Friday, June 18 |
09:00 - 09:20 |
Anders Kock: Barycentric calculus, and the log-exp bijection ↓ In terms of synthetic differential geometry, it makes sense to compare the infinitesimal structure of a space and of its tangent bundle. This hinges of the possibility to form certain affine combinations (barycentic calculus) of the algebra maps from A to B, where A and B are arbitrary commutative rings. (Online) |
09:30 - 09:50 |
Kadri Ilker Berktav: Higher structures in physics ↓ This is a talk on higher structures in geometry and physics. We, indeed, intend to overview the basics of derived algebraic geometry and its essential role in encoding the formal geometric aspects of certain moduli problems in physics. Throughout the talk, we always study objects with higher structures in a functorial perspective, and we shall focus on algebraic local models for those structures. With this spirit, we will investigate higher spaces and structures in a variety of scenarios. In that respect, we shall also mention some of our works in this research direction. (Online) |
10:00 - 10:20 |
Rowan Poklewski-Koziell: Frobenius-Eilenberg-Moore objects in dagger 2-categories ↓ A Frobenius monad on a category is a monad-comonad pair whose multiplication and comultiplication are related via the Frobenius law. Street has given several equivalent definitions of Frobenius monads. In particular, they are those monads induced from ambidextrous adjunctions. On a dagger category, much of this comes for free: every monad on a dagger category is equivalently a comonad, and all adjunctions are ambidextrous. Heunen and Karvonen call a monad on a dagger category which satisfies the Frobenius law a dagger Frobenius monad. They also define the appropriate notion of an algebra for such a monad, and show that it captures quantum measurements and aspects of reversible computing. In this talk, we will show that these definitions are exactly what is needed for a formal theory of dagger Frobenius monads, with the usual elements of Eilenberg-Moore object and completion of a 2-category under such objects having dagger counterparts. This may pave the way for characterisations of categories of Frobenius objects in dagger monoidal categories and generalisations of distributive laws of monads on dagger categories. (Online) |
10:30 - 10:50 |
Tarmo Uustalu: Monad-comonad interaction laws (co)algebraically ↓ I will introduce monad-comonad interaction laws as mathematical objects to describe how an effectful computation (in the sense of functional programming) can run in an environment serving its requests. Such an interaction law is a natural transformation typed
T X \times DY \to R (X \times Y)
for T, R monads and D a comonad on a cartesian (or symmetric monoidal) category C subject to two equations. I will show what interaction laws amount to in terms of functors between the categories of (co)algebras of these (co)monads. I will explain that interaction laws are measuring morphisms in [C, C] as a duoidal category wrt. Day convolution and composition and show how the (co)algebraic perspective helps describe the Sweedler hom of T, R, i.e., the universal D interacting with T R$-residually.
This is based on joint works with Dylan McDermott, Shin-ya Katsumata, Exequiel Rivas, Niels Voorneveld. (Online) |
11:00 - 11:20 |
Nicolas Blanco: Bifibrations of polycategories and MLL ↓ Polycategories are structures generalising categories and multicategories by letting both the domain and codomain of the morphisms to be lists of objects. This provides an interesting framework to study models of classical multiplicative linear logic. In particular the interpretation of the connectives ise given by objects defined by universal properties in contrast to their interpretation in a *-autonomous category.
In this talk, I will introduce the notion of bifibration of polycategories and I will present how the universal properties of the connectives can be recovered as specific bifibrational properties.
I will illustrate this approach through the examples of finite dimensional Banach spaces and contractive maps. These form a *-autonomous category which structure is given by lifting the compact closed structure of the category of finite dimensional vector spaces. This lifting is made possible by considering the fibrational properties of the forgetful functor between the underlying polycategories. (Online) |
11:30 - 11:50 |
Simona Paoli: Weakly globular double categories and weak units ↓ Weakly globular double categories are a model of weak 2-categories based on the notion of weak globularity, and they are known to be suitably equivalent to Tamsamani 2-categories. Fair 2-categories, introduced by J. Kock, model weak 2-categories with strictly associative compositions and weak unit laws. In this talk I will illustrate how to establish a direct comparison between weakly globular double categories and fair 2-categories and prove they are equivalent after localisation with respect to the 2-equivalences. This comparison sheds new light on weakly globular double categories as encoding a strictly associative, though not strictly unital, composition, as well as the category of weak units via the weak globularity condition.
