Last Update: 2024-04-29. For updates or additions to this page, please send a note to Robin Cockett.


Attendees, Abstracts, and Slides

(1) Geoff Cruttwell (Mount Allison, Canada)Notes

    Title: Algebraic Geometry: more paths on the mountain
    Abstract: At FMCS 2022, I gave a tutorial on one way to approach the basic ideas of algebraic geometry (specifically affine schemes) via category theory. In this tutorial I'll review these ideas, before adding additional ideas and thoughts about projective geometry.

(2) David Spivak (Topos, USA) Slides

    Title: Tutorial on polynomial functors
    Abstract: In this 2-hour session, we'll discuss the theory and applications of the category Poly of polynomial functors in one variable and the natural transformations between them. As for theory, Poly is complete and cocomplete, cartesian closed, and equipped with infinitely many monoidal closed structures and various orthogonal factorization systems. Polynomial comonads are precisely the same as small categories, and the associated double category Cat^#:=Comod(Poly) of polynomial comonoids and bicomodules includes multivariate polynomial functors as a special case (a locally-full sub-double-category). Moreover, the formal monads in Cat^# model all sorts of lower and higher categories, e.g. categories, symmetric monoidal categories, double categories, symmetric infinity categories, etc. Priyaa Srinivasan and I recently showed that Poly is also a linear distributive category with some interesting structures.As for applications, Cat^# can model databases and data migrations, various flavors of machines (e.g. Moore and Mealy machines) and wiring diagrams, cellular automata, and various functional programming constructs.

(3) C.B. Aberle (Topos and CMU, USA)

    Title: Parametricity via Cohesion
    Abstract: Parametricity is a key metatheoretic property of type systems, which implies strong uniformity properties of the structure of types within systems possessing it. In recent years, various systems of dependent type theory have emerged with the aim of expressing such parametric reasoning in their internal logic. More recently still, such internal parametricity has been applied to solving various problems arising from the complexity of higher-dimensional coherence conditions in type theory. As a first step toward the unification, simplification, and extension of these methods for internalizing parametricity, I argue that there is an essentially modal aspect of parametricity, intimately connected with the topos-theoretic concept of cohesion. On this basis, I describe a general categorical semantics for modal parametricity, develop a corresponding framework of axioms in dependent type theory, and demonstrate the practical utility of these axioms by using them to painlessly derive classical parametricity results as well as more intricate examples such as induction principles for higher inductive types.

(4) Priyaa Srinivasan (Topos, USA)

    Title: What kind of linearly distributive categories do Polynomial functors form? Slides
    Abstract: In this tutorial, I will introduce Poly, the category of polynomial functors and natural transformations as a linearly distributive category (LDC). Currently well-known examples are both symmetric and *-autonomous, and are structurally quite complex. Poly is an accessible yet a rich example of non-symmetric and non *-autonomous isomix LDC. More excitingly, inside Poly we find simple yet non-trivial examples of linear duals, linear monoids, linear comonoids and linear bialgebras. By studying Poly as an LDC, we will also discover new properties of LDCs such as bi-closure. This tutorial will be a tour of LDCs in the Poly land.


    Title: Communicating Relational Thinking Slides
    Abstract: In this talk I will give an overview of our recently completed experiment at the Topos Institute on communicating categorical thinking to a STEM-oriented audience outside mathematics. Angeline, Brendan, Paul and myself recently authored a free online textbook titled "Relational thinking: from Abstractions to Applications". In this talk, I will walk through the contents of the book and equally importantly the open source technologies behind the book. This talk is an invitation to the audience not only to read the book but also to create their own inclusive material around category theory and math leveraging the newly available technologies.

(5) Dorette Pronk (Dalhousie, Canada)Slides

    Title: Double Category Sites for all Grothendieck Topoi
    Abstract: In earlier work, Darien DeWolf and I introduced Ehresmann sites as a double categorical representation for etendues. This representation was not entirely satisfactory: we could give maps between sites that would give maps between topoi and we could identify the maps satisfying a double categorical version of the Comparison Lemma, but there were not enough ​"roofs" to allow for a bicategory of fractions. In this talk I will introduce joint work with Darien DeWolf and Julia Ramos Gonzalez where we generalize the notion of Ehresmann site. These generalized sites can be used to represent all Grothendieck topoi and the new Comparison Lemma maps do admit a bicategory of fractions representation for the 2-category of Grothendieck topoi.  I will also characterize the sites that among these new sites that represent \'etendues. The main techniques used in this work are related to the connections between categories with factorization systems and double categories with certain fibration properties.

