.
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: Operads with homological stability
Abstract: For a carefully constructed operad M of surfaces, Tillmann showed that algebras over M group complete to infinite loop spaces. This result relies, in part, on Harer’s homological stability theorem for mapping class groups of surfaces. We will review Tillmann’s result and provide a more general framework which shows that operads satisfying a certain homological stability condition detect infinite loop spaces. This is joint work with M. Basterra, I. Bobkova, K. Ponto, and U. Tillmann.
Title: Ribenboim’s generalized power series and weighted Rota-Baxter categories (slides)
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.
