# Schedule for: 21w5251 - Tangent Categories and their Applications (Online)

Beginning on Monday, June 14 and ending Friday June 18, 2021

All times in Banff, Alberta time, MDT (UTC-6).

Monday, June 14 | |
---|---|

08:45 - 09:00 |
Introduction and Welcome by BIRS Staff ↓ A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions. (Online) |

09:00 - 09:05 |
Group Photo ↓ Turn on your cameras if you would like to appear in the group photo (Online) |

09:05 - 09:15 |
Welcome and introduction to the workshop ↓ Kristine Bauer, Geoff Cruttwell, and Robin Cockett will welcome you and describe how the workshop is organized. (Online) |

09:15 - 10:00 |
Rick Blute: Syntax and Semantics of Differentiation ↓ This talk will be an introduction to differential linear logic and its associated categorical notion, differential categories. Differential linear logic due to Ehrhard & Regnier, is an extension of linear logic via the addition of an inference rule modelling differentiation. It was inspired by models of linear logic discovered by Ehrhard, where morphisms have a natural smooth structure. A differential category is an additive symmetric monoidal category with a coalgebra modality and a differential combinator, satisfying a number of coherence conditions. In such a category, one should imagine the morphisms in the base category as being linear maps and the morphisms in the coKleisli category as being smooth. We will look at several examples as well as some of the directions that the subject has gone since its inception. (Online) |

10:15 - 11:00 |
Jean-Simon Lemay: The World of Differential Categories: A Tutorial on Cartesian Differential Categories ↓ In this tutorial talk, we will provide an introduction to Cartesian differential categories, as well as discussing examples, the term calculus, and applications. We will also take a look at the geography of the theory of differential categories and discuss the various connections and constructions between each of the stages. (Online) |

11:15 - 12:00 |
Robin Cockett: The Faa Di Bruno Construction and Skew Enrichment ↓ This tutorial will introduce the Faa Di Bruno construction and segue into viewing Cartesian differential categories as skew enriched (following Garner and Lemay) . The aim is to explain the first embedding theorem of Cartesian Differential Categories into the coKleisli category of a (tensor) differential category. (Online) |

15:00 - 15:05 |
Group Photo ↓ Second group photo for the evening session attendees. Please turn your cameras on if you would like to be in the picture. (Online) |

15:05 - 15:50 |
Geoffrey Cruttwell: Introduction to tangent categories ↓ In this talk I'll introduce the idea of a tangent category, which can be seen as a minimal categorical setting for differential geometry. I'll discuss a variety of examples, and then focus on how analogs of vector spaces and (affine) connections can be defined in any tangent category. Time-permitting, I'll also briefly describe a few other structures that can be defined in a tangent category, including differential forms and (ordinary) differential equations and their solutions. (Online) |

16:00 - 16:45 |
Ben MacAdam: An introduction to differential bundles ↓ This tutorial will show how algebraic structure in tangent categories can capture geometric differential structure by considering the relationship between vector bundles and differential bundles in the category of smooth manifolds. Vector bundles are fibered vector spaces that are also fibre bundles, so they are not essentially algebraic in the sense of Freyd. Differential bundles, however, are coalgebras for the weak comonad induced by the vertical lift on the tangent bundle satisfing a universal property.
We will begin by showing that Cockett and Cruttwell's original characterization of a differential bundle is equivalent to the current definition. Then, we will show that the functor from vector bundles to differential bundles is an isomorphism of categories. (Online) |

17:00 - 17:45 |
Richard Garner: Weil spaces, and the embedding theorem for tangent categories ↓ The purpose of this tutorial is to introduce the enriched perspective on tangent categories: they are precisely categories (with certain colimits) enriched in the cartesian closed category of "Weil
spaces". Here a "Weil space" is more or less what an algebraic geometer would call a "formal deformation problem": a nicely-behaved functor from a category of Weil algebras (= local Artinian algebras) into Sets. We also sketch how the enriched perspective on tangent categories allows us to prove an embedding theorem: every tangent category embeds fully and faithfully into a representable tangent category. (Online) |

