Speakers

Title: Koszul duality of algebras for operads

Abstract: Ginzburg-Kapranov and Getzler-Jones exhibited a duality between algebras for an operad O and coalgebras (with divided powers) for a “Koszul dual” cooperad BO, taking the form of an adjoint pair of functors between these categories. Instances of this duality include that between Lie algebras and cocommutative coalgebras, as in Quillen’s work on rational homotopy theory, and bar-cobar duality for associative (co)algebras, as in the work of Moore. I will review this formalism and discuss the following basic question: on what subcategories of O-algebras and BO-coalgebras does this duality adjunction restrict to an equivalence? I will discuss an answer to this question and explain the relation to a conjecture of Francis and Gaitsgory.

Recording (Passcode: u.2c?5w?)

Title: Regular logic in a `double category of relations’

Abstract: In this talk, we will present a definition of a `double category of relations’, inspired by that of a bicategory of relations’ due to Carboni and Walters. Roughly, a `double category of relations’ is a cartesian equipment whose horizontal bicategory satisfies a certain discreteness condition. We will then sketch how any such structure yields a sound interpretation of regular logic.

Title: The Eckmann-Hilton argument in duoidal categories

Abstract: We will go over the basics of duoidal categories, illustrating with a number of examples. As monoidal categories provide a context for monoids, duoidal categories provide one for duoids and bimonoids. Our main goal is to discuss a number of versions of the classical Eckmann-Hilton argument which may be formulated in this setting. As an application we will obtain the commutativity of the cup product on the cohomology of a bimonoid with coefficients in a duoid, an extension of a familiar result for group and bialgebra cohomology.

The talk borrows on earlier work in collaboration with Swapneel Mahajan on the foundations of duoidal categories (2010). The main results are from ongoing work with Javier Coppola. We also rely on work of Richard Garner and Ignacio López-Franco (2016).

Title: Iterated Traces

Abstract: The trace of a matrix does not seem like an operation that should be iterated, but if we step back and think of trace as an operation on endomorphisms (or almost endomorphisms) that is invariant under cyclic permutation this becomes more plausible.  I’ll make sense of iterated traces in monoidal bicategories, describe independence of order for iterated traces, and connect this result to (disguised) examples that have appeared in the literature.  

Title : Graded (co-)Differential Categories

Abstract : Co-differential categories are additive symmetric monoidal categories where you can differentiate certain kind of morphisms with the help of a combination of a monad, monoids and a deriving transformation.

Recently, J.-S. Pacaud Lemay proved that the category of finite-dimensional vector spaces cannot be endowed with a nontrivial structure of co-differential category. And this is very unlikely that it will work better with some familiar categories such as the category of finite sets and relations or the category of all Hilbert spaces.

In this talk, I will introduce a generalisation of co-differential categories: graded co-differential categories, which basically consist in putting the word “graded” everywhere in the axioms of a co-differential category. I will explain how the preceding categories can be endowed with the structure of a graded co-differential category as example of a general categorical construction which use symmetric powers.

This is joint work in progress with Jean-Simon Pacaud-Lemay.

Title: An (∞,2)-categorical pasting theorem

Abstract: Power’s 2-categorical pasting theorem, asserting that any pasting diagram in a 2-category has a unique composite, is at the basis of the 2-categorical graphical calculus, which is used extensively to develop the theory of 2-categories. In this talk we discuss an (∞,2)-categorical analog of the pasting theorem, asserting that the space of composites of any pasting diagram in an (∞,2)-category is contractible. This result, which is joint with Hackney—Ozornova—Riehl, rediscovers independent work by Columbus.

Title: Describing principal bundles and pushing TQFTs forward

Abstract: In the first part of this talk we will discuss several ways, how principal bundles over a manifold can be described. The main two of them are maps from the base manifold into the group’s classifying space and assignments of group elements to the codimension one structures of a special decomposition, called fine stratification. Both of them provide an equivalence of categories to the category of principal bundles. Having all these equivalent descriptions, one can translate geometric constructions with principal bundles into discrete combinatorial constructions. In the second part of this talk I will outline, starting from a pushforward construction for equivariant Topological Quantum Field Theories, how this can be used to define a pushforward construction for defect Topoplogical Quantum Field Theories.
This is joint work with Gregor Schaumann, as part of my Master’s project.

Titre : From trees to infinity-operads
Abstract : The first goal of this talk will be to introduce the dendroidal world, which generalises the simplicial world. Instead of working with the category Delta, we will work with the category Omega, a category of trees, introduced by Moerdijk and Weiss. Presheaves on Delta are called dendroidal sets and generalise simplicial sets. I will explain how operads and infinity-operads appear in this setting. In a second part, I will introduce a notion of homology for infinity-operads, using a bar construction.
This is a joint work with Ieke Moerdijk.

Title: Measurement in the symplectic setting.

Abstract: In previous work, we have given generators for affine Lagrangian relations over an arbitrary field; exploiting the graphical calculus for affine relations. In the case of a finite field of odd prime order d, we have shown that this is isomorphic to qudit stabilizer circuits. In this talk, the question of measurement in this symplectic setting will be addressed. We show that the CPM construction applied to affine Lagrangian relations yields affine coisotropic relations; which can be obtained by adding the discard relation to affine Lagrangian relations. By splitting the decoherence maps in affine coisotropic relations, we obtain a two-sorted presentation for classical/stabilizer quantum circuits. We show that this is equivalent to adding injection and coinjection relations to affine Lagrangian relations. Time permitting, we will discuss the connection of this work to classical and quantum additive codes.

Recording (Passcode: 58n&dKr$)

Title: RO(C_2)-graded coefficients of C_2-Eilenberg-MacLane spectra

Abstract: In non-equivariant topology, the ordinary homology of a point is described by the dimension axiom and is quite simple – namely, it is concentrated in degree zero. The situation in G-equivariant topology is different. This is due to the fact that Bredon homology – the equivariant counterpart of the ordinary homology – is naturally graded over RO(G), the ring of G-representations. Whereas the equivariant dimension axiom describes the part of the Bredon homology of a point which is graded over trivial representations, it does not put any requirements on the rest of the grading – in which the homology may be quite complicated.

The RO(G)-graded Bredon homology theories are represented by G-Eilenberg-MacLane spectra, and thus the Bredon homology of a point is the same thing as coefficients of these spectra. During the talk I will present the method of computing the RO(C_2)-graded coefficients of C_2-Eilenberg-MacLane spectra based on the Tate square. As demonstrated by Greenlees, the Tate square gives an algorithmic approach to computing the coefficients of equivariant spectra. In the talk we will discuss how to use this method to obtain the RO(C_2)-graded coefficients of a C_2-Eilenberg-MacLane spectrum as a RO(C_2)-graded abelian group. We will also present the multiplicative structure of the C_2-Eilenberg-MacLane spectrum associated to the Burnside Mackey functor. This allows us to further describe the RO(C_2)-graded coefficients of any C_2-Eilenberg-MacLane spectrum as a module over the coefficients of the C_2-Eilenberg-MacLane spectrum of the Burnside Mackey functor. Finally, we will discuss the RO(C_2)-graded ring structure of coefficients of spectra associated to ring Mackey functors.