(6) Kristine Bauer (Calgary, Canada)

(7) Jean-Simon Lemay (Macquarie, Australia)SlidesSlides (no pause)

    Title: Drazin Inverses in Categories
    Abstract: Drazin inverses are a special kind of generalized inveres that have been extensively studied with many applications in ring theory, semigroup theory, and matrix theory. Drazin inverses can also be defined for endomorphisms in any category.  In this tutorial, I will give an introduction to Drazin inverses from a categorical perspective. The talk is based on the paper which is joint work with Robin Cockett and Priyaa Srinivasan.

(8) Rick Blute (Ottawa, Canada)

    Title: An Introduction To Quantum Sets And The Linearly Distributive Version
    Abstract: We'll begin by looking at the following classical observation, which is a special case of Gelfand duality. The category of finite-dimensional commutative C*-algebras and ∗-homomorphisms is equivalent to the opposite of the category of finite sets and functions. In keeping with the theory of categorical quantum mechanics, the theory of quantum sets involves lifting this result to general monoidal categories, where C*-algebras are replaced by dagger Frobenius algebras. This work began with a paper by Coecke, Pavlovic and Vicary., which we'll review in detail. We'll also look at the related notion of involutive monoid defined by Vicary.

(9) Robin Cockett (Calgary, Canada)

(10) Marcello Lanfranchi (Dalhousie, Canada)Slides

    Title: Display systems are things of the past! Tangent display maps are the future!
    Abstract: In the category of smooth manifolds, the existence of pullbacks between two maps is a delicate matter. Consequently, when working with a family of bundles in a tangent category, lots of care is put into deciding which pullbacks should exist. Often, one would like a desired property to be preserved by pullbacks and by the tangent bundle functor. This desire brought Cockett and Cruttwell to introduce the notion of a tangent display system: a collection of morphisms stable under pullback and the tangent bundle functor. From the perspective of differential geometry, this is quite an odd assumption. Instead of requiring an intrinsic property of a bundle, one needs to carry the structure of a tangent display system and assume the bundle to be an element of the display system.
The solution? Tangent display maps. In this talk, I will present the notion of a tangent display map and prove that tangent display maps form the maximal tangent display system. I will also discuss an important sufficient condition under which tangent display maps are closed under retracts. Using this result and employing an important characterization due to Ben MacAdam, I will give sufficient condition under which all display bundles of a tangent category are automatically tangent display maps.

This is a joint work with Geoff Cruttwell

(11) Florian Schwarz (Calgary, Canada)Slides

    Title: Differential bundles in tangent (infinity) categories
    Abstract: Tangent categories generalize differential geometry in a functorial way. Applications of their infinity categorical version, tangent infinity categories include Goodwillie functor calculus. Differential bundles are one of the most important structures appearing in tangent categories. They can be used for connections and dynamical systems. A special case of them, differential objects, provide an adjunction between cartesian differential categories and tangent categories. While differential objects have been generalized to tangent infinity categories, differential bundles are still in the process to be generalized. In this talk I will present the current status of the generalization process. The key step to the generalization of differential bundles to infinity categories is a functorial characterization of differential bundles in “ordinary” categories I recently obtained. The next steps after this are to transfer this functorial characterization into infinity categories.

(12) Susan Niefield (Union College, USA)Slides

    Title: Adjoints in double categories
    Abstract: We present three double categories of quantales.  The first  is strict, the second is pseudo, and the third is actiualy a double bicategory.  Along the way we encounter quantale-values relations, projective modules, companions, conjoins, adjoint bimodules, and Cauchy completeness.  The strict bicategory is Cauchy (i.e. every object is Cauchy complete). The pseudo one is not, but this is corrected using the Kleisli bicategory of a graded monad as part of the double bicategory.

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

    Title: Fundamental Groupoids of Graphs
    Abstract: We examine several different approaches to defining fundamental groupoids for graphs with respect to X-homotopy. We begin with a definition for reflexive graphs, and then move to general graphs. We then consider a fundamental groupoid based on an edge-centered view of graphs. Since edges come with a natural action of Z/2 based on reversing their direction, we can define an equivariant edge fundamental groupoid. During this talk, we will explore these definitions (with many pictures) and I will explain some connections between them.