Tuesday, June 15 | |
---|---|

09:00 - 09:30 |
Thomas Ehrhard: Differentiation in probabilistic coherence spaces ↓ Probabilistic coherence spaces are a model of classical linear logic but not a model of differential linear logic. Nevertheless differentiation is a perfectly meaningful operation in this model. I will explain its meaning, some of its properties and present a tentative categorical axiomatization of this operation. (Online) |

09:45 - 10:15 |
Michele Pagani: Automatic differentiation in PCF ↓ We study the correctness of automatic differentiation (AD) in the context of a higher-order, Turing-complete language (PCF with real numbers), both in forward and reverse mode. Our main result is that, under mild hypotheses on the primitive functions included in the language, AD is almost everywhere correct, that is, it computes the derivative or gradient of the program under consideration except for a set of Lebesgue measure zero. Stated otherwise, there are inputs on which AD is incorrect, but the probability of randomly choosing one such input is zero. Our result is in fact more precise, in that the set of failure points admits a more explicit description: for example, in case the primitive functions are just constants, addition and multiplication, the set of points where AD fails is contained in a countable union of zero sets of non-null polynomials. (Online) |

10:30 - 11:15 |
Marie Kerjean: From categorical models of differentiation to topologies in vector spaces. ↓ Differential categories have a rich relation with proof theory and linear logic. In this talk, we will focus on models interpreting differential linear logic in topological vector spaces, and specifically for models interpreting the involutive linear negation of classical linear logic.
We will survey the main ingredients that can make a category with smooth functions over topological vector spaces cartesian closed. We also review the main limitations to reaching *-autonomy in topological vector spaces. If time permits, we will explore how chiralities, models of polarized linear logic, are especially appropriate in this context, and facilitate the search for cartesian closedness and *-autonomy. (Online) |

11:30 - 12:00 |
Lionel Vaux: A groupoid of permutation trees (with applications to the Taylor expansion of λ-terms) ↓ We introduce a groupoid of trees whose objects are (labelled, planar, rooted) trees, and whose morphisms are trees with permutations attached to internal nodes: we obtain a morphism from T to T' exactly when T' is obtained by permuting the subtrees of each node in T inductively, according to permutations given by the morphism. The degree of a tree is then defined as the cardinality of its group of automorphisms.
We are interested in the effect of tree substitution on the degree of trees: tree substitution is a variant of the usual operadic composition of trees, parameterized by a selection of the leaves to be substituted.
This study is motivated by an approach to the Taylor expansion of λ-terms recently developed by Federico Olimpieri and myself. In particular, up to a mild generalisation of the above setting, the coefficient of a resource term occurring in the Taylor expansion of a pure λ-term is exactly the inverse of the degree of its syntactic tree. (Online) |

12:45 - 13:45 |
Gather Town Discussion: An open book on tangent categories ↓ J. Gallagher suggests a discussion of an open book on tangent categories, something like the HoTT book. This discussion is for anyone who wants to collaborate. (Online) |

15:00 - 15:45 |
Rory Lucyshyn-Wright: An introduction to connections in tangent categories ↓ This tutorial will be an introduction to the notion of connection introduced in [1] in the setting of tangent categories and its equivalent characterizations in [2]. Building on two formulations of connections on vector bundles that are due to Ehresmann and to Patterson [3], respectively, Cockett and Cruttwell [1] defined a notion of connection in the abstract setting of tangent categories. Equivalent definitions of connections in tangent categories were developed in [2] using biproducts in the additive category of differential bundles over an object of a tangent category, leading also to an economical definition of connections in tangent categories as vertical connections with the property that a certain cone is a limit cone. In this tutorial, we shall survey these equivalent definitions of connection and some aspects of their equivalence. [1] J. R. B. Cockett and G. S. H. Cruttwell, Connections in tangent categories, Theory Appl. Categ. 32 (2017), 835-888. [2] R. B. B. Lucyshyn-Wright, On the geometric notion of connection and its expression in tangent categories, Theory Appl. Categ. 33 (2018), 832-866. [3] L.-N. Patterson, Connexions and prolongations, Canad. J. Math. 27 (1975), 766-791. |

