The stable homotopy hypothesis predicts that "stable" n-groupoids should model stable homotopy n-types. Stability on the categorical side is usually interpreted as a symmetric monoidal structure with invertible objects, while on the topological side these are spectra (instead of spaces) with homotopy groups concentrated between dimensions 0 and n. Proving this hypothesis is the first step in comparing spectra and higher categories rather than the goal. I will discuss a very special case of going further than the hypothesis when n=1 and 2 (the dimensions in which the stable homotopy hypothesis is a theorem rather than just conjecture), namely what categorical objects correspond to the truncated sphere spectrum. We will recover some classical algebra for n=1, and a complicated generalization of that algebra for n=2 that I don't think we really understand yet.

I will talk about some recent developments in the framework of "black box causal reasoning". In this minimal setting, we assume access to some abstract process and attempt to describe, quantify, or prove properties about the causal relationships between its inputs and outputs. This works both for first-order processes, which can capture e.g. a device shared by multiple agents, or higher-order processes, which can capture the universe in which those agents live. This higher-order picture leads naturally to the structure of a *-autonomous category. Whereas first order processes (e.g. quantum gates) only have two natural notions of composition (in series and in parallel), higher-order processes have a rich and nuanced notion of composition that must avoid inconsistency in the composition of causal paths. I will show how provability in the internal logic of a *-autonomous category gives sufficient conditions for causal consistency, then discuss several avenues of extension toward a complete characterisation

The ménagerie of categorical models of dynamical systems is becoming a veritable zoo, but what makes all these animals tick? In this talk, I will introduce a new specimen: a symmetric monoidal category of continuous-time open Markov processes with general state spaces. I will explain how this category is obtained from a category of "continuous-time coalgebras" opindexed by polynomials, and describe how this recipe also gives categories of nondeterministic systems in arbitrary (continuous) time. These new specimens are motivated by the cybernetic question of how to model systems that are continuously performing approximate Bayesian inference. I will therefore sketch why their better-known cousins weren't quite up to the job, and show that our new SMC admits Bayesian inversion. Finally, I will attempt to make contact with the the "open games" branch of categorical cybernetics, asking what makes the shapes of our structures seem so similar-but-different, and how we might begin to understand systems nested within systems.

Suppose that we are given a collection of endofunctors (e.g., loop space functors) on a pointed homotopy theory $A$. We will call such a homotopy theory stable if it is stable under these functors; that is, each functor is an auto-equivaence. Alternatively, one can consider this as an action of a monoidal category, and in this case stable means the monoidal category acts by auto-equivalences. In this talk, we discuss actions of monoidal categories on relative categories, and applications in stable homotopy theory. Given a monoidal category $I$ and an $I$-relative category $A$ (that is, a relative category with an $I$-action), the (co)stabilization of $A$ is an $I$-relative category that is universal with respect to the property that every object of $I$ acts by auto-equivalences (on homotopy category). We introduce a notion of $I$-equivariance for functors between $I$-relative categories and give constructions of stabilization and costabilization in terms of (weak) ends and coends in a $2$-category of $I$-relative categories and $I$-equivariant relative functors. Several examples existing in the literature, including various categories of spectra and cohomology theories with exotic gradings, can be seen as particular instances of this setting after fixing $A$ and the $I$-action on it. In particular, categories of sequential spectra, coordinate free spectra, genuine equivariant spectra, genuine parameterized spectra (indexed by vector bundles), and cohomology theories with various exotic gradings can be obtained in terms of weak ends. On the other hand, the costabilization of a relative category with respect to an action gives a stable relative category akin to a version of the Spanier-Whitehead category. This is a joint work with Özgün Ünlü.

Lawvere's Elementary Theory of the Category of Sets (ETCS) was conceived as an alternative to ZFC that represents more accurately how mathematicians actually do mathematics. But can ETCS do everything that ZFC can? I will present some evidence that yes, it can. Specifically, I will sketch how the beginning of the theory of large cardinals looks in ETCS, describing both the similarities and the differences between the two approaches. No prior familiarity with ETCS will be assumed.