(14) Samuel Desrochers (Ottawa, Canada)Slides

    Title: With enough structure, the list functor is a polynomial
    Abstract: Given a set X, one can form the set L(X) of all finite lists with elements in X. Since the set of lists of length n is equivalent to X^n, if you sort the lists by length, we find that L(X) is equivalent to a disjoint sum of finite powers of X. This looks a bit like… a polynomial? Indeed, the functor L (which maps X to L(X)) is an example of a polynomial functor! In this talk, I’ll explain how these concepts (list objects and polynomial functors) are generalized to an arbitrary category C, and we’ll see that the list object functor L is still a polynomial functor… as long as you make certain assumptions about C.

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

    Title: An introduction to V-graded categories, functor categories, and bifunctors, without symmetry
    Abstract: Categories graded by a monoidal category V generalize both V-enriched categories and V-actegories. The theory of (V-)graded categories demands no assumptions on the base V and is in some ways more accessible than that of enriched categories, as V-graded categories admit an elementwise formulation in terms of graded morphisms. V-graded categories were introduced by Richard Wood, who called them large V-categories, and they have been studied and used by several authors, including Kelly, Labella, Schmitt, and Street (who called them procategories), Garner (who described tangent categories as certain graded categories), Levy (who called them locally graded categories), and McDermott and Uustalu (who employed them in the study of computational effects).
In Part I of this two-part talk, we introduce (left) V-graded categories and methods for working with them, and we discuss examples of V-graded categories. We establish a formalism of commutative diagrams for V-graded categories by embedding each V-graded category into a V-actegory, freely, as an instance of a construction that generalizes optics. We discuss the dual notions of right V-graded category and V-cograded category, and we introduce a notion of V-W-bigraded category, noting that V itself is canonically V-V-bigraded. We establish a contravariant change-of-base process for V-graded categories with respect to opmonoidal functors, and we provide a method for presenting V-graded categories by generators and relations.
In Part II, we discuss the speaker's recent results providing a theory of V-graded functor categories and bifunctors in the absence of a symmetry (or any further structure) on V. These results are formulated in terms of a given pair of monoidal categories V and W, assumed strict monoidal by appeal to coherence. Given a left V-graded category A and a V-W-bigraded category C, we show that left V-graded functors from A to C are the objects of a right W-graded category [A,C], while if B is right W-graded category then right W-graded functors from B to C are the objects of a left V-graded category [B,C]. For example, taking W = V and C = V, every left V-graded category A determines a right V-graded category [A,V]. Given a left V-graded category A and a right W-graded category B, we define a V-W-bigraded category AB whose objects are pairs consisting of an object of A and an object of B, and we show that V-W-bigraded functors AB --> C are in 2-natural bijective correspondence with left V-graded functors A --> [B,C] and also with right W-graded functors B --> [A,C]. We employ these results to establish broad classes of examples of graded categories of structures in bigraded categories.

(16) Alexanna Little (Calgary, Canada)Slides

    Title:Semantics of CaMPL - Showing the powerset functor is monoidal
    Abstract: Categorical Message Passing Language (CaMPL) is a concurrent functional programming language being developed by a team led by Dr. Robin Cockett at the University of Calgary. The semantics of non-deterministic CaMPL programs are given by the powerset functor, and a key step in defining the semantics this way is showing the powerset functor is monoidal. In this talk, we will explore a definition of moniodal functors and show that the powerset functor satisfies this definition.

(17) Melika Norouzbeygi (Calgary, Canada)Slides

    Title: Type Classes in CaMPL
    Abstract: Overloading, which allows functions to exhibit different behaviors based on the types involved, is one of the most useful facilities of a typed programming language. In the Haskell programming language, this facility is provided by type classes. Type classes offer a systematic solution for overloading, providing uniform operations for arithmetic, equality, and displaying values, and so on. In addition, higher-order type classes are used to implement important more advanced structures such as functors and monads. Categorical Message Passing Language (CaMPL) is a concurrent programming language based on a categorical semantic given by a linear actegory. CaMPL has with type inference for both sequential and concurrent programs. The sequential side of CaMPL is a functional-style programming language, while the concurrent side supports message passing between processes along channels with concurrent types called protocols. In this presentation, first the importance of type classes will be discussed, then CaMPL will be introduced, and its different features will be explained. Finally, the process of integrating type classes into both sequential and concurrent tiers of CaMPL will be discussed.

