Last Update: 2022-06-13.
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Attendees, Abstracts, and Slides

(1) Geoff Cruttwell (Mount Allison, Canada) Slides
Title: Algebraic geometry: a different path up the mountain
Abstract; Algebraic geometry is a central area of mathematics, yet can often be very difficult to get into, partly due to the intimidating nature of some of its key definitions (eg., the definition of a scheme). However, if you have some basic knowledge of category theory (eg., opposite category, monos and epis, pullbacks), there are some shortcuts one can take to relatively quickly get insight into some of the key ideas of the subject. I'll present this point of view in this tutorial.

(2) Ben MacAdam (Calgary, Canada)

(3) Priyaa Srinivasan (NIST, USA) Part 1 Part 2
Title: Title: Dagger linear logic for categorical quantum mechanics
Abstract: Dagger monoidal and dagger compact closed categories are the standard settings for Categorical Quantum Mechanics (CQM). These settings of CQM are categorical proof theories of compact dagger linear logic and are motivated by the interpretation of quantum systems in the category of finite dimensional Hilbert spaces. In this tutorial, I will describe a new non-compact framework called Mixed Unitary Categories (MUCs) with examples built on linearly distributive and *-autonomous categories which are categorical proof theories of (non-compact) multiplicative linear logic.
One of the motivations to develop a non-compact framework is to accommodate arbitrary dimensional systems in CQM in a structurally seamless manner. The notion of complimentary observables lies at the heart of quantum mechanics: two quantum observables A and B are complementary if measuring one increases the uncertainty regarding the value of the other. I will show that complementary observables and classical non-linearity are related by proving that every complementary pair of observables can be viewed as the exponential modalities - ! and ? - of linear logic "compacted" into the unitary core of the MUC, thereby exhibiting a complementary system as arising via the compaction of distinct systems of arbitrary dimensions. The machinery to arrive at this result involves linear monoids, linear comonoids, linear bialgebras and dagger- exponential modalities.
This talk is based on my thesis.
Topics to be covered: Linear logic, Linearly distributive categories, Dagger LDCs, Mixed Unitary categories, Linear duals, Linear monoids, Linear comonoids, Linear bialgebras, Dagger exponential LDCs.

(4) Dorette Pronk (Dalhousie, Canada) Slides
Title: Double Fibrations
Abstract In this talk I will present joint work with Geoff Cruttwell, Michael Lambert, and Martin Szyld [1]. We introduce a notion of double fibration as a particular kind of pseudo category structure in a suitable category of fibrations. Another way to look at this structure is as a double functor between (pseudo) double categories with certain properties; namely, the ones that make it an internal fibration (as defined in [7]) in a suitable 2-category of double categories. This construction is shown to generalize various existing notions of fibration: the discrete double fibrations in [2] and the monoidal fibrations in [3] and [5].
Furthermore, we generalize the double category of elements construction given by Paré in [6] to obtain a representation theorem establishing a correspondence between double fibrations and Span(Cat)-valued double pseudo-functors as indexing functors, or "indexed double categories" (for a suitable double 2-category Span(Cat)). This generalizes the result for discrete double fibrations given by Lambert in [2]. When considering monoidal categories as a special kind of double categories, our representation theorem also induces the equivalence between monoidal fibrations and monoidal indexed categories as in [3,5]. Finally, the “double Grothendieck construction” introduced in Definition 5.3 of [5] can be seen as an instance of our construction.
[1] Geoff Cruttwell, Michael Lambert, Dorette Pronk and Martin Szyld, Double Fibrations, preprint
[2] Michael Lambert, Discrete double fibrations, TAC 37 (2021).
[3] Joe Moeller and Christina Vasilakopoulou, Monoidal Grothendieck construction, TAC 35 (2020).
[4] David Jaz Myers, Double categories of open dynamical systems, Electronic Proceedings in Theoretical Computer Science 333 (2021).
[5] Michael Shulman, Framed bicategories and monoidal fibrations, TAC 20 (2008).
[6] Robert Paré, Yoneda theory for double categories, TAC 25 (2011).
[7] R. Street, Fibratiions and Yoneda’s lemma in a 2-category, in LNM 420, Springer, 1974

(5) Kristine Bauer (Calgary, Canada)

(6) Jonathan Gallagher (Howard Hughes Laboratories, USA)

Title: Introduction to differential programming
Abstract: The idea of differential programming is simple: one writes programs that represent smooth functions, and hence may be differentiated. Differential programming is one of the oldest areas of computer science. Yet recently, due to the use of gradient based optimization in machine learning, combined with increases in parallel compute, the study of differential programming has increased rapidly. This tutorial will introduce differential programming with some basic results, applications, and theory.