Reference: S. Paoli, Weakly globular double categories and weak units, arXiv:2008.11180v1 (Online) |
13:00 - 13:20 |
Chad Nester: Concurrent Material Histories ↓ The resource-theoretic interpretation of symmetric monoidal categories allows us to express pieces of material history as morphisms. In this talk we will see how to extend this to capture concurrent interaction.
Specifically, we will see that the resource-theoretic interpretation extends to single object double categories with companion and conjoint structure, and that in this setting material history may be decomposed into interacting concurrent components.
As an example, we will show how transition systems with boundary (spans of reflexive graphs) can be equipped to generate material history in a compositional way as transitions unfold. Some directions for future work will also be proposed. (Online) |
13:30 - 13:50 |
Cole Comfort: A graphical calculus for Lagrangian relations ↓Symplectic vector spaces are the phase space of linear mechanical systems. The symplectic form describes, for example, the relation between position and momentum as well as current and voltage. The category of linear Lagrangian relations between symplectic vector spaces is a symmetric monoidal subcategory of relations which gives a semantics for the evolution -- and more generally linear constraints on the evolution -- of various physical systems.
We give a new presentation of the category of Lagrangian relations over an arbitrary field as a `doubled' category of linear relations. More precisely, we show that it arises as a variation of Selinger's CPM construction applied to linear relations, where the covariant orthogonal complement functor plays of the role of conjugation. Furthermore, for linear relations over prime fields, this corresponds exactly to the CPM construction for a suitable choice of dagger. We can furthermore extend this construction by a single affine shift operator to obtain a category of affine Lagrangian relations. Using this new presentation, we prove the equivalence of the prop of affine Lagrangian relations with the prop of qudit stabilizer theory in odd prime dimensions. We hence obtain a unified graphical language for several disparate process theories, including electrical circuits, Spekkens' toy theory, and odd-prime-dimensional stabilizer quantum circuits. (Online) |
14:00 - 14:20 |
Nuiok Dicaire: Localization of monads via subunits ↓ Given a “global” monad, one wishes to obtain “local” monads such that these locally behave like the global monad. In this talk, I will provide an overview of how subunits can be used to provide a notion of localisation on monads. I will start by introducing subunits, a special kind of subobject of the unit in a monoidal category. Afterwards, I will provide two equivalent ways of understanding the localisation of monads. The first involves a strength on subunits, while the second relies on the formal theory of graded monads. I will also explain how to construct one from the other. (Online) |
14:30 - 14:50 |
Jean-Simon Lemay: Linearizing Combinators ↓ Bauer, Johnson, Osborne, Riehl, and Tebbe (BJORT) showed that the Abelian functor calculus provides an example of a Cartesian differential category, where the differential combinator is defined using linearization. From the Cartesian differential category perspective, the BJORT construction is backwards. In any Cartesian differential category it is always possible to define the notion of a linear map and to linearize a map using the differential combinator. BJORT constructed their differential combinator using an already established notion of linear map and linearization.
In this talk, we reverse engineer BJORT's construction by abstracting the notion of linear approximation by introducing linearizing combinators. Every Cartesian differential category comes equipped with a canonical linearizing combinator obtained by differentiation at zero. Conversely, a differential combinator can be constructed à la BJORT from a system of linearizing combinators in context. Therefore, linearizing combinators provide an equivalent alternative axiomatization of Cartesian differential categories.
This is joint work with Robin Cockett. (Online) |
15:00 - 15:20 |
Sacha Ikonicoff: Divided power algebras with derivation ↓ Classical divided power algebras are commutative associative algebras endowed with `divided power' monomial operations. They were introduced by Cartan in the 1950's in the study of the homology of Eilenberg-MacLane spaces, and appear in several branches of mathematics, such as crystalline cohomology and deformation theory.
In this talk, we will investigate divided power algebras with derivation, and identify the most natural compatibility relation between a derivation and the divided power operations. The work of Keigher and Pritchard on formal divided power series (also called Hurwitz series) suggests a certain `power rule'. We will prove, using the framework of operads, that this power rule gives a reasonable definition for a divided power algebra with derivation. We will extend this result to a more general notion of divided power algebras, such as restricted Lie algebras, with derivation. (Online) |