(18) Saina Daneshmand (Calgary, Canada)Slides

    Title: Type Inference for CaMPL
    Abstract: CaMPL is a statically typed programming language. In statically typed languages, the type of each variable must be explicitly declared or inferred by the compiler before the code is executed. CaMPL features a powerful type inference system, allowing the compiler to automatically deduce the types of expressions without explicit type declarations from the programmer. If the types are explicitly declared, it ensures that the inferred types are compatible with the declared types. This ensures that the types are valid with regard to each other, whether they are explicitly declared or not. I will introduce the type checking and type inference rules for various constructs of this language. Through detailed examples, I will demonstrate how type inference in CaMPL can identify type errors and ensure that types are used consistently and correctly.

(19) Durgesh Kumar (Calgary, Canada)Slides

    Title: The Category of Lagrangian Relations
    Abstract: We will first motivate the study of Lagrangian relations from Stabilizer mechanics. Our starting point will be the correspondence between Weyl-Heisenberg Operators and vectors in the Symplectic Vector Space Z_p^2n. Stabilzer states corresponds to symplectomorphisms of this space. The graph of these symplectomorphisms is a Lagrangian relation. Our main focus will be to define the category of Lagrangian Relations. We will see why Lagrangian relations is closed to composition by using what are called co-isotroipc reductions of Symplectic Spaces. We will end the talk by defining the prop of Lagrangian relations and discussing some results.

(20) Adrian Tadic (Calgary, Canada)

    Title: Introduction the abstract machine for CaMPL
    Abstract: An abstract machine is a formal model of computation that specifies precisely how programs written for it are executed independent of any implementation details. This is particularly helpful when designing new programming languages, as one can write highly portable simulations of abstract machines which can actually execute programs. In my talk, I will go over the abstract machine which underlies the CaMPL programming language designed by Cockett et al. This machine has two components -- a sequential component which handles individual processes and a concurrent component which handles communication between processes. I will discuss both during my talk, but I will focus on explaining the novel method the concurrent side of the machine uses to handle message passing. I will conclude the talk with a few remarks about the implementation of the machine.

(21) Katrina Honigs (Simon Fraser, Canada)Slides

    Title: McKay for reflection groups and semiorthogonal decompositions
    Abstract: The classical McKay correspondence is concerned with the finite subgroups G of SL(2,C). There is a bijection between irreducible representations and the exceptional divisors of the minimal resolution C^2/G. Bridgeland, King and Reid showed this correspondence can be recast and extended as an equivalence of derived categories of coherent sheaves. When this framework is extended to finite subgroups of GL(2,C) generated by reflections, the equivalence of categories becomes a semiorthogonal decomposition whose components are, conjecturally, in bijection with irreducible representations of G. This correspondence has been verified in recent work of Potter and of Capellan for a particular embedding of the dihedral groups D_n in GL(2,C). I will discuss joint work in progress toward verifying this decomposition in further cases.

(22) Shayesteh Naeimabadi (Ottawa, Canada)

    Title:Cartesian Linear Bicategories
    Abstract: In this talk, after briefly reviewing the notions of locally ordered cartesian bicategories and linear bicategories, we present our initial thoughts on extending the notion of cartesian bicategories to locally ordered linear bicategories, which we call Cyclic Cartesian linear bicategories. We will then explain why this structure does not work properly by demonstrating that the linear bicategory Rel of sets and relations is Cyclic Cartesian but does not have the linear bicategorical product. Subsequently, we introduce an alternative definition called Cartesian linear bicategories which works properly and allows us to extend the structure to general linear bicategories.

(23) Martin Frankland (Regina, Canada)

    Title: Beck modules
    Abstract: Beck modules were introduced in the 1960s as a convenient notion of coefficient module for cohomology. In the first lecture, we will cover the basics of Beck modules, some examples, and their role in cohomology theories. In the second lecture, we will explore some categorical aspects: fibered category of Beck modules (a.k.a. tangent category), representing ring(oid)s, and simplicial homotopy theory.

(24) Jonathon R. Funk (CUNY, USA)Slides

    Title: Toposes and C*-algebras
    Abstract: We define and study a certain left-cancellative category and topos associated with a C*-algebra. The topos we define is inspired by and to some extent resembles what is done in pseudogroup and inverse semigroup theory, while recognizing that for a C*-algebra there are distict and novel points of departure from the semigroup constructions.  We work under the hypothesis we call a supported C*-algebra, which means that the algebra has enough projections in a certain sense. We shall establish a topos interpretation of the so called polar decomposition of an operator. This intepretation is part of a correspondence between quotients of a torsion-free generator of the topos of the C*-algebra, and certain subcategories of the left-cancellative category of the algebra.