(7) Jean-Simon Lemay (Kyoto, Japan) Slides

Title: Tangent Categories and Algebraic Geometry
Abstract: In Geoff's tutorial, we learned the basics of algebraic geometry. In this tutorial, I will explain the tangent category story found in algebraic geometry.

(8) Rick Blute (Ottawa, Canada) Part1 Part2
Title: Introduction to (locally posetal) bicategories and linear bicategories
Abstract: Bicategories are categorical structures which allow morphisms between morphisms. So in general, one has objects (0-cells) and for each pair of 0-cells, X and Y, a category Hom(X,Y) rather than a set. The objects of these categories are 1-cells and the arrows of these categories are 2 cells. One has composition and identities as in an ordinary category and these are subject to a great many coherence conditions.
It turns out that much of the interesting structure and a lot of interesting examples are already present when one only considers those bicategories for which the Hom categories are posets. Rel, the category of sets and relations, is an example, as well as the category Q-Rel of Q-valued relations where Q is a quantale. This category is the basis for the theory of Monoidal Topology, which we will discuss in some detail.
Linear bicategories are bicategories with 2 compositions, which should be thought of as a bicategorical analogue of the 2 multiplicative connectives of linear logic. We'll introduce the definition and a few examples, and a later talk by Rose Kudzman-Blais will give some general constructions.

(9) Sacha Ikonicoff (Calgary, Canada) Slides

Title: Cartesian Differential Monads
Abstract: Cartesian Differential Categories are defined to introduce and study the notion of differential from calculus in a category theory point of view. In a Cartesian Differential Category, morphisms between objects can be "derived", and this differentiation operation must satisfy a list of properties, including a version of the chain rule. The most predominant source of Cartesian Differential Categories is obtained by studying the free (co)algebras of a (co)monad equipped with a heavy structure – a differential storage structure – via the concept of (co)Kleisli category.
In this talk, we will introduce the notion of a Cartesian Differential (co)Monad on a Category with finite biproducts, which gives the lightest apparatus on a (co)monad which allows us to define a Cartesian Differential Category structure on its (co)Kleisli category. We will then list quantity of examples of such monads, most of which could not be given a differential storage structure, thus motivating our new construction.
This joint work with J-S Pacaud Lemay is available on the ArXiv:

(10) Robin Cockett (Calgary, Canada)

(11) Amolak Ratan (Calgary, Canada) Slides
Title: Categories of Kirchhoff Relations
Abstract: It is known that the category of affine Lagrangian relations over a field F, of integers modulo a prime p (with p > 2) is isomorphic to the category of stabilizer quantum circuits for p-dits. Furthermore, it is known that electrical circuits (generalized for the field F) occur as a natural subcategory of affine Lagrangian relations. The purpose of this talk is to provide a characterization of the relations in this subcategory -- and in important subcategories thereof -- in terms of parity-check and generator matrices as used in error detection.
In particular, we introduce the subcategory consisting of Kirchhoff relations to be (affinely) those Lagrangian relations that conserve total momentum or equivalently satisfy Kirchhoff's current law. We characterize these Kirchhoff relations in terms of parity-check matrices and, study two important subcategories: the deterministic Kirchhoff relations and the lossless relations. This is joint work with Robin Cockett and Shiroman Prakash.

(12) Peter Selinger (Dalhousie, Canada) Slides

Title: The combinatorial game theory of Hex
Abstract: Hex is a strategy game for two players, invented in 1942 by Piet Hein and later rediscovered by John Nash. It is characterized by extremely simple rules that give rise to a surprising amount of strategic depth. Combinatorial game theory is a formalism for the study of sequential perfect information games, introduced by Conway and Berlekamp, Conway, and Guy in the 1970s and 1980s. In this tutorial, I will introduce the basic definitions and properties of combinatorial game theory. I will then describe a recently introduced variant of combinatorial game theory that is appropriate to analyzing Hex positions, and give some examples of how it can be applied to solve previously unsolved problems.