The Bogomolov multiplier of a group is the subgroup of its Schur multiplier, of classes with vanishing restrictions to all abelian subgroups. Bogomolov multiplier plays an important role in birational algebraic geometry (Noether problem) and in topology (groups of singular submanifolds).

In the talk explain how Bogomolov multiplier appears in the study of soft symmetries of modular tensor categories. Here soft means that a symmetry does not move objects. More precisely, I will interpret Bogomolov multiplier as a normal subgroup of the group of soft autoequivalences of the Drinfeld double.

We give an elementary introduction to some ideas in double category theory by concentrating on one particular example, the double category of rings with homomorphisms, and bimodules. Along the way we discover a number of “new” morphisms of rings, interesting in their own right. We also look at interesting double adjunctions in this context.

No previous knowledge of double categories is assumed, just some familiarity with categories, rings and modules.

One can imagine a database schema as a category and an instance or state of that database as a functor I: C-->Set; the category of these is denoted C-Set. One can think of a data-migration functor, a way of moving data between schemas C and D, as a parametric right adjoint C-Set --> D-Set. In database speak, these are D-indexed "unions of conjunctive queries".

Scene change. The usual semiring of polynomials in one variable with cardinal coefficients, polynomials such as p = y^3 + 3y + 2, can be categorified to Poly, the category of polynomial functors, where + and x are the categorical coproduct and product. Composition of polynomials (p o q) gives a monoidal operation on this category for which the identity polynomial, y, is the unit. Ahman and Uustalu showed in 2016 that, up to isomorphism, the comonoids in (Poly, o, y) are precisely categories (!), and Garner sketched a proof in a recent video that bimodules between polynomial comonoids are parametric right adjoints between copresheaf categories. Recall that these are precisely the data-migration functors described above. In the talk, I will describe this circle of ideas.

I propose that in 2021 a great transition is upon us; distances that were measured in days are now measured in zoom-hiccups. The speed of data migration—if that's indeed a valid way to model it—is much faster than ever, driving dominance into the hands of those who move data: roughly speaking, computerized processes. Researchers use biomimicry to formalize as many aspects of human intelligence as they can, much of which is then installed as automated software systems that run constantly. I call the automated speed-up of bio-inspired intellectual processes AI, and I'm not judging it as good or bad, but I do consider it immensely important. I propose that we as mathematicians have the ability to shape the course of AI. Mathematics becomes technology, and I believe we'll fare better if that technology is based on elegant principles rather than made ad-hoc. Polynomial functors are my entry point, and this talk can serve as an invitation to others to join in whatever capacity appeals to them. To respect the standards of academic talks, I will mainly restrict my discussion to mathematics and its applications, rather than to speculation.

Partial evaluations are a way to encode, in terms of monads, operations which have been computed only partially. For example, the sum "1+2+3+4" can be evaluated to "10", but also partially evaluated to "3+7", or to "6+4". Such structures can be defined for arbitrary algebras over arbitrary monads, and even 2-monads, and can be considered the 1-skeleton of a simplicial object called the bar construction. The higher simplices of the bar construction can be interpreted as ways to compose partial evaluations. Recent research has shown that, while for cartesian monads partial evaluations form a category, for weakly cartesian monads the compositional structure is more complex, and in particular it does not in general form any of the standard higher-categorical structures. Moreover, partial evaluations return known concepts of "partially evaluated operations" in the following contexts: - For the free cocompletion monad, where the operation is taking the colimit, partial evaluations correspond to left Kan extensions; - For probability monads, where the operation is taking the expected value, partial evaluations correspond to conditional expectations.

The research presented in this talk has been carried out jointly with Carmen Constantin, Tobias Fritz, Brandon Shapiro, and Walter Tholen.

The talk will be in English, but you are welcome to ask questions in Spanish if anything is not clear.

