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(2) Ben MacAdam (Calgary, Canada)

(3) Priyaa Srinivasan (NIST, USA) Part 1 Part 2

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

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)

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

(8) Rick Blute (Ottawa, Canada) Part1 Part2

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

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: https://arxiv.org/abs/2108.04304

(10) Robin Cockett (Calgary, Canada)

(11) Amolak Ratan (Calgary, Canada) Slides

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

(13) Martin Frankland (Regina, Canada) Slides

(14) Marcello Lanfranchi (Dalhousie, Canada) Slides

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

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

(18) Colleen Delaney (Indiana, USA)

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

(20) Laura Scull (Fort Lewis College, USA) Slides

This is joint work with Tien Chih of MSU Billings and Fort Lewis College undergraduate students.

(21) Chad Nester (Tallinn, Estonia) Slides

(Joint work with Guillaume Boisseau and Mario Roman)

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

(23) Deni Salja (Dahhousie, Canada) Slides

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

(25) Amelie Comtois (Ottawa, Canada) Slides

(26) Mario Roman (Tallinn, Estonia) Slides

(27) Elena Di Lavore (Tallinn, Estonia) Slides

(28) Samuel Desrochers (Ottawa, Canada) Slides

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

(joint work with Jason Parker)

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

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

(31) Shayester Naeimabadi (Ottawa, Canada) Slides

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

(33) Jeff Egger (Mount Allison, Canada)

(36) Brenda Johnson (Union College, USA) (off-site) Slides

(35) Cole Comfort (Oxford, UK) Slides

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

(39) Jean-Baptiste Vienney (Marseille, France) Slides

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

(41) Xiaoning Bian (Dalhousie, Canada) Slides

(42) Fahimeh Bayeh (Dalhousie, Canada) Slides

(43) Don Stanley (Regina, Canada)

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