(13) Martin Frankland (Regina, Canada) Slides

Title: Modules over bialgebroids and Beck modules
Abstract: In his 1967 thesis, Beck proposed a notion of module over an object in a category. This provided a natural notion of coefficient module for André-Quillen (co)homology of any algebraic structure, generalizing the original case of commutative rings. In some cases, such as groups or Lie algebras, Beck modules are encoded by a bialgebra. The comultiplication then induces a well-behaved tensor product of modules. In work in progress with Raveen Tehara, we investigate "bialgebras with many objects" as a more general framework to encore Beck modules, where the tensor product of modules is still available. We will look at examples that fit into this framework but not that of bialgebras.

(14) Marcello Lanfranchi (Dalhousie, Canada) Slides
Abstract: One of the main questions I posed to my supervisor Geoffrey Cruttwell when I applied for the PhD program, was whether non-commutative geometry could be described using the language of tangent categories. Recently, my supervisor showed me his research on tangent category theory applied to algebraic geometry. This new work on algebraic geometry, allows me to reformulate my question in the following terms: can we extend this tangent category construction, defined for commutative algebras, to general associative algebras?
In this talk, I present an answer to this question showing how the construction studied by my supervisor 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, and then I will briefly describe the construction on commutative algebras. I will then define the concept of operads and algebras over operads. Following that, I will show how to construct a canonical tangent structure on the category of algebras over an operad. Thereafter, I will discuss the corresponding tangent structure over the opposite category, showing its geometrical meaning. 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.

(15) Florian Schwarz (Calgary, Canada) Slides

(16) Pawel Sobocinski (Tallinn, Estonia) Slides
Title: Graphical Relational Algebras
Abstract: This tutorial will start with Graphical Linear Algebra, a diagrammatic calculus for linear (aka additive) relations -- those relations between vector spaces that are also linear subspaces. The primitives of the calculus are closely related to the algebraic structure present in abelian bicategories, in the sense of Carboni and Walters.
The calculus is visually close to classical diagrammatic circuit notations in various application domains, for example signal flow graphs in engineering and control theory. Further work explored various extensions of the calculus in order to increase expressivity in terms of the class of relations that can be denoted, including affine relations, additive relations on the rig of natural numbers, and polyhedral relations, as well as introducing a general technique for adding a notion of state to the calculus. We will go through these extensions and showcase some applications, including reasoning about non-passive electrical circuits, and concurrent models of computation such as Petri nets.
[1] Carboni, Aurelio, and Robert FC Walters: Cartesian bicategories I. Journal of pure and applied algebra 49.1-2 (1987): 11-32.
[2] Filippo Bonchi, Pawel Sobocinski, Fabio Zanasi: Interacting Hopf Algebras. CoRR abs/1403.7048 (2014)
[3] Filippo Bonchi, Pawel Sobocinski, Fabio Zanasi: A Categorical Semantics of Signal Flow Graphs. CONCUR 2014: 435-450
[4] Brendan Fong, Pawel Sobocinski, Paolo Rapisarda: A categorical approach to open and interconnected dynamical systems. LICS 2016: 495-504
[5] Filippo Bonchi, Pawel Sobocinski, Fabio Zanasi: Full Abstraction for Signal Flow Graphs. POPL 2015: 515-526
[6] Filippo Bonchi, Joshua Holland, Dusko Pavlovic, Pawel Sobocinski: Refinement for Signal Flow Graphs. CONCUR 2017: 24:1-24:16
[7] Filippo Bonchi, Robin Piedeleu, Pawel Sobocinski, Fabio Zanasi: Graphical Affine Algebra. LICS 2019: 1-12
[8] Filippo Bonchi, Joshua Holland, Robin Piedeleu, Pawel Sobocinski, Fabio Zanasi: Diagrammatic algebra: from linear to concurrent systems. Proc. ACM Program. Lang. 3(POPL): 25:1-25:28 (2019)
[10] Filippo Bonchi, Alessandro Di Giorgio, Pawel Sobocinski: Diagrammatic Polyhedral Algebra. FSTTCS 2021: 40:1-40:18
[11]João Paixão, Lucas Rufino, Pawel Sobocinski: High-level axioms for graphical linear algebra. Sci. Comput. Program. 218: 102791 (2022)

(17) Chris Heunen (Edinburgh, UK) Slides

Title: Categories like Hilbert spaces
Abstract: Hilbert spaces are the mathematical foundation of quantum theory. We will consider the abstract structure of the monoidal dagger category they form. We will discuss a characterisation of this category, and identify within these axioms conceptual ingredients for quantum computation, such as dual objects and Frobenius structures to model entanglement and measurement. Spatiotemporal structure is discussed not only via the graphical calculus, but also in terms of subunits and sheaves.

