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Title: “Sheaf representation of monoidal categories”

Abstract: Wouldn’t it be great if monoidal categories were nice and easy? They are! We will discuss how a monoidal category embeds into a “nice” one, and how a “nice” monoidal category consists of global sections of a sheaf of “easy” monoidal categories. Here “nice” means that the idempotent subobjects of the tensor unit have joins that are respected by tensor products, and “easy” means that the topological space of which these subobjects are the opens is local. This theorem subsumes sheaf representation results for toposes, its proof is entirely concrete, and it cleanly separates “spatial” and “temporal” directions of monoidal categories. We will focus mostly on explaining these statements and placing them in context. Joint work with Rui Soares Barbosa and others.

Title: The etale category associated to an algebraic theory

Abstract: Equational algebraic theories are typically used in

mathematics to specify notions such as group, ring, lattice, and so on.

In computer science, by contrast, they are often used to specify

programming language features such as input/output, state, stack, and so

on.

For the mathematical applications, one typically cares about set-based

models. For the computer science applications, greater weight is placed

on the set-based comodels (i.e., models in Set^op), which can be seen as

state machines providing an environment for interpreting the given

language features. This is an idea due to Plotkin, Power and Shkaravska

(in various combinations).

The objective of this talk is to explain that:

1) the category of set-based comodels of an algebraic theory is always a

presheaf category;

2) the category of topological comodels of an algebraic theory is always

a category of continuous presheaves on an etale topological category.

By way of illustration, we explain how certain etale topological

categories, well-known from the study of C*-algebras, arise in this

manner.

Title: Pictures of finite limits

Abstract: In (classical) Lawvere theories, the central role is played by categories with finite products. For example, the free category with finite products on one object (FinSet^op) is the Lawvere theory of the empty algebraic theory, and the free category with finite products on a signature of an algebraic theory has a concrete description as a category of terms.

In recent joint work with Ivan di Liberti, Fosco Loregian and Chad Nester, we developed a Lawvere-style approach to algebraic theories with partially defined operations. It turns out that in this setting, instead of categories with finite products, the relevant concept is discrete cartesian restriction categories. We developed the technology to describe free such categories. It is known that splitting idempotents yields categories with finite limits. After describing the main steps of the above narrative I will focus on giving examples of the resulting string diagrammatic calculus for free categories with finite limits.

Title: Homotopy invariance and derivatives

This talk involves joint work with Matthew Burke, Michael Ching, Brenda Johnson and Sarah Yeakel

Abstract: Goodwillie’s functor calculus provides a tower of polynomial approximations to functors of homotopical categories which resemble the Taylor series in ordinary calculus of functions. These towers are often useful because they contain a lot of data which is interesting to homotopy theorists, or because the tower make computations simpler.

Practitioners of differential category theory may naturally ask whether the resemblance to the Taylor series is indicative that Goodwillie calculus is related to some kind of categorical differentiation. For functors of abelian categories, this was answered in the affirmative in BJORT 2017, which showed that the category of abelian categories is itself a differential category (up to chain homotopy equivalence). For homotopy functors, this is answered by work of myself, Matthew Burke and Michael Ching proving that there is a tangent structure on the infinity category of infinity categories.

The purpose of this talk is to review the role of homotopy in both of these contexts and provide an introduction to these ideas for newcomers. In the case of abelian functor calculus, BJORT 2017 sought to establish a type of higher order chain rule up to homotopy. From this chain rule we were able to extract two kinds of operads. However, higher coherence data makes the extraction process difficult – a process which is reflected in my joint work with Johnson and Yeakel. In the case of homotopy functors, higher coherence data is already encoded in infinity categories. What is required in that case is ensuring that tangent structures can be defined to be homotopy invariant. To do this, one needs to examine a related model structure in which the category Weil – which is used to define tangent structures – is itself cofibrant. In this talk I’ll attempt to compare and contrast the abelian and homotopy calculus stories.