(25) Cole Comfort (Univ. Loraine, France)Slides

   Title: A complete equational theory for Gausian quantum circuits
   Abstract: In this talk, I will discuss recent work with Robert I. Booth and Titouan Carette where we give a generators and equations presentation for Gaussian quantum circuits with formal infinitely squeezed states. It is well-known that the category of (infinite dimensional) Hilbert spaces is not compact closed, thus it does not admit a ZX-calculus style presentation in terms of interacting Hopf/Frobenius algebras. To sidestep this issue, we use an alternative ``phase-space'' semantics in terms of subcategory of affine Lagrangian relations between finite dimensional complex vector spaces. This allows Gaussian quantum circuits to be regarded, in some sense, as infinite dimensional generalisation of the stabiliser ZX-calculus.

(26) Aaron Fairbanks (Dalhousie, Canada)Slides

    Title: Representable PROs
    Abstract: The definition of monoidal category intimidates people. The good news is, there is an equivalent definition that is relatively self-explanatory. Not only is this definition beginner-friendly, but in my opinion it makes thinking about monoidal categories easier. There are also similar tricks for thinking about symmetric monoidal categories, bicategories, and doubly weak double categories. If you don't know anything about monoidal categories, come along to learn what they are. If you know everything about monoidal categories, come along to unlearn what they are.

(27) Samuel Steakley (Calgary, Canada)Slides

    Title: The Free Cornering as a Functor
    Abstract: The free cornering [A] of a monoidal category A has been proposed as a categorical model of concurrent computations. [A] is a certain one-object double category whose vertical category is essentially a copy of A, and which also has freely added corner cells that make it a proarrow equipment. The free cornering is recommended by the fact that it enjoys recourse to the graphical calculus of proarrow equipments and their rich theory as a tool in formal category theory, and that Chad Nester has demonstrated connections with linear actegories and message passing logic. In this talk, we will introduce the free cornering and present work in progress on its properties as a functor on the category of strict monoidal categories, motivated by the prospect of comparing the free cornering with categorical models of higher-order quantum theory.

(28) Jean Baptiste Vienney (Ottawa, Canada)Slides

    Title: Sections, retractions and algebraic structures
    Abstract: How can we quotient a monad? Maybe simply by quotienting a monoid in the monoidal category of endofunctors. How to quotient a monoid in a monoidal category? A first idea is to use a coequalizer preserved by the tensor product. But it is unlikely that the coequalizer will be preserved by the tensor product in an endofunctor category. We can also think of quotients in terms of sections and retractions. The intuition is to replace equivalence classes by canonical representatives. It gives another way to quotient monoids in a monoidal category, and this one applies nicely to monads. For instance, it produces the multiset monad as a quotient of the list monad. Similar results apply to algebras in universal algebra, algebras over a monad and algebras for an operad. We can also obtain sub algebraic structures in each of these situations. All of this has a very constructive feeling, and it can be explained with string diagrams. This is joint work with Ralph Sarkis.

(29) Matthew Di Meglio (Edinburgh, UK)Slides

    Title:Abelian groups are to abelian categories as Hilbert spaces are to what?
    Abstract:The notion of abelian category is an elegant distillation of the fundamental properties of the category of abelian groups, comprising a few simple axioms about products and kernels. While the categories of real and complex Hilbert spaces and bounded linear maps are not abelian, they satisfy almost all of the abelian category axioms. Heunen and Kornell’s recent characterisation (https://doi.org/10.1073/pnas.2117024119) of these categories of Hilbert spaces is reminiscent of the Freyd–Mitchell embedding theorem, which says that every abelian category has a full, faithful and exact embedding into the category of modules over a ring. The axioms are similar, but incorporate the extra structure of a dagger—an identity-on-objects involutive contravariant endofunctor—which encodes adjoints of bounded linear maps. By keeping only the axioms that directly parallel the ones for abelian categories, we arrive at a nice class of dagger categories, which I call rational dagger categories, that enjoy many of the same properties as the categories of Hilbert spaces mentioned above. The name alludes to their unique enrichment in the category of rational vector spaces. In this talk, I will give a gentle introduction to rational dagger categories, highlighting the parallels with abelian categories. I will not assume prior familiarity with dagger categories, instead introducing the relevant concepts as needed. This talk is based on a recent preprint (https://arxiv.org/abs/2312.02883).