(18) Colleen Delaney (Indiana, USA)

Title: Fusion Categories and Applications
Abstract: Fusion categories are monoidal categories with nice finiteness properties that allow them to be described combinatorially. They appear naturally in representation theory, low-dimensional topology, and quantum physics.
The classification and structure theory of fusion categories – which can be viewed as "quantu" generalizations of finite groups - is a quickly evolving frontier in modern algebra that is being pushed forward by mathematicians and physicists alike.
The first lecture will focus on the theory of fusion categories: how they are defined, what methods we use to study them, and the general state of our knowledge. We will see that diagrammatics, computer algebra software, and additional structures on fusion categories play important roles.
In the second lecture we will discuss some applications of fusion categories with an emphasis on topological quantum computation, making connections to Chris Heunen’s tutorial where possible. Time permitting we will also explain some connections to higher category theory.

(19) Susan Niefield (Union College, USA) Slides
Title: Locally Non-Posetal Linear Bicategories
Abstract: In this tutorial we consider biclosed bicategories which are not locally posetal. Examples include the bicategory Span of sets, spans, and their morphisms, as well as the bicategories Quant and Qtld of quantales and quantaloids, respectively (with bimodules as 1-cells and bimodule homomorphisms as 2-cells, in both cases). Examining the properties of Quant which induce a second bicategorical structure, we indicate how to generalize this example to obtain other non-locally posetal linear bicategories.

(20) Laura Scull (Fort Lewis College, USA) Slides
Title: The Fundamental Groupoid in the Category of Graphs
Abstract: Interpreting the idea of homotopy in the discrete case of graphs leads to some interesting and unexpected results. In this talk I will focus particularly on defining a fundamental groupoid for graphs, based on the notion of $times$-homotopy, and look at what it shares and how it differs from the fundamental group of the graph as a topological space.
This is joint work with Tien Chih of MSU Billings and Fort Lewis College undergraduate students.

(21) Chad Nester (Tallinn, Estonia) Slides

Title: Cornering Optics
(Joint work with Guillaume Boisseau and Mario Roman)
Abstract: Optics in a monoidal category model phenomena of interest in functional programming, the theory of open games, and the semantics of machine learning. The prevailing categorical account of optics views them as certain coends, and to reason about optics in this framework is to reason about coends. This can be quite painful.
We show that the category of optics in a monoidal category arises as a full subcategory of the horizontal cells of the free cornering (free double category with companion and conjoint structure) on the monoidal category in question. The category of optics inherits the string diagram calculus of the free cornering. This approach to optics is dramatically simpler than the coend-centric one.
In the literature on quantum circuits there is a notion of "comb diagram", which works like a circuit diagram with some number of holes in it. It has been noted that comb diagrams may be modelled as a sort of "multi-optic" in a monoidal category. Our approach to optics in the free cornering extends to account for comb diagrams.
This suggests that the free cornering of a monoidal category is a setting to consider when optics or comb diagrams are in play.

(22) Geoff Vooys (Dalhousie, Canada) Slides
Title: Equivariant Tangent Categories on Varieties

Abstract: In recent work a general formulation for an equivariant category over a variety has been developed based on descent-theoretic techniques as indexed by (a certain family of) pseudofunctors. Properties of the equivariant category itself may be deduced from structural aspects of these pseudofunctors, their pseudonatural transformations, and even modifications between them. In particular, this means that various structural properties of the equivariant category may be studied and probed by way of pseudofunctors, pseudonatural transformations, and modifications. In this talk we will introduce the resolution and descent based approach to equivariance, connect it to a fibration-based approach, and then discuss when the equivariant category is a tangent category. This is part of joint work with Dorette Pronk.

(23) Deni Salja (Dahhousie, Canada) Slides
Title: Pseudo-colimits of Diagrams of Internal Categories
Abstract: Computing a pseudo-colimit of a (small) diagram of categories can be done by localizing the category of elements by its cartesian arrows (Exercise 6.f. of SGA4). My masters thesis is about an internal description of the Grothendieck construction and Ore-localization that could be used to compute pseudo-colimits of diagrams of internal categories. This can be used to calculate hom-groupoids in the bicategory of orbifolds respresented by etale groupoids.
In this talk I'll review the usual definitions of the Grothendieck construction and Ore-localization, discuss the ambient structure used to internalize them, and share the internal definitions in my thesis along with the ideas of how proofs are translated to the internal setting(s).