16:00 - 16:45 |
Richard Garner: The free tangent category on an affine connection ↓ The purpose of this talk is to sketch a construction of the free tangent category containing an object M with a connection on its tangent bundle. It turns out that the maps of this tangent category are completely determined by the calculus of multilinear maps on the tangent bundle; and that this calculus can be encoded by a certain kind of operad, which comes endowed with an operation of covariant derivative O(n)->O(n+1) and constants T in O(2) (torsion) and R in O(3) (curvature), with as axioms the chain rule, the two Bianchi identities and the Ricci identity. Any such operad generates a tangent category; the free such operad generates the free tangent category on an affine connection. This is work-in-progress with Geoff Cruttwell. |

17:00 - 17:45 |
Ben MacAdam: New tangent structures for Lie algebroids and Lie groupoids ↓ The tangent bundle on a smooth manifold is, in a sense, sufficient structure for Lagrangian mechanics. In a famous note from 1901, Poincare reformulated Lagrangian mechanics by replacing the tangent bundle with a Lie algebra acting on a smooth manifold [1, 2]. Poincare's formalism leads to the Euler-Poincare equations, which capture the usual Euler-Lagrange equations as a specific example. In 1996, Weinstein sketched out a general program building on Poincare's ideas to formulate mechanics on Lie groupoids using Lie algebroids [3], which motivates the work of Martinez et al. [4,5], Libermann [6], and the recent thesis by Fusca [7]. In this talk, we will look at Weinstein's program through the lens of tangent categories, using recent advances in involution algebroids. We will use the fact that, in smooth manifolds, Lie algebroids are the same thing as involution algebroids. This means Lie algebroids can be reformulated as a certain category of tangent functors from Weil algebras into smooth manifolds. This tangent structure on the category of Lie algebroids agrees with Martinez's presentation of Lie algebroids as generalized tangent bundles. We propose that tangent categories provide the proper algebraic framework to describe a theory of Lagrangian mechanics that extends to Weinstein's program when using these tangent bundles. This talk draws on joint work with Matthew Burke and Richard Garner. [1] Poincaré H. Sur une forme nouvelle des équations de la mécanique. CR Acad. Sci. 1901;132:369-71. [2] Marle CM. On Henri Poincaré’s note “Sur une forme nouvelle des équations de la Mécanique”. Journal of geometry and symmetry in physics. 2013;29:1-38. [3] Weinstein A. Lagrangian mechanics and groupoids. Fields Institute Proc. AMS. 1996;7:207-31. [4] Martínez E. Lagrangian mechanics on Lie algebroids. Acta Applicandae Mathematica. 2001 Jul;67(3):295-320. [5] de León M, Marrero JC, Martínez E. Lagrangian submanifolds and dynamics on Lie algebroids. Journal of Physics A: Mathematical and General. 2005 Jun 1;38(24):R241. [6] Libermann P. Lie algebroids and mechanics. Archivum mathematicum. 1996;32(3):147-62. [7] Fusca D. A groupoid approach to geometric mechanics (Doctoral dissertation, University of Toronto). |

Wednesday, June 16 | |
---|---|

09:00 - 09:45 |
Tom Goodwillie: Functor calculus ↓ Functor calculus is a way of organizing the interplay between homotopy theory and stable homotopy theory. Its name reflects an analogy with differential calculus. There are derivatives, $n$th derivatives, and Taylor polynomials in functor calculus.
Functors between homotopical categories (categories which, like the category of topological spaces, have a suitable structure for “doing homotopy theory”) can be thought of as resembling smooth maps between manifolds. Homotopical categories that are stable correspond to manifolds that are vector spaces. I will sketch the high points of functor calculus with this geometric analogy in mind.
Until recently the relation with smooth geometry has existed mostly as a suggestive analogy. It is being pursued in detail now by Bauer, Burke, Ching, and others using the framework of tangent structures on categories. (Online) |

10:00 - 10:45 |
Brenda Johnson: An example of a cartesian differential category from functor calculus ↓ In this talk, I will provide an introduction to abelian functor calculus, a version of functor calculus inspired by classical constructions of Dold and Puppe, and of Eilenberg and Mac Lane. I will then explain how the analog of a directional derivative in abelian functor calculus gives rise to the structure of a cartesian differential category for a particular category of functors of abelian categories. (Online) |

