# Atlantic Category Theory Seminar

Information about previous seminars can be found here.

### Winter 2022 schedule

• January 18th, 2022 [note]
JS Lemay, Mount Allison University

Title: Monoidal Closed but not Compact Closed

Abstract:
Every compact closed category is monoidal closed where the internal-hom is given as $A \multimap B = A^* \otimes B$. In the special case of self-dual compact closed category, where every object is its own dual $A^* = A$, then we have that the internal-hom is precisely the tensor product $A \multimap B = A \otimes B$. However, is the converse true? Starting with a symmetric monoidal category X such that $A \multimap B = A \otimes B$, is X compact closed? At first glance this seems like this might be true... But if one tries to prove the snake equations, one runs into troubles! So the answer is more likely no... but what's the counter-example? Unfortunately, no one seems to have a natural counter-example... In this talk, I'll discuss this odd problem in more detail and hopefully someone smarter than me in the audience will come up with a counter-example. This will be an easy and relaxed talk to start the term, so everyone should be able to follow along!

• February 1st, 2022
Mitchell Riley, Wesleyan University

Title: Combining Bunched Type Theory with Dependent Types

Abstract:
Categories of interest often have more than one monoidal closed structure, and "bunched type theories" have been developed to capture this situation. In this talk, I will present a novel combination of bunched type theory with dependent types, giving a much more expressive theory. Every type will have both a linear and non-linear aspect, and the various type formers interact with both aspects. The motivation for this work was to find a way to formalise arguments from stable homotopy theory in type theory, but we also anticipate applications to the theory of quantum programming languages.
jww. with Dan Licata

• February 8th, 2022
Michael Lambert, Mount Allison University

Title: Improvements to 'Double Categories of Relations'

Abstract:
In this talk I'll discuss how the overabundance of axioms in my previous characterization of double categories of relations (ACT 2021) can be cut down to a manageable number.

• February 15th, 2022 [video]
Theo Johnson-Freyd, Dalhousie University

Title: Categorified algebraic closure

Abstract:
A famous theorem of Deligne's says that any (abelian, $\mathbb{C}$-linear) symmetric monoidal category satisfying certain mild size constraints admits a symmetric monoidal functor to the category sVec of super vector spaces. Deligne used this result to classify such symmetric monoidal categories in terms of representation theories of algebraic groups — this is the "Tannakian Duality". A few years ago, I pointed out that Deligne's theorem has a neat interpretation: it says that the symmetric monoidal category sVec is the "algebraic closure" of the symmetric monoidal category Vec of vector spaces; that the extension Vec $\subset$ sVec is "Galois"; and that the Tannakian Duality is a categorified version of the Galois correspondence. In this talk, I will explain the statement of Deligne's theorem and my interpretation, and mention some aspects of Deligne's proof.

• March 1st, 2022
Martin Szyld, Dalhousie University

Title: Double Fibrations (or functors are categories)
Joint work with Geoff Cruttwell, Michael Lambert, and Dorette Pronk

Abstract:
I will give the definition of a double fibration as a (pseudo) category object in the 2-category of fibrations of categories whose source and target are cleavage-preserving. I will show how, much like functors are category objects in an arrow category, double fibrations correspond to (strict) double functors between (pseudo) double categories that satisfy some properties. We have shown that these properties are precisely the ones making the double functor an internal fibration in the 2-category of double categories. We have also shown a "Grothendieck construction theorem", that is an equivalence of categories between double fibrations and indexed double categories. The latter are given by a type of lax functor D --> Span(Cat) from a double category D, where Span(Cat) is a 3-dimensional structure that we call a double 2-category. Two examples that are unified by the theory of double fibrations can also help to understand it, so we will look at them. One is that (as the name indicates) ours is the non-discrete case of [1], where the Grothendieck construction is that given by Bob Paré in [2] for a lax functor D --> Span(Set) between double categories. Also, seeing monoids as 1-object categories, we can recover the results for monoidal fibrations and monoidal indexed categories [3,4]. Being 3-dimensional, this is all a bit "wild" so I don't plan to give the full details, but I hope to still be able to convey "why" Span(Cat) is the receiving structure here that makes things work.

[1] DISCRETE DOUBLE FIBRATIONS, MICHAEL LAMBERT, TAC 37 (2021).
[2] YONEDA THEORY FOR DOUBLE CATEGORIES, ROBERT PARE, TAC 25 (2011).
[3] FRAMED BICATEGORIES AND MONOIDAL FIBRATIONS, MICHAEL SHULMAN, TAC 20 (2008).
[4] MONOIDAL GROTHENDIECK CONSTRUCTION, JOE MOELLER AND CHRISTINA VASILAKOPOULOU, TAC 35 (2020).