(24) Nathan Hayden (Waterloo, Canada/Tallin Estonia) Slides
Title: Peirce’s 1883 presentation of relations, linear distributivity, and its corresponding graphical calculus
Abstract: Peirce presented a version of the calculus of relations in 1883 that is noteworthy for emphasizing relative composition and its dual (what he calls relative sum) as well as for stating the corresponding linearly distributive and negation laws. He went on in the next decade to develop a graphical calculus of relations in his Existential Graphs. In this talk I’ll present the corresponding graphical version of this early variant of the calculus of relations and compare Peirce’s graphical syntax to modern versions of proof nets and circuit diagrams for linearly distributive categories.

(25) Amelie Comtois (Ottawa, Canada) Slides
Title: Constructing the Tensor Product in the Category of Sup-Lattices
Abstract: The category Sup, of sup-lattices and sup-preserving functions, is of fundamental importance in several fields, most notably, monoidal topology. The tensor product of two sup-lattices classifies functions that are sup-preserving in each variable, in much the same way that the tensor product of two vector spaces classifies bilinear maps. We will observe two constructions of this tensor product and, time-permitting, describe the tensor of two sup-enriched categories.

(26) Mario Roman (Tallinn,  Estonia) Slides
Title: Monoidal Streams for Dataflow Programming
Abstract; We introduce monoidal streams: a generalization of causal stream functions to monoidal categories. In the same way that streams provide semantics to dataflow programming with pure functions, monoidal streams provide semantics to dataflow programming with theories of processes represented by a symmetric monoidal category. At the same time, monoidal streams form a feedback monoidal category, which can be used to interpret signal flow graphs.  As an example, we study a stochastic data flow language.

(27) Elena Di Lavore (Tallinn, Estonia) Slides
Title: Monoidal width
Abstract: We introduce monoidal width as a measure of the difficulty of decomposing morphisms in monoidal categories. By instantiating monoidal width and two variations a suitable category of cospans of graphs, we capture existing notions, namely branch width, tree width and path width. By changing the category of graphs, we are also able to capture rank width. Through these and other examples, we propose that monoidal width: (i) is a promising concept that, while capturing known measures, can similarly be instantiated in other settings, avoiding the need for ad-hoc domain-specific definitions and (ii) comes with a general, formal algebraic notion of decomposition using the language of monoidal categories.

(28) Samuel Desrochers (Ottawa, Canada) Slides
Title: What recursive functions can be constructed in certain categorical settings?
Abstract: In a general Cartesian category, we can simulate the natural numbers by using a natural numbers object (NNO): an object equipped with a rule for defining morphisms recursively. To see how this concept compares to our intuition for natural numbers, we would like to better understand all the operations one can define on an NNO using this rule, possibly along with other rules. One approach to doing so is to construct the initial object in the 2-category of categories that obey such rules. In this talk, I’ll describe this “initial category” both for Cartesian categories with NNO and categories with NNO and finite limits, and I’ll explain the link between these categories and primitive recursive arithmetic.

(29) Rory Lucyshyn-Wright (Brandon, Canada)