The following are references for the topics discussed in the talk:

arxiv.org/abs/1810.06037

arxiv.org/abs/2009.07302

arxiv.org/abs/2101.04531

The foundations of probability, statistics, and information theory are slowly undergoing a potentially dramatic change of perspective through the language of category theory via string diagrams [3]. This point of view has been abstracted to finite-dimensional C*-algebras via a stochastic variant of the Gelfand—Naimark theorem and quantum Markov categories. Through this abstraction, one can immediately analyze concepts such as Bayesian inversion in non-classical settings, which in fact has recently been done in finite dimensions [2]. What can be said for more general (possibly infinite dimensional) C*-algebras? In this talk, I will review some background on Markov categories, provide some motivation for their study, introduce our main quantum example, and then I'll provide a small refresher on C*-tensor products. Then, I'll explain why all C*-algebras do not form a quantum Markov category, and I will provide some suggestions for an alternative framework [1].

Main reference:

[1] arxiv.org/abs/2001.08375 (especially Remark 3.12 and Question 3.25).

Additional references of potential interest:

[2] arxiv.org/abs/2005.03886 (on Bayesian inversion in quantum mechanics)

[3] arxiv.org/abs/1908.07021 (on classical Markov categories)

It is an old observation of Kelly and Street that the "calculus of mates" for adjunctions is naturally expressed using a double category of functors and adjunctions. This was generalized by Cheng, Gurski, and Riehl to mates for multivariable adjunctions, such as the tensor-hom adjunction of a closed monoidal category or the tensor-hom-cotensor adjunction of an enriched category, using a cyclic multi double category. I will explain how the latter is more naturally viewed as a poly double category (that is, an internal category in polycategories), and how it arises as a special case of a "double Chu construction".

A subfactor is a unital inclusion of simple von Neumann algebras/factors $A\subset B,$ and we study it via its standard invariant $\cC,$ which corresponds to a unitary tensor category (UTC). We will review some subfactor reconstruction techniques by Popa, and Guionnet-Jones-Shlyakhtenko (GJS), highlighting that subfactors have quantum symmetries which are encoded by UTC-actions. Namely, we reinterpret the inclusion $A\subset B$ as encoding an action of its standard invariant $\cC$ on $A,$ and reconstruct the overfactor $B$ as a generalized crossed-product by this UTC-action.

Large scale work of many researchers worldwide has recently culminated in the classification of C*-algebras, which is now at the level of Connes' classification of injective factors. Nowadays, C*-algebras is at a similar state to that of von Neumann algebras after Jones introduced the index for subfactors in the early 80s. Thereafter, great interest has arisen in constructing and classifying UTC-actions on C*-algebras, aiming to understand their structure from the viewpoint of quantum symmetries.

We will see that every UTC $\cC$ acts on some simple, unital separable and monotracial C*-algebra constructed only from $\cC$ by adapting diagrammatic and free probabilistic techniques from GJS. Using a 'Hilbertification' technique, we recover the UTC-action constructed by Brothier-Harglass-Penneys of $\cC$ on the free group factor $L\mathbb{F}_\infty.$ This is joint work with Hartglass. Finally, we will review some recent developments and obstructions to the existence of UTC actions on (classifiable) C*-algebras.

The reflection equation, an analogue of the Yang-Baxter equation, appeared from the boundary quantum field theory due to Cherednik in the 80's. It came to known to have strong connection to quantization of homogeneous spaces, and in particular symmetric spaces through Tannaka-Krein type duality. In this talk I will review the basic categorical paradigm behind this correspondence, and an analogue of Kohno-Drinfeld rigidity theorem that gives a classification of monoidal categorical structure (ribbon braided module category) quantizing compact symmetric spaces in terms of eigenvalues of solution to reflection equation. Based on joint works with Kenny De Commer, Sergey Neshveyev, and Lars Tuset.