11:00 - 11:45 |
Eric Finster: The Nilpotence Tower ↓ Much like the theory of affine schemes and commutative rings, the
theory of (higher) topoi leads a dual life: one algebraic and one
geometric. In the geometric picture, a topos is a kind of
generalized space whose points carry the structure of a category.
Dually, in the algebraic point of view, a topos may be thought of
as the "ring of continuous functions on a generalized space with
values in homotopy types".
In this talk, I will explain the connection between Goodwillie's
calculus of functors and this algebro-geometric picture of the
theory of higher topoi. Specifically, I will describe how one can
view the topos of n-excisive functors as an analog of the
commutative k-algebra k[x]/xⁿ⁺¹, freely generated by a nilpotent
element of order n+1.
More generally, I will show how every left exact localization E → F
of topoi may be extended to a tower of such localizations
E ⋯ → Fₙ → Fₙ₋₁ → ⋯ F₀ = F
which we refer to as the Nilpotence Tower, and whose values at an
object of E may be seen as a generalized version of the Goodwillie
tower of a functor with values in spaces. Under the analogy with
scheme theory described above, this construction corresponds to the
completion of a commutative ring along an ideal, or, geometrically,
to the filtration of the formal neighborhood of a subscheme by it's
n-th order sub-neighborhood. I will also explain how, in addition
to the homotopy calculus, the orthogonal calculus of Michael Weiss
can be seen as an instance of this same construction.
This is joint work with M. Anel, G. Biedermann and A. Joyal. (Online) |

13:00 - 13:45 |
Gather Town Discussion: Differential and/or tangent categories and non-commutative geometry ↓ Jean-Baptiste Vienney suggests a discussion about non-commutative geometry and category theory. This topic is wide open: everything from brainstorming to references to the complete story are welcome. (Online) |

15:00 - 15:45 |
Kristine Bauer: Tangent Infinity Categories ↓ This is joint work with M. Burke and M. Ching. In this talk, I will present the definition of a tangent infinity category as a generalization of Leung's presentation of tangent categories as Weil-modules. A key example of a tangent structure on the infinity category of infinity categories is an extension of Lurie’s tangent bundle functor. We call this the Goodwillie tangent structure, since it encodes the theory of Goodwillie calculus. The differential objects in this tangent infinity category are precisely the stable infinity categories. Following Cockett-Cruttwell these form a cartesian differential category. I will explain that the derivative in this CDC is the same as the BJORT derivative for abelian functor calculus, showing that the Goodwillie tangent structure is an extension of BJORT. (Online) |

16:00 - 16:45 |
Michael Ching: Dual tangent structures for infinity-toposes ↓ I will describe two tangent infinity-categories whose objects are the infinity-toposes: one algebraic and one geometric. The algebraic version is a restriction to infinity-toposes of the Goodwillie tangent structure defined by Bauer, Burke and myself, in which the tangent bundle consists of the stabilizations of slice infinity-toposes. The geometric structure is dual to the algebraic with tangent bundle functor given by an adjoint to that of the Goodwillie structure. There is a useful analogy to tangent structures on the category of commutative rings and its opposite (the category of affine schemes).
The main prerequisites for this talk are some familiarity with infinity-categories and with tangent categories. I will introduce infinity-toposes and explain how the two tangent bundle functors are defined. I will also show that the geometric tangent structure can be viewed as an extension of the Goodwillie structure (as well as its dual) by looking at injective infinity-toposes and their infinity-categories of points. (Online) |

17:00 - 17:45 |
André Joyal: The (higher) topos classifying $\infty$ -connected objects ↓ Joint work with Mathieu Anel, Georg Biedermann and Eric Finster.
I will present an application of Goodwillie’s calculus to higher topos theory. The (higher) topos which classifies $\infty$-connected objects is formally the "dual" of the (higher) logos $S[U_\infty]$ freely generated by an $\infty$-connected object $U_\infty$. The logos $S[U_\infty]$ is a left exact topological localisation of the logos $S[U] = Fun[Fin, S]$ freely generated by an object $U$. We show that a functor
$ Fin \to S$ belongs to $S[U_\infty]$ if and only if it is $\infty$-excisive if and only if it is the right Kan extension of its restriction to the subcategory of finite n-connected spaces $C_n \subset Fin$ for every $n \geq 0$. There is a morphism of logoi from $S[U_\infty]$ to the category of Goodwillie towers of functors $Fin \to S$, but we do not know if it is an equivalence of categories. We also consider the logos $S[U_\infty′ ]$ freely generated by a pointed $\infty$-connected object $U'_\infty$ . (Online) |