Title: Diagrammatic presentations of enriched monads and theories for a subcategory of arities
(joint work with Jason Parker)
Abstract: Building on Moggi’s insight that monads model computational effects [1], the program of algebraic computational effects of Plotkin and Power begins with the idea that computational effects are “realised by families of operations, with a monad being generated by their equational theory” [2]. As such monads are usually enriched over a closed category V, this program requires a robust theory of presentations of enriched monads by operations and equations. Work of Kelly, Power, and Lack [3, 4] provides a framework for presentations of enriched α-ary monads on a locally presentable V-category C over a locally presentable closed category V, where the arities of the operations are α-presentable objects of C for a regular cardinal α. Recent generalizations involve working with a given subcategory of arities J in a V-category C and considering enriched monads, theories, and pretheories defined relative to J [5, 6, 7]. In particular, Bourke and Garner [6] employ small subcategories of arities in locally presentable V-categories in the case where V is locally presentable, but in this case the arities are still α-presentable for some α. The Kelly-Power-Lack approach to presentations has recently been generalized by Parker and the speaker [7] to apply to small eleutheric subcategories of arities in locally bounded V-categories [8] over a locally bounded V, thus removing the assumption of local presentability and so admitting a host of new examples in closed categories of relevance in computer science, topology, and analysis. Neither of the frameworks in [6] and [7] subsumes the other, and one may argue that none of the above frameworks entirely achieves the practical objective of presenting enriched monads directly in terms of individual operations, instead requiring the user to construct a signature internal to C or a pretheory enriched in V. In this talk, we establish a common extension of the above frameworks for presentations of enriched monads, and on this basis we introduce a flexible formalism for directly describing enriched algebraic structure borne by an object of a V-category C in terms of what we call parametrized J-ary operations and diagrammatic equations, for a suitable subcategory of arities J. We introduce the notion of diagrammatic J-presentation, and we show that each such presentation presents a J-ary (or J-nervous) 33V-monad whose algebras may be described equivalently as objects of C equipped with specified parametrized operations, satisfying specified diagrammatic equations. By definition, a J-ary variety is a V-category of algebras for a diagrammatic J-presentation, and we show that the category of J-ary varieties is dually equivalent to the category of J-ary V-monads on C.
We work in an axiomatic setting based primarily on the assumption that free algebras for J-pretheories exist, and we establish a result to the effect that our axioms are in fact equivalent to the requirement that the given subcategory of arities supports presentations in an axiomatic sense. We show that our results on presentations of enriched monads are applicable in a wide variety of contexts in which V need not be locally presentable, such as in locally bounded closed categories V and various categories C enriched over such V. In particular, among locally bounded closed categories one finds various convenient categories of relevance in programming language semantics, topology, analysis, and differential geometry, including all concrete quasitoposes [9] and various cartesian closed categories of topological spaces, such as compactly generated spaces.
We discuss examples of diagrammatic J-presentations that illustrate their applicability for computational effects, including the global state algebras of Plotkin and Power [2] as well as various parametrized syntactic theories introduced by Staton for reasoning about algebraic effects [10]. We also discuss examples of diagrammatic J-presentations in category theory, including presentations for internal categories and monoidal internal categories. Lastly, we define the tensor product of diagrammatic J-presentations, which is relevant for combining algebraic computational effects (cf. [11]).
[1] E. Moggi, Notions of computation and monads, Inf. and Comp. 93 (1991), 55–92.
[2] G. Plotkin and J. Power, Notions of computation determine monads, Lecture Notes in Comput. Sci., vol. 2303, Springer, 2002, 342–356.
[3] G. M. Kelly and A. J. Power, Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads, J. Pure Appl. Algebra 89 (1993), 163–179.
[4] S. Lack, On the monadicity of finitary monads, J. Pure Appl. Algebra 140 (1999), 65–73.
[5] R. B. B. Lucyshyn-Wright, Enriched algebraic theories and monads for a system of arities, Theory Appl. Categ. 31 (2016), 101–137.
[6] J. Bourke and R. Garner, Monads and theories, Adv. Math. 351 (2019), 1024–1071.
[7] R. B. B. Lucyshyn-Wright and J. Parker, Presentations and algebraic colimits of enriched monads for a subcategory of arities, Preprint, arXiv:2201.03466, 2022.
[8] R. B. B. Lucyshyn-Wright and J. Parker, Locally bounded enriched categories, Theory Appl. Categ. 38 (2022), 684–736.
[9] E. J. Dubuc, Concrete quasitopoi, Lecture Notes in Math., vol. 753, Springer 1979, 239–254.
[10] S. Staton, Instances of computational effects: an algebraic perspective, 28th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2013), 2013, 519–528.
[11] M. Hyland, G. Plotkin, and J. Power, Combining effects: sum and tensor, Theoret. Comput. Sci. 357 (2006), 70–99.