Title: Divided power algebras over an operad

Abstract: Divided power algebras were introduced by Cartan in the study of the homology of Eilenberg-MacLane spaces. The more general notion of divided power algebra over an operad was introduced by Fresse in the study of the homotopy of simplicial algebras over an operad. The aim of this talk is to characterise divided power algebras over an operad as defined by Fresse in terms of monomial operations and relations, following the classical definition of Cartan. We will establish such a characterisation, and show some examples of refinement obtained by fixing the operad, and the characteristic of the base field.

Title: Ehrhard’s exponential modalities are free!

Abstract: Ehrhard introduced models of linear logic based on “Finiteness spaces” in 2005. Priyaa Srinivasan, Cole Comfort and I used one of these models (finiteness matrices) as a model for our version of infinite dimensional quantum mechanics. To model certain quantum phenomena, we needed free exponential modalities and set out to prove that Ehrhard’s are free … to discover (belatedly) that they are, indeed, free thanks to JS pointing us at Christine Tasson’s PhD. thesis!

The talk introduces the linear logic models based on finiteness spaces and develops what we expected to be the free exponential modalities therein (which is the E_\inft of Tasson, Tabareau, and Mellies) … but which we belatedly discovered fails to be such.

Title: Complementarity in dagger linearly distributive categories

Abstract: Complementarity is key feature that distinguishes quantum from classical mechanics. Two physical variables are complementary if measurement of one variable leads to maximum uncertainty about the value of the other, and vice versa. Algebraically, complementarity is described as two commutative dagger Frobenius Algebras interacting by the Hopf Law in a dagger symmetric monoidal category. The goal of this talk to set up complementarity within the framework of dagger linearly distributive categories. As an example of our algebraic description of complementarity in this setting, I will show that splitting certain kind of idempotents on exponential modalities (! and ?) gives rise to complementary observables.

Title: The Grothendieck Construction for Double Categories

Abstract: I will describe what a double index functor is and describe a Grothendieck construction of a double category of elements and show that it forms a lax double colimit for the diagram. I will discuss how these colimits are related to other notions of 2-categorical colimits and discuss the relationship with other double categorical Grothendieck constructions.

Title: Lie integration as a left Kan extension

Abstract: Charles Ehresmann first introduced Lie groupoids as groupoids in the category of smooth manifolds through his initial work in sketch theory. His student Pradines developed Lie algebroids as the “infinitesimal approximation” of a Lie groupoid (extending the Lie group-Lie algebra correspondence). The question of what Lie algebroids could be “integrated” into a source-simply-connected Lie groupoid was one of the significant problems in differential geometry and Lie theory, and was resolved by Crainic and Fernandes in their paper “On the integrability of Lie brackets.”

In this talk, we will use Kelly’s enriched sketches to show that Lie integration is a left Kan extension. The construction follows in three steps: we first show that Lie algebroids are equivalent to involution algebroids, we next show that involution algebroids are *sketchable*, and finally we show that the theory of involution algebroids has a left-exact inclusion into the theory of smooth groupoids. A useful consequence of this result is that in presentable tangent categories (such as models of synthetic differential geometry) we can see that there is an adjunction between involution algebroids and smooth groupoids.

This talk is based on joint work with Matthew Burke.

Title: Props and Distributive Laws

Abstract:

In this talk, I will review Lack’s technique of composing props, and give examples thereof.

Many well-known concrete structures are presented by props; for example, (FinSet,+) is presented by the prop for the free commutative monoid. And by composing this prop with its opposite category using Lack’s technique, we obtain either bicommutative bialgebras or bicommutative Frobenius algebras, which are presentations for spans or cospans of finite sets, respectively—depending on the direction we compose things.

This modular process of building up props continues, eventually yielding much more complex structures. I intend to continue this process of building up props to the point where a fragment of the ZH calculus is obtained.