Thursday, June 17 | |
---|---|

09:00 - 09:45 |
Jonathan Gallagher: Differential programming, probably ↓ Differential and tangent categories have been applied to providing the
semantics of differential programming languages. As interest in
differential programming langauges continues to grow due to
applications in machine learning, many differential programming
languages are being extended with features for probabilistic
programming and in some cases quantum programming. In this talk, we
will investigate structures on top of differential and tangent
categories that allow modelling probabilistically extended programming
languages. To do this, we will develop some of the basics of
functional analysis and distribution theory in the context of
differential categories. We will also develop different approaches to
encoding probabilistic computations in a differential language. (Online) |

10:00 - 10:45 |
Bruno Gavranovic: (with Paul Wilson) Categorical Foundations of Gradient-Based Learning ↓ We propose a categorical foundation of gradient-based machine learning algorithms in
terms of lenses, parametrised maps, and reverse derivative categories.
This foundation provides a powerful explanatory and unifying framework: it encompasses a variety of gradient
descent algorithms such as ADAM, AdaGrad, and Nesterov momentum,
as well as a variety of loss functions such as as MSE and Softmax cross-entropy, shedding new light on their similarities and differences.
Our approach also generalises beyond neural networks (modelled in categories of smooth maps),
accounting for other structures relevant to gradient-based learning such as boolean circuits.
Finally, we also develop a novel implementation of gradient-based learning in
Python, informed by the principles introduced by our framework. (Online) |

11:00 - 11:45 |
Mario Alvarez-Picallo: Soundness for automatic differentiation via string diagrams ↓ Reverse-mode automatic differentiation, especially in the presence of complex language
features, is notoriously hard to implement correctly, and most implementations focus on
differentiating straight-line imperative first-order code. Generalisations exist, however,
that can tackle more advanced features; for example, the algorithm described by Pearlmutter
and Siskind in their 2008 paper can differentiate (pure) code containing closures.
We show that AD algorithms can benefit enormously from being translated into the language
of string diagrams in two steps: first, we rephrase Pearlmutter and Siskind's algorithm as
a set of rules for transforming hierarchical graphs; rules which can -and indeed have been-
be implemented correctly and efficiently in a non-trivial language. Then, we sketch a proof
of soundness for it by reducing its transformations to the axioms of Cartesian reverse
differential categories, expressed as string diagrams. (Online) |

12:00 - 12:15 | MITACS Presentation (Online) |

13:00 - 13:45 |
Gather Town Discussion (tentative): Generalized Weil modules and degree n functors ↓ K. Bauer suggests this discussion. A persistent question has come up regarding functor calculus and tangent categories. Basically, the question is "Why Weil modules and not e.g. N[x]/x^n". This is a reasonable alternative, and the purpose of this discussion is to explore what might happen if one used these generalized Weil modules as a model for the tangent structure. (Online) |

15:00 - 15:20 |
Dorette Pronk: Exponentials and Enrichment for Orbispaces ↓ Orbifolds are defined like manifolds, by local charts. Where manifold charts are open subsets of Euclidean space, orbifold charts consist of an open subset of Euclidean space with an action by a finite group (thus allowing for local singularities). However, a more useful way to represent them is in terms of proper étale groupoids (which we will call orbispaces) and the maps between them are obtained as a bicategory of fractions of the 2-category of proper étale groupoids with respect to the class of essential equivalences. In recent work with Bustillo and Szyld we have shown that in any bicategory of fractions the hom-categories form a pseudo colimit of the hom categories of the original bicategory.
We will show that this result can be extended to our topological context: for topological groupoids the hom-groupoids can again be topologized and under suitable conditions on the spaces these groupoids form both exponentials and enrichment. We will show that taking the appropriate pseudo colimit of these hom-groupoids within the 2-category of topological groupoids gives us a notion of hom-groupoids for the bicategory of orbispaces. When the domain orbispace is orbit compact, we see show that this groupoid is proper and satisfies the conditions to be an exponential. When we further cut back our morphisms between orbispaces to so-called admissible maps, we obtain a proper étale groupoid that is essentially equivalent to the pseudo colimit and hence is also the exponential. Furthermore, we show that the bicategory of orbit-compact orbispaces is enriched over orbispaces: the composition is given by a map of orbispaces rather than a continuous functor.
This work rephrases the result from [Chen] in terms of groupoid representations for orbifolds and strengthens his result on enrichment: he expressed this in terms of a map between the quotient spaces of the mapping orbispaces, where we are able to give this in terms of a map between the orbispaces.
I will end the talk with several examples of mapping spaces. This is joint work with Laura Scull and started out as a project of the first Women in Topology workshop.
[Chen] Weimin Chen, On a notion of maps between orbifolds I: function spaces, Communications in Contemporary Mathematics 8 (2006), pp. 569-620. (Online) |