(30) Rose Kudzman-Blais (Ottawa, Canada) Slides
Title: Constructing linear bicategories
Abstract: Linearly distributive categories were introduced to model the tensor/par fragment of linear logic, without resorting to the use of negation. Linear bicategories are the bicategorical version of linearly distributive categories. Essentially, a linear bicategory has two forms of composition, each determining the structure of a bicategory, and the two compositions are related by a linear distribution.
After the initial paper on the subject, there was little further work as there seemed to be a lack of examples. The main goal of this paper is to demonstrate that there are in fact a great many examples, which are mainly obtained by extending familiar constructions from the (ordinary) bicategorical setting. While it is standard in the field of monoidal topology that the category of quantale-valued relations is a bicategory, we begin by showing that if the quantale is a Girard quantale, we obtain a linear bicategory. We further show that Q-Rel for Q a unital quantale is a Girard quantaloid if and only if Q is a Girard quantale. The tropical and arctic semiring structures fit together into a Girard quantale, so this construction is likely to have multiple applications. More generally, we define LD-quantales, which are sup-lattices with two quantale structures related by a linear distribution, and show that Q-Rel is a linear bicategory if Q is an LD-quantale.
We then consider several standard constructions from bicategory theory, and show that these lift to the linear bicategory setting and produce new examples of linear bicategories. In particular, we consider quantaloids. We first define the notion of a linear quantaloid Q and then consider linear Q-categories and linear monads in Q, where Q as a linear quantaloid. Every linear quantaloid is a linear bicategory.
We want to develop non-locally posetal examples as well. We consider Loc, the bicategory whose objects are locales, 1-cells are bimodules and two-cells are bimodule homomorphisms. This bicategory turns out to be what we call a Girard bicategory, which are in essence a closed version of linear bicategories.

(31) Shayester Naeimabadi (Ottawa, Canada) Slides
Title: Constructing cartesian linear bicategories
Abstract: Linear bicategories were introduced by Cockett, Koslowski and Seely as bicategories with two horizontal compositions corresponding to the two logical connectives of linear logic. In particular, a 1-object linear bicategory is a linearly distributive category, i.e. a model of the tensor-par fragment of linear logic. The main example the authors introduce is the usual locally posetal bicategory Rel of sets and relations equipped with a second composition in addition to the usual one.
The notion of cartesian bicategory, due to Carboni and Walters, was introduced largely as an abstraction of the locally posetal bicategory of sets and relations. In this talk, we introduce the notion of cartesian linear bicategory and show that Rel is an example.
Blute, together with Kudzman-Blais and Niefield, show that Q-Rel the bicategory of quantale-valued relations is a linear bicategory when the quantale is a Girard quantale, or more generally, what they call an LD-quantale. While the categories Q-Rel are abstractions of the category of sets and relations, they do not in general form cartesian bicategories. We begin with the obvious observation that when the quantale is in fact a locale, then Q-Rel is a cartesian bicategory. We then show that for certain locales, in particular completely distributive lattices, then Q-Rel is a cartesian linear bicategory.

(32) Thomas Vandeven (Otttawa, Canada) Slides
Title: Monoidal Topology on Linear Bicategories
Abstract: An integral step in the generation of lax algebras in the setting of monoidal topology is the lax extension of Set functors and monads to lax functors and lax monads on Q-Rel, where Q is a quantale. In the case of locally posetal linear bicategories, a linear functor is a pair consisting of a lax functor and an oplax functor satisfying 4 linear strengths. Analogous to lax extension, we define linear extension of Set functors to linear functors on Q-Rel, where Q is a Girard quantale. Next, we define the linear Barr extension, which we will use to generate exciting new examples of linear functors on Rel, such as the linear ultrafilter functor.

(33) Jeff Egger (Mount Allison, Canada)
(36) Brenda Johnson (Union College, USA) (off-site) Slides
Title: Localizations of Model Categories from Functor Calculus
Abstract: Tom Goodwillie's calculus of homotopy functors provides a means of approximating (in a homotopical sense) a functor of topological spaces with an $n$-excisive functor that is analogous to approximating a real-valued function with a degree $n$ Taylor polynomial. The calculus of homotopy functors has inspired the creation of many other types of functor calculi in algebraic and topological contexts, including manifold calculus, orthogonal calculus, and abelian functor calculus. Work of David Barnes, Georg Biedermann, Rosona Eldred, Oliver R\"ondigs, Niall Taggart, and others has established ways to use the approximations of particular functor calculi to create model structures on functor categories, and then study the relevant functor calculi via these model structures. In this talk, we will discuss a general framework for building functor calculus model structures. This is work in progress with Lauren Bandklayder, Julie Bergner, Rhiannon Griffiths, and Rekha Santhanam.

