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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.
Title: A Complete Classification of the Toffoli Gate with Ancillary bits (second attempt)
Abstract: We provide a complete set of identities for the symmetric monoidal category, TOF, generated by the Toffoli gate and computational ancillary bits. We do so by demonstrating that the functor which evaluates circuits on total points is an equivalence into the full subcategory of sets and partial isomorphisms with objects finite powers of the two element set. The structure of the proof builds — and follows the proof of Cockett et al. — which provided a full set of identities for the cnot gate with computational ancillary bits. Thus, first it is shown that TOF is a discrete inverse category in which all of the identities for the cnot gate hold; and then a normal form for the restriction idempotents is constructed which corresponds precisely to subobjects of the total points of TOF. This is then used to show that TOF is equivalent to the full subcategory of sets and partial isomorphisms in which objects have cardinality 2^n for some n in N.
Title: Cartan’s calculus for Sector Forms
Abstract: Cartan’s calculus of differential forms refers to a collection of operators on differential forms – the exterior derivative, Lie derivative, and interior product – and identities that they satisfy. The calculus of forms is particularly useful in mechanics. In this talk, we show that Cartan’s calculus of forms may be formulated in terms of J.E. White’s sector forms. Definitions for an interior product and Lie Derivative of sector forms will be presented, it will be shown these satisfy the usual identities from Cartan’s calculus.
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.