15:30 - 15:50 |
Susan Niefield: Linear Bicategories: Quantales and Quantaloids ↓ Linear bicategories were introduced by Cockett, Koslowski and Seely as the
bicategorical version of linearly distributive categories. Such a bicategory B
has two forms of composition related by a linear distribution. In this talk, we
consider locally ordered linear bicategories of the form Q-Rel, i.e., relations
valued in a quantale Q; as well as those B which are Girard bicategories.
The latter provide examples which are not locally ordered; and they have
the same relation to linear bicategories as ∗-autonomous categories have to
linearly distributive categories. Examples include the bicategories Quant and
Qtld, whose 1-cell are bimodules and objects are quantales and quantaloids,
respectively.
This is joint work with Rick Blute. (Online) |

16:00 - 16:20 |
Bryce Clarke: Lenses as algebras for a monad (Cancelled) ↓ Lenses are a family of mathematical structures used in computer science to specify bidirectional transformations between systems. In many instances, lenses can be understood as morphisms equipped with additional algebraic structure, and admit a characterisation as algebras for a monad on a slice category. For example, very well-behaved lenses between sets were shown to be algebras for a monad on Set / B, while c-lenses between categories (better known as split opfibrations) are algebras for a monad on Cat / B. Delta lenses are another kind of lens between categories which generalise both of these previous examples, however they have only been previously characterised as certain algebras for a semi-monad. In this talk, I will improve this result to show that delta lenses also arise as algebras for a monad, and discuss several interesting consequences of this characterisation. (Online) |

16:30 - 16:50 |
Simon Fortier-Garceau: Causality, interventions and counterfactuals in Structural Causal Models ↓ In the field of causal inference, Pearl’s structural causal model (SCM) is the standard model in which interventional and counterfactual reasoning are regarded as essential components of causal reasoning. We will take a brief look at examples of interventions and counterfactuals in SCMs, and then describe a faithful representation of SCMs as presheaves on a graph with matching family operators. (Online) |

17:00 - 17:20 |
Priyaa Srinivasan: Exponential modalities and complementarity ↓ Exponential modality of linear logic has been used as a de facto structure for modelling infinite dimensional spaces. In this talk, I will explain the connection between exponential modalities of linear logic and complementarity observables in quantum mechanics. Two quantum observables A and B are complimentary if measuring one increases the uncertainty regarding the value of the other. In the category of finite dimensional Hilbert spaces and linear maps, complimentary observables are represented by a pair of special commutative dagger Frobenius algebras interacting to produce two Hopf algebras. The connection between complimentary observables and exponential modality can be observed by formulating quantum systems within mixed unitary categories and by characterizing the notion of quantum measurements within this framework. Mixed unitary categories provide a categorical semantics for dagger linear logic. In this talk, I will show that the exponential modalities - ! and ? - of linear logic may be "compacted" into the usual notion of complementary observables in a dagger monoidal category, thereby exhibiting a complementary system as arising via the compaction of distinct systems of arbitrary dimensions. |

Friday, June 18 | |
---|---|

09:00 - 09:20 |
Anders Kock: Barycentric calculus, and the log-exp bijection ↓ In terms of synthetic differential geometry, it makes sense to compare the infinitesimal structure of a space and of its tangent bundle. This hinges of the possibility to form certain affine combinations (barycentic calculus) of the algebra maps from A to B, where A and B are arbitrary commutative rings. (Online) |