• March 8th, 2022 [slides]
Jason Parker, Brandon University

Joint work with Rory Lucyshyn-Wright

Abstract:
Several structure-semantics adjunctions and monad-theory equivalences have been established in category theory. Lawvere developed a structure-semantics adjunction between Lawvere theories and tractable Set-valued functors, which was subsequently generalized by Linton, while Dubuc established a structure-semantics adjunction between V-theories and tractable V-valued V-functors for a symmetric monoidal closed category V. It is also well known (and due to Linton) that there is an equivalence between Lawvere theories and finitary monads on Set. Generalizing this result, Lucyshyn-Wright (2016) established a monad-theory equivalence for eleutheric systems of arities in arbitrary closed categories. Building on earlier work by Nishizawa and Power, Bourke and Garner (2019) subsequently proved a general monad-theory equivalence for arbitrary small subcategories of arities in locally presentable enriched categories. However, neither of these equivalences generalizes the other, and there has not yet been a general treatment of enriched structure-semantics adjunctions that specializes to those established by Lawvere, Linton, and Dubuc.

• March 15th, 2022 [slides] [video]
Geoffrey Cruttwell, Mount Allison University

Title: Algebraic geometry and Tangent categories

Abstract:
Tangent categories axiomatize the existence of a "tangent bundle" endofunctor on a category. The canonical example is the category of smooth manifolds, but many other examples exist, including some recent examples from functor calculus.

In this series of two talks, I'll describe a tangent category structure in algebraic geometry (specifically, on the category of affine schemes) and explain why I think it's interesting.

The first talk will be introductory: I'll introduce/review tangent categories and affine schemes, and then look at the tangent category structure of affine schemes. In the second talk, we'll see how concepts you can define in any tangent category (such as vector fields, differential bundles, and connections) apply to this particular example, and how these concepts relate to existing notions in algebraic geometry.

This talk is based on joint work with JS Lemay.

• March 22nd, 2022 [slides] [video]
Geoffrey Cruttwell, Mount Allison University

Title: Algebraic geometry and Tangent categories

Abstract:
Tangent categories axiomatize the existence of a "tangent bundle" endofunctor on a category. The canonical example is the category of smooth manifolds, but many other examples exist, including some recent examples from functor calculus.

In this series of two talks, I'll describe a tangent category structure in algebraic geometry (specifically, on the category of affine schemes) and explain why I think it's interesting.

The first talk will be introductory: I'll introduce/review tangent categories and affine schemes, and then look at the tangent category structure of affine schemes. In the second talk, we'll see how concepts you can define in any tangent category (such as vector fields, differential bundles, and connections) apply to this particular example, and how these concepts relate to existing notions in algebraic geometry.

This talk is based on joint work with JS Lemay.

• March 29th, 2022
Geoffrey Vooys, Dalhousie University

Title: Persistent Cohomology for Grothendieck Toposes

Abstract:
Persistent (co)homology of spaces is a tool used by topological data analysists to give information about a topological space in terms of a filtration of vector spaces on the space. At a higher level, persistent cohomology is used to give information and features about a topological space as it persists over time. In more recent years, persistent cohomology on a topological space was generalized to a persistent sheaf cohomology by way of Cech cohomology for sheaves.

In this talk we will discuss and show how to generalize persistent cohomology on a topological space to give a notion of persistent cohomology for Grothendieck toposes both in terms of a Cech persistent cohomology and a right derived functor persistent cohomology ---this gives us two different ways of describing how geometric information can persist through a filtration of sheaves on a site. As an application and time-permitting, we will also describe a relative Comparison Lemma for sites and its interaction with persistent Cech cohomology on Grothendieck toposes, and/or a persistent Artin Comparison Theorem for finite type schemes over the complex numbers.
This is joint work with Dorrete Pronk and Deni Salja.

• April 5th, 2022 [video]
Marcello Lanfranchi, Dalhousie University

Abstract:
A few weeks ago, my supervisor Geoffrey Cruttwell presented his research on tangent category theory applied to algebraic geometry in two ATCAT seminar talks. With this work, we now know that algebraic geometry, differential geometry and synthetic geometry are models for the axiomatic theory of tangent categories.

One of the main questions I posed to Geoff when I applied for the PhD program, was whether non-commutative geometry could be described using the language of tangent categories. My background in theoretical physics makes me care about non-commutative geometry because it could be a valid mathematical framework to describe general relativity in a way that is compatible with quantum mechanics. Geoff and Lemay's work on commutative algebras allowed me to reformulate my question in the following terms: can we extend this construction to general associative algebras?

In this talk, I present an answer to this question showing how the construction presented by Geoff can be extended to non-commutative geometry and more generally to algebras of (algebraic symmetric) operads.

The talk will be structured as follows: I will start by giving the main motivation for the talk, then I will briefly recall the main definitions and constructions of tangent category theory and of Geoffâ€™s construction. I will then give the main definitions and results of operad theory. Therefore, I will show how to construct a canonical tangent structure on the category of algebras over an operad. Then, I will discuss the corresponding tangent structure over the opposite category. Finally, I will give some of the results that I found so far that extend the constructions of the commutative case.

This work is in collaboration with my supervisors Geoffrey Cruttwell and Dorette Pronk. I also would like to thank J.S. Lemay for the great discussions and ideas he shared with me about his work and mine.

• May 3rd, 2022
Luuk Stehouwer, Max Planck Institute for Mathematics

Title: Group actions on categories and their fixed points

Abstract:
Actions of a group on a set can be categorified to actions of a group on a category. The fixed point set becomes a fixed point category. I will give examples coming from representation theory, topology and physics. If time permits, I will sketch a generalization to actions of 2-groups on 2-categories.

Updated May 3, 2022 by Frank Fu