We often understand our world piece by piece, weaving local models and perspectives into a bigger picture. In any category, amalgamating parts into a single object can be captured through the notion of colimit. Cospans provide a convenient syntax for performing this weaving, building colimits piece by piece. Through the use of decorated and structured cospans, this approach can extend to categories where colimits may be complicated to compute, or may not even exist at all. In this talk, I'll provide a perspective on both the power and limits of cospans as a tool for composition, meditating on why, how, and when cospans can contribute.

We consider the notions of Fibration of categories, (pseudo)Filtered category, and the axioms for a category of Fractions. A basic fact involving them is: given a Fibration, if the arrows of the base category are (pseudo)coFiltered, then the cartesian arrows satisfy Fractions. This is a Proposition in SGA 4 (Exp. VI, Prop. 6.4) whose proof is left to the reader as an exercise, and I want to start this talk by solving this exercise. Let me tell you why.

Each of the three "F" notions above has been considered for bicategories, or at least for 2-categories. I will start with what may be the easiest one to understand, that of Filtered: in a Filtered bicategory, in addition to asking for cones for two objects and for two parallel arrows, we add a third axiom asking for cones for parallel 2-cells. I will present the definitions of Filtered and pseudoFiltered bicategory, a set of axioms for a bicategory of Fractions, and some properties of Fibrations of bicategories that all fit this same pattern. We arrived at these notions when proving a "bicategory version" of the Proposition in SGA 4, in fact a small generalization that I will present.

This result is part of an ongoing collaboration with P. Bustillo and D. Pronk, we're working on showing some basic properties of the bicategorical localization by fractions which are known in dimension 1. If time permits, I hope to mention how we ended up here within our current work and how this result can be applied here.

En los últimos años, la teoría de funtores en biconjuntos ha demostrado ser una herramienta muy útil para abordar problemas relacionados con grupos finitos y sus representaciones. En esta plática comenzaré por la definición de funtor en biconjuntos y la motivación que llevó a esta definición. Después, presentaré la completación aditiva de la categoría de biconjuntos, veremos que los objetos en esta categoría pueden escribirse como fracciones y que, de hecho, en muchos sentidos se comportan como tales. Veremos también que esta categoría es aditiva, monoidal simétrica, autodual y con una descomposición de tipo Krull-Schmidt para los objetos. Si el tiempo lo permite, veremos su relación con la categoría de Burnside, introducida por Lindner en 1976.

Composition in ordinary 1-categories is strictly unital, and it's known that every weak 2-category is equivalent to a 2-category with strict units, a useful theorem that makes 2-categories easier to work with. But what does it mean for an n-category, or even an infinity-category, to have strict units? In this talk I give an accessible introduction to infinity categories, and use lots of examples to illustrate the theory of strict units for infinity-categories. This is joint work with Eric Finster and David Reutter (arXiv:2007.08307).

The homotopy hypothesis is a well-known bridge between topology and category theory. Its most general formulation, due to Grothendieck, asserts that topological spaces should be "the same" as infinity-groupoids. In the stable version of the homotopy hypothesis, topological spaces are replaced with spectra.

In this talk we will review the classical homotopy hypothesis, and then focus on the stable version. After discussing what the stable homotopy hypothesis should look like on the categorical side, we will use the Tamsamani model of higher categories to provide a proof. This is based on joint work with Moser, Ozornova, Paoli and Verdugo.

Multiplicative gerbes can be understood as monoid objects on the 2-category of gerbes. We take this point of

view on the 2-category of topological gerbes in order to define appropriate representations of these multiplicative gerbes.

We take an explicit model for topological gerbes using Graeme Segal's cohomology of topological groups and we show that with this model

we may replicate several constructions done over multiplicative gerbes over finite groups.

Initially, the centre of a monoidal category was used independently by V. Drinfeld in connection with Hopf algebras and by A. Joyal and the speaker to study the category of framed tangles. We were influenced by lectures of Yu. Manin in Montréal and the work of D. Yetter and V. Turaev. In this talk I will give a gentle introduction to how the construction can be viewed in various ways. In particular, it is a limit construction which can be performed in other places besides the cartesian monoidal 2-category of categories. Some examples from different fields will be provided.