09:30 - 09:50 |
Kadri Ilker Berktav: Higher structures in physics ↓ This is a talk on higher structures in geometry and physics. We, indeed, intend to overview the basics of derived algebraic geometry and its essential role in encoding the formal geometric aspects of certain moduli problems in physics. Throughout the talk, we always study objects with higher structures in a functorial perspective, and we shall focus on algebraic local models for those structures. With this spirit, we will investigate higher spaces and structures in a variety of scenarios. In that respect, we shall also mention some of our works in this research direction. (Online) |

10:00 - 10:20 |
Rowan Poklewski-Koziell: Frobenius-Eilenberg-Moore objects in dagger 2-categories ↓ A Frobenius monad on a category is a monad-comonad pair whose multiplication and comultiplication are related via the Frobenius law. Street has given several equivalent definitions of Frobenius monads. In particular, they are those monads induced from ambidextrous adjunctions. On a dagger category, much of this comes for free: every monad on a dagger category is equivalently a comonad, and all adjunctions are ambidextrous. Heunen and Karvonen call a monad on a dagger category which satisfies the Frobenius law a dagger Frobenius monad. They also define the appropriate notion of an algebra for such a monad, and show that it captures quantum measurements and aspects of reversible computing. In this talk, we will show that these definitions are exactly what is needed for a formal theory of dagger Frobenius monads, with the usual elements of Eilenberg-Moore object and completion of a 2-category under such objects having dagger counterparts. This may pave the way for characterisations of categories of Frobenius objects in dagger monoidal categories and generalisations of distributive laws of monads on dagger categories. (Online) |

10:30 - 10:50 |
Tarmo Uustalu: Monad-comonad interaction laws (co)algebraically ↓ I will introduce monad-comonad interaction laws as mathematical objects to describe how an effectful computation (in the sense of functional programming) can run in an environment serving its requests. Such an interaction law is a natural transformation typed
$T X \times DY \to R (X \times Y)$
for $T$, $R$ monads and $D$ a comonad on a cartesian (or symmetric monoidal) category $C$ subject to two equations. I will show what interaction laws amount to in terms of functors between the categories of (co)algebras of these (co)monads. I will explain that interaction laws are measuring morphisms in $[C, C]$ as a duoidal category wrt. Day convolution and composition and show how the (co)algebraic perspective helps describe the Sweedler hom of $T$, $R$, i.e., the universal $D interacting with $T$ $R$-residually.
This is based on joint works with Dylan McDermott, Shin-ya Katsumata, Exequiel Rivas, Niels Voorneveld. (Online) |

11:00 - 11:20 |
Nicolas Blanco: Bifibrations of polycategories and MLL ↓ Polycategories are structures generalising categories and multicategories by letting both the domain and codomain of the morphisms to be lists of objects. This provides an interesting framework to study models of classical multiplicative linear logic. In particular the interpretation of the connectives ise given by objects defined by universal properties in contrast to their interpretation in a *-autonomous category.
In this talk, I will introduce the notion of bifibration of polycategories and I will present how the universal properties of the connectives can be recovered as specific bifibrational properties.
I will illustrate this approach through the examples of finite dimensional Banach spaces and contractive maps. These form a *-autonomous category which structure is given by lifting the compact closed structure of the category of finite dimensional vector spaces. This lifting is made possible by considering the fibrational properties of the forgetful functor between the underlying polycategories. (Online) |

11:30 - 11:50 |
Simona Paoli: Weakly globular double categories and weak units ↓ Weakly globular double categories are a model of weak 2-categories based on the notion of weak globularity, and they are known to be suitably equivalent to Tamsamani 2-categories. Fair 2-categories, introduced by J. Kock, model weak 2-categories with strictly associative compositions and weak unit laws. In this talk I will illustrate how to establish a direct comparison between weakly globular double categories and fair 2-categories and prove they are equivalent after localisation with respect to the 2-equivalences. This comparison sheds new light on weakly globular double categories as encoding a strictly associative, though not strictly unital, composition, as well as the category of weak units via the weak globularity condition.
Reference: S. Paoli, Weakly globular double categories and weak units, arXiv:2008.11180v1 (Online) |