(30) Geoff Vooys (Dalhousie, Canada)Slides

    Title: A Grothendieck Topology for Gluing Differential Bundles
    Abstract: One of the most important techniques in algebraic geometry is the gluing of sheaves; in fact, the celebrated Gabriel-Rosenberg Theorem states that quasi-separated schemes can be recovered up to isomorphisms from equivalences of quasi-coherent sheaves.
Recent work of Geoff Crutwell and JS Lemay on tangent categories in algebraic geometry, we know that there is an opposite equivalence DBun(X) \simeq QCoh(X)^{op} for any scheme X. Putting these observations together suggests using the differential bundles in a tangent category to build a Grothendieck topology in a way that mirrors the manner in which quasi-coherent sheaves are glued on schemes. In this talk we'll discuss some progress towards making this more than just a pipe dream, describe some technical difficulties that arise, and also describe some situations where this does work. This talk is based on joint work with JS Lemay.

(31) Robert Morissette (Dalhousie, Canada)

(32) Rose Kudzman-Blais (Ottowa, Canada)Slides

    Title: Medial Linearly Distributive Categories
    Abstract: Linearly distributive categories (LDCs) were introduced by Cockett and Seely to model multiplicative linear logic. Of particular interest are cartesian LDCs, which model the "and/or" of intuitionistic logic. This raises the question of whether there is an adjunction relating symmetric and cartesian LDCs, similar to the relationship for monoidal categories described by Fox’s theorem. As we will discuss in this talk, it turns out that we need to restrict our attention to a subclass: medial linearly distributive categories. In these LDCs, the tensor and par are additionally linked by a medial transformation. The medial inference rule has been studied in the context of deep inference as it allows contraction and weakening to be presented in atomic forms and in duoidal categories as it connects the two monoidal structures present.

(33) David Sprunger (Indiana State,USA)

    Title: Training neural networks with quiver representations
    Abstract: Armenta and Jodoin (2021) model neural networks abstractly with a slight modification of a quiver representation that they call a network representation. They show that the space of network representations for a given quiver form a moduli space and calculate its dimension. I will discuss some in-progress work (joint with Michael Henry) developing new training algorithms for neural networks, as a direct result of our failure to equip this space of quiver representations with a differential structure.

(34) Amelie Comtois (Ottawa, Canada)

    Title: The equivalence between V-graded categories and bicategories with a local discrete vibration
    Abstract: Categories graded by a monoidal category V, or simply V-graded categories, manage to generalize both V-enriched categories and V-actegories without requiring any additional properties of the base category V, making them particularly interesting structures. Still, more generally, categories graded by a bicategory V may be defined succinctly as categories enriched in the local cocompletion of V. Kelly, Labella, Schmitt, and Street identified a correspondence (on objects) between V-graded categories and local discrete fibrations over V, which are bicategories equipped with a strict homomorphism into V that is locally a discrete fibration. This correspondence was extended to a 2-equivalence as part of an unpublished collaboration between Cockett, Niefield, and Wood. The equivalence provides an alternative and elementary description of V-graded categories that makes no reference to bicategories or 2-cells. When V is strict monoidal, we thus obtain a particularly simple description of V-graded categories as ordinary categories with additional structure. This presentation is based on joint work with Richard Blute and Rory Lucyshyn-Wright.

(35) Sam Winnick (Waterloo, Canada)Slides

    Title: Interactions between de Morgan duality, associativity, reflexivity, and enrichment
    Abstract: We consider a unital operation on a category without imposing associativity let alone braidedness, and reveal interactions between other possible structures that the category may possess. We will begin by recalling a few properties of adjoint functors that respect the operation, and then specialize to the case where the category is biclosed and the functors are the left and right negation with respect to some dualizing object. In particular, we show that if objects are determined by their global elements, in a sense we formalized through enrichment, then a (possibly non-coherent) kind of associativity is forced by the enriched Yoneda embedding. We give separating examples throughout, and explain how this ties into the story of star-autonomy.