(35) Cole Comfort (Oxford, UK) Slides
Title: Graphical Symplectic Algebra
Abstract: Inspired by the graphical calculi for affine and linear relations, we give presentations in terms of string diagrams for the props of linear/affine Lagrangian/coisotropic relations. Owing to their symplectic nature, these props give semantics for various classes of mechanical systems; notably stabilizer quantum circuits, as well as passive electrical circuits, depending on which base field is chosen.
We show that there are two constructions, both interpreted as "adding discarding", which produce coisotropic relations from Lagrangian relations. By splitting an idempotent in coisotropic relations, one obtains a two sorted prop: the original sort corresponds to the mechanical system and the new one corresponds to the linear system. In the case of stabilizer circuits, the former is the quantum system, and the latter is the classical system. In the case of stabilizer circuits, the new sort is interpreted as carrying the measured value of a current or a voltage. We also give a presentation for this two sorted prop.

(36) Sam Robertson (Calgary, Canada)
(37) Alexanna Little (Calgary, Canada) Slides
(38) Rachel Hardeman Morrill (Calgary, Canada) Slides
Title: Universal Covers in A-Homotopy Theory
Abstract: A-homotopy theory is a homotopy theory developed for graphs. We would like to know if this homotopy relation gives the weak equivalences of a model structure on the category of graphs. In order to do this, we are mimicking a strategy found in the homotopy theory of topological spaces that involves covering spaces and lifting properties. In this talk, I will discuss the universal covers I developed for graphs with no 3 or 4-cycles and the covering graphs obtained from quotienting these universal covers, as well as some related results. Note that Scull and Chih proved similar results for x-homotopy theory, a sister homotopy theory for graphs.

(39) Jean-Baptiste Vienney (Marseille, France) Slides
Title: Graded codifferential categories and some cousins
Abstract: Codifferential categories are a categorical setting for differentiation. They require an exponential which is too big to fit in the categories of finite-dimensional vector spaces. However, using a graded family of smaller exponentials provides a solution to this problem: we will define graded codifferential categories.
Natural transformations involving graded exponentials are even more explicit about conservation of resources than the non-graded ones. We will take advantage of this to present a categorical axiomatisation of Koszul complexes and Hasse derivatives under a slightly new guise.
The idea of graded codifferential was initiated reading JS Lemay's paper "Why FHilb is Not an Interesting (Co)Differential Category". I had introduced this definition to work in the category of all Hilbert spaces. JS suggested that it also unlock the case of FHilb. I also thank for their help my supervisors Rick Blute and Phil Scott while visiting Ottawa for the end of my masters program.

(40) Peng Fu (Dalhousie, Canada) Slides
Title: A biset-enriched categorical model for Proto-Quipper with dynamic lifting.
Abstract: Quipper and Proto-Quipper are a family of quantum programming languages that, by their nature as circuit description languages, involve two runtimes: one at which the program generates a circuit and one at which the circuit is executed, normally with probabilistic results due to measurements. Accordingly, the language distinguishes two kinds of data: parameters, which are known at circuit generation time, and states, which are known at circuit execution time. Sometimes, it is desirable for the results of measurements to control the generation of the next part of the circuit. Therefore, the language needs to turn states, such as measurement outcomes, into parameters, an operation we call dynamic lifting. The goal of this paper is to model this interaction between the runtimes by providing a general categorical structure enriched in what we call "bisets". We demonstrate that the biset-enriched structure achieves a proper semantics of the two runtimes and their interaction, by showing that it models a variant of Proto-Quipper with dynamic lifting. The present paper deals with the concrete categorical semantics of this language, whereas a companion paper [FKRS2022a] deals with the syntax, type system, operational semantics, and abstract categorical semantics.

(41) Xiaoning Bian (Dalhousie, Canada) Slides
Title: Generators and relations for 2-qubit Clifford+T operators
Abstract: We give a presentation by generators and relations of the group of Clifford+T operators on two qubits. The proof relies on an application of the Reidemeister-Schreier theorem to an earlier result of Greylyn, and has been formally verified in the proof assistant Agda.

(42) Fahimeh Bayeh (Dalhousie, Canada) Slides
Title: Category of Quantum Domains
Abstract: In this talk, I will introduce a category of quantum domains. This category is useful for developing models for quantum programming language which deal with "progressive information". This is a work in progress and is joint work with Dr. Peter Selinger and Dr. Abraham Westerbaan.

(43) Don Stanley (Regina, Canada)

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