13:00 - 13:20 |
Chad Nester: Concurrent Material Histories ↓ The resource-theoretic interpretation of symmetric monoidal categories allows us to express pieces of material history as morphisms. In this talk we will see how to extend this to capture concurrent interaction.
Specifically, we will see that the resource-theoretic interpretation extends to single object double categories with companion and conjoint structure, and that in this setting material history may be decomposed into interacting concurrent components.
As an example, we will show how transition systems with boundary (spans of reflexive graphs) can be equipped to generate material history in a compositional way as transitions unfold. Some directions for future work will also be proposed. (Online) |

13:30 - 13:50 |
Cole Comfort: A graphical calculus for Lagrangian relations ↓ Symplectic vector spaces are the phase space of linear mechanical systems. The symplectic form describes, for example, the relation between position and momentum as well as current and voltage. The category of linear Lagrangian relations between symplectic vector spaces is a symmetric monoidal subcategory of relations which gives a semantics for the evolution -- and more generally linear constraints on the evolution -- of various physical systems. We give a new presentation of the category of Lagrangian relations over an arbitrary field as a `doubled' category of linear relations. More precisely, we show that it arises as a variation of Selinger's CPM construction applied to linear relations, where the covariant orthogonal complement functor plays of the role of conjugation. Furthermore, for linear relations over prime fields, this corresponds exactly to the CPM construction for a suitable choice of dagger. We can furthermore extend this construction by a single affine shift operator to obtain a category of affine Lagrangian relations. Using this new presentation, we prove the equivalence of the prop of affine Lagrangian relations with the prop of qudit stabilizer theory in odd prime dimensions. We hence obtain a unified graphical language for several disparate process theories, including electrical circuits, Spekkens' toy theory, and odd-prime-dimensional stabilizer quantum circuits. |

14:00 - 14:20 |
Nuiok Dicaire: Localization of monads via subunits ↓ Given a “global” monad, one wishes to obtain “local” monads such that these locally behave like the global monad. In this talk, I will provide an overview of how subunits can be used to provide a notion of localisation on monads. I will start by introducing subunits, a special kind of subobject of the unit in a monoidal category. Afterwards, I will provide two equivalent ways of understanding the localisation of monads. The first involves a strength on subunits, while the second relies on the formal theory of graded monads. I will also explain how to construct one from the other. (Online) |

14:30 - 14:50 |
Jean-Simon Lemay: Linearizing Combinators ↓ Bauer, Johnson, Osborne, Riehl, and Tebbe (BJORT) showed that the Abelian functor calculus provides an example of a Cartesian differential category, where the differential combinator is defined using linearization. From the Cartesian differential category perspective, the BJORT construction is backwards. In any Cartesian differential category it is always possible to define the notion of a linear map and to linearize a map using the differential combinator. BJORT constructed their differential combinator using an already established notion of linear map and linearization.
In this talk, we reverse engineer BJORT's construction by abstracting the notion of linear approximation by introducing linearizing combinators. Every Cartesian differential category comes equipped with a canonical linearizing combinator obtained by differentiation at zero. Conversely, a differential combinator can be constructed à la BJORT from a system of linearizing combinators in context. Therefore, linearizing combinators provide an equivalent alternative axiomatization of Cartesian differential categories.
This is joint work with Robin Cockett. (Online) |

15:00 - 15:20 |
Sacha Ikonicoff: Divided power algebras with derivation ↓ Classical divided power algebras are commutative associative algebras endowed with `divided power' monomial operations. They were introduced by Cartan in the 1950's in the study of the homology of Eilenberg-MacLane spaces, and appear in several branches of mathematics, such as crystalline cohomology and deformation theory.
In this talk, we will investigate divided power algebras with derivation, and identify the most natural compatibility relation between a derivation and the divided power operations. The work of Keigher and Pritchard on formal divided power series (also called Hurwitz series) suggests a certain `power rule'. We will prove, using the framework of operads, that this power rule gives a reasonable definition for a divided power algebra with derivation. We will extend this result to a more general notion of divided power algebras, such as restricted Lie algebras, with derivation. (Online) |