(36) Sacha Ikonicoff (Strasbourg, France)Slides

    Title: Abelianisation and differential structures
    Abstract: A Generalised Cartesian Differential Category (GCDC), as defined by Geoff Cruttwell, is a category in which each object can be linearised in a suitable way, and equipped with a differential combinator akin to the one of Cartesian Differential Categories (CDC). Like in a CDC, the differential combinator of a GCDC associates to each morphism from A to B a directional derivative ; except this time, the directional derivative goes from the product of A with its linearisation L(A) to the linearisation L(B) of B.
While the archetypal example of a GCDC is given by smooth maps between open subsets of Euclidian spaces, one may ask for algebraic examples of such structures.
In this talk, we will show that any category equipped with an appropriate notion of abelianisation functor is also equipped with a GCDC structure, where the abelianisation plays the role of the linearisation. In increasing order of generality, this comprises the category of groups, any semi-abelian category, and any category containing a semi-additive, reflective subcategory.
This is part of ongoing joint work with JS Lemay and Tim Van der Linden.

(37) César Bardomiano (Ottawa, Canada)Slides

    Title: The language of a model category
    Abstract: Quillen model categories are a cornerstone for modern homotopy theory. These categories, originally devised to capture homotopical properties of categories like topological spaces, simplicial sets or chain complexes, have gained relevance for giving a way to construe higher categories which are of great importance, for example, in algebraic topology and geometry. In this talk, we will see that model categories also have logical information on their own in the following sense: Given any model category, we can associate to it a class of first-order formulas referring to the fibrant objects of the category. For example, the associated language of the category of small categories, equipped with its canonical model structure, coincides with language for categories defined by Blanc [1] and Freyd [2], whose central feature is that it respects the equivalence principle. Similarly, the language we associate to a model category respects the appropriate version of the equivalence principle: two homotopically equivalent objects satisfy the same formulas and replacing parameters by homotopically equivalent ones does not change the validity of a formula. Finally, we will show that for M and N two Quillen equivalent model categories, their associated languages are, suitably, equivalent.

(38) Rachel Hardeman Morill (Calgary, Canada)

    Title: Path Object Categories and Graphs
    Abstract: In classical homotopy theory, two spaces are considered homotopy equivalent if one space can be continuously deformed into the other. This theory, however, does not respect the discrete structure of graphs with their vertices and edges. For this reason, a discrete homotopy theory for graphs is needed. A path object category is a structure associated to Moore paths and a natural starting place for defining a homotopy theory. In this talk, I will discuss what a path category is, how path categories can be used to define a discrete homotopy theory for graphs, and what kind of structure a path category gives. This work was done in collaboration with Laura Scull and Robin Cockett.

(39) Jack Jia (Dalhousie, Canada)Slides

    Title: The Monster Lie Algebra
    Abstract: The Moonshine conjecture, which reveals the unexpected connection between the monster group and the j-invariant, was proven by Borcherds in 1992. A crucial step of the proof is the construction of the monster Lie algebra, which is acted on by the monster group naturally. I will describe Borcherds' construction of this Lie algebra via a quantization functor.

(40) Matthew Alexander (Regina,Canada)

(41) Manak Singh (Regina, Canada)

   Title: Homotopy operations on simplicial commutative algebras
    Abstract: Homotopy operations are natural transformations between homotopy functors. On simplicial commutative algebras over finite fields, these operations form an algebra where the multiplication is defined by composition laws. For the field with char of 2, the algebra obtained is a Koszul algebra. For fields with an odd prime char, it is yet to be established if their associated algebras are Koszul. I will introduce this problem and talk about why this might be important.

(42) Federica Pasqualone (Carnegie Mellon, USA)Slides

    Title: Differential Calculus on Prefactorization Algebras: Starter Pack
    Abstract: This talk will provide an introduction to prefactorization algebras, by investigating their categorical and physical foundations. In short, PFAs are tools for modelling observables of a quantum field theory.
The first part of the talk contains a general overview of the underlying logical structure, in the second part the focus will shift to the actual calculations we are able to perform in this framework, including their relation with observables and other formalisms.

(43) Karen Little (Calgary, Canada) Slides

   Title: Awesome Mathematical Knitting Constructions
    Abstract: This talk will explore physical constructions of two dimensional manifolds (Torus, Möbius, Klein bottle) created through knitting by using their inherent properties.
Although no physical knitting experience is required or provided, I will supplement the talk with a selection of knitted projects as examples of each concept.


Last Update: 2024-04-29. For updates or additions to this page, please send a note to Robin Cockett.

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