One beautiful result one learns early when studying the functorial approach to TQFTs is that in two dimensions, such theories are the same as commutative Frobenius algebras. One furthermore learns that in functorial TQFTs, state spaces are necessarily finite-dimensional. There are several ways to overcome this restriction, and in two dimensions, one possibility is to equip the surfaces with an area. The most famous example of such a QFT is 2d Yang Mills theory for a compact gauge group. Surprisingly, one finds that a very similar classification to the 2d TQFT still holds. Time permitting, I will also discuss defects in 2d area dependent QFTs, of which Wilson line observables in 2d Yang Mills are an example. This is joint work with Lorant Szegedy.

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). This affects the way that transitions between charts need to be described, and it is generally rather cumbersome to work with atlases. It has been shown in [Moerdijk-P] that one can represent orbifolds by groupoids internal to the category of manifolds, with etale structure maps and a proper diagonal, I.e., combined source-target map (s,t): G_1 -> G_0 x G_0. We have since generalized this notion further to orbispaces, represented by proper etale groupoids in the category of Hausdorff spaces. Two of these groupoids represent the same orbispace if they are Morita equivalent. However, Morita equivalences are generally not pseudo-invertible in this 2-category, so we consider the bicategory of fractions with respect to Morita equivalences.

For a pair of paracompact locally compact orbigroupoids G and H, with G orbit-compact, we want to study the mapping groupoid [G, H] of arrows and 2-cells in the bicategory of fractions. The question we want to address is how to define a topology on these mapping groupoids to obtain mapping objects for the bicategory of orbispaces. This question was addressed in [Chen], but not in terms of orbigroupoids, and with only partial answers.

We will present the following results:

1. When the orbifold G is compact, we define a topology on [G,H] to obtain a topological groupoid OMap(G, H), which is Morita equivalent to an orbigroupoid. To obtain a Morita equivalent orbigroupoid, we need to restrict ourselves to so-called admissible maps to form AMap(G,H), and

Orbispaces(K × G, H) is equivalent to Orbispaces(K, AMap(G, H)).

So AMap(G,H) is an exponential object in the bicategory of orbispaces.

2. We will also show that AMap(G,H) thus defined provides the bicategory of orbit-compact orbispaces with bicategorical enrichment over the bicategory of orbispaces: composition can be given as a generalized map (an arrow in the bicategory of fractions) of orbispaces.

In this talk I will discuss how this work extends the work done by Chen and I will show several examples. This is joint work with Laura Scull.

[Chen] Weimin Chen, On a notion of maps between orbifolds I: function spaces, Communications in Contemporary Mathematics 8 (2006), pp. 569-620.

[Moerdijk-P] I. Moerdijk, D.A. Pronk, Orbifolds, sheaves and groupoids, K-Theory 12 (1997), pp. 3-21.

Some ideas from quantum theory are beginning to percolate back to classical probability theory. For example, the master equation for a chemical reaction network - also known as a stochastic Petri net - describes particle interactions in a stochastic rather than quantum way. If we look at this equation from the perspective of quantum theory, this formalism turns out to involve creation and annihilation operators, coherent states and other well-known ideas — but with a few big differences.

Networks can be combined in various ways, such as overlaying one on top of another or setting two side by side. We introduce `network models' to encode these ways of combining networks. Different network models describe different kinds of networks. We show that each network model gives rise to an operad, whose operations are ways of assembling a network of the given kind from smaller parts. Such operads, and their algebras, can serve as tools for designing networks. Technically, a network model is a lax symmetric monoidal functor from the free symmetric monoidal category on some set to Cat, and the construction of the corresponding operad proceeds via a symmetric monoidal version of the Grothendieck construction.

In this talk we will discuss Petri nets from a categorical perspective. A Petri net freely generates a symmetric monoidal category whose morphisms represent its executions. We will discuss how to make Petri nets "open" i.e. equip them with input and output boundaries where resources can flow in and out. Open Petri nets freely generate open symmetric monoidal categories: symmetric monoidal categories which can be glued together along a shared boundary. The mapping from open Petri nets to their open symmetric monoidal categories is functorial and this gives a compositional framework for reasoning about the executions of Petri nets.

Sea C una categoría esférica abeliana. Denote por N_C la subcategoría plena consistente de objetos despreciables, es decir, objetos tales que la traza de cualquiera de sus endomorfismos se anula. En esta charla, propondremos una definición en nivel derivado de categoría y anillo de fusión para la categoría C. Como ejemplo, mostraremos que esta definición produce los mismos anillos de fusión que se obtienen de las categorías de representaciones de los grupos cuánticos grande y pequeño.

En esta platica se expondrán algunas ideas acerca del papel que podría jugar la noción de entropia de entrelazamiento, proveniente de la teoría cuántica en física, en el estudio de ciertas categorías tensoriales. En particular, veremos cómo ayuda este enfoque a entender la estructura de la categoría de representaciones de un grupoide de Lie propio.

Markov categories have recently gained prominence as a categorical approach to probability and statistics. In this talk, I will argue that Markov categories provide a very general theory of information flow, and that this theory generalizes probability theory in a manner analogous to how topos theory generalizes set theory.

In the first part, I will sketch some theorems of probability and statistics which have already been developed synthetically in terms of Markov categories, including a version of the Blackwell-Sherman-Stein theorem which seems to be new even when instantiated in the traditional measure-theoretic framework. In the second part, I will sketch the vast and largely unexplored landscape of Markov categories on which these synthetic results can be instantiated. Some basic knowledge of monoidal categories and discrete probability should be enough to follow the talk.

Many operator algebras describe "noncommutative spaces" admitting intrinsic symmetries that cannot be described in a classical way, through a group action. However, many of these "noncommutative symmetries" can be understood in a broader spectrum via the use of 2-groups or even 2-groupoids. While a groupoid can be seen as a category where all morphisms are invertible, a 2-groupoid is nothing but a 2-category (or bicategory) where all morphisms and 2-morphisms are invertible. A 2-group is just a 2-groupoid with a single object. To understand how these act on objects of other bicategories, like C*-algebras, we use the theory of bicategories and look at weak functors from a 2-groupoid to certain bicategories of C*-algebras. This gives us a good flexibility and allows to understand several sort of new symmetries and also rediscover known ones from a different point of view.

Open dynamical systems allow scientists to modularly specify complicated models as the composition of primitive systems. For example, an ecologist may model a complicated ecosystem as a composition of simple interactions between species. We will implement two open dynamical systems doctrines --- composition as machines and composition as resource sharers --- using the attributed C-Set framework in Catlab. We will showcase several features of attributed C-Sets including functorial data migration and (co)limits of C-Sets in order to streamline this modular modeling framework.

I will present a new formalism for Petri nets based on polynomial-style finite-set configurations and etale maps. The processes of a Petri net P are etale maps G -> P from graphs. The main feature of the formalism is that Petri nets have elements --- they are the elements you see in the pictures. This makes the definition more representable (in the categorical sense of the word) than previous definitions. The main result I want to arrive at is that P-processes form a symmetric monoidal Segal space, which is the free prop-in-groupoids on P, but most of the time will be spent just explaining Petri nets, their markings and firings, the token game, processes, and the basic idea of concurrency and causality --and how these notions look in the new formalism.

Reference: "Elements of Petri nets and processes" [ArXiv:2005.05108]

We will explain studies of 2-dimensional topological order in terms of tensor networks and subfactors arising from commuting squares. Appearance of braiding strucuture from 3-dimensional topological quantum field theory is demonstrated from a viewpoint of tensor categories. We will give higher relative commutants of a subfactor as spaces on which Hamiltonian acts.