May 13: Simon Henry, University of Ottawa
Generalized polygraphs
Polygraphs (also called computads) have been originally introduced for 2-categories by Street, and later generalized to strict infinity categories (Street, Power, Burroni) and to many different other structures by many people. They are the "free models" of some given higher algebraic theories but where "free" is taken in a very specific and iteratively defined way. They generally appear in connection to coherence and strictification problems for these higher structures.
I will present a general notion of "polygraphs for a dependently sorted algebraic theory" that covers all the known examples and captures this general idea of free models. The definition itself is fairly straightforward, but at this level of generality the category of polygraphs we obtain have very poor properties, so the key point is to understand how various assumptions on the theory allows to recover various classically expected or desired properties of these categories of polygraphs - I will in particular start discussing the question of when these are presheaves categories.
The long term goal is to develop a more general understanding of how properties of the category of polygraphs relates to homotopy theoretic properties of the corresponding higher structures through coherence and strictification problems, but there is still a lot of work left to get there.
(Work in progress, joint with Daniel De Almeida Souza)
January 14: Dorette Pronk, Dalhousie University
Generalized Ehresmann Sites and Sites with a Factorization System
Grothendieck topoi are regular categories and hence have an epi-mono factorization system. In recent work, Darien DeWolf and Julia Ramos González and I have shown that this can be translated into the following property of sites: every Grothendieck topos can be described as sheaves on a site with an orthogonal factorization system where all arrows in the left class are covering maps and all arrows in the right class are monics; we call such sites CM-sites. Furthermore, there are enough CM-sites to describe the 2-category of topoi as the bicategory of left fractions of CM-sites with factorization preserving, covering preserving, covering-flat morphisms with respect to the factorization preserving Comparison Lemma maps.
Building on this result, we have extended the results from [1] and [2] and introduced double categorical sites for all topoi, which we call generalized Ehresmann sites. The idea is that the horizontal arrows in these sites are all coverings and the vertical part of the site is posetal and has assigned covering families of vertical arrows.
When we take all functors that preserve the factorization systems between the CM-Sites and all double functors between the generalized Ehresmann sites, we obtain a 2-adjunction between the resulting 2-categories. This adjunction becomes a bi-equivalence if we restrict ourselves to the generalized Ehresmann sites where each vertical component has a maximal object.
This provides the context to define maps of generalized Ehresmann sites as those corresponding to the covering preserving and covering flat morphisms between CM-sites. The notion of being covering preserving is easy to translate, but the notion of covering flatness leads us to an interesting question: in the definition of covering flatness for Grothendieck sites with a factorization system, can we restrict our attention to diagrams indexed by a category that has a factorization system? To answer this question in the affirmative, we will revisit an older result on categories with factorization systems, presented in [4] and [5]. The free category with an SFS and an OFS for a given category is its arrow category, and its universal property will allow us to transfer covering flatness from arrows between Grothendieck sites to the induced arrows between generalized Ehresmann sites.
[1] D. DeWolf, and D. Pronk, A double categorical view on representations of étendues, Cah. Topol. Géom. Différ. Catég. 61 (2020), no. 1, 3--56.
[2] M.V. Lawson, and B. Steinberg, Ordered Groupoids and Étendues, Cah. Topol. Géom. Différ. Catég. 45 (2004), no. 2, 82--108.
[3] Miloslav Štěpán, Factorization systems and double categories, Theory and Applications of Categories, 41 (2024), No. 18, 551--592.
[4] M. Korostenski, and W. Tholen, Factorization systems as Eilenberg-Moore algebras, JPAA 85 (1993) no. 1, 57–72.
[5] M. Grandis, On the monad of proper factorisation systems in categories, JPAA 171 (2002), no. 1, 17–26.
January 28: Theo Johnson-Freyd, Dalhousie University
How to build a Hopf algebra.
Let $(\mathcal{C},1_{\mathcal{C}})$ be a pointed $(\infty,2)$-category. Every left-adjunctible morphism $X\to 1_{\mathcal{C}}$ supplies an algebra object in $\Omega\mathcal{C}:=\text{End}{\mathcal{C}}(1{\mathcal{C}})$. I will explain a "square" of this construction. Suppose that $(\mathcal{C},1_{\mathcal{C}})$ is a pointed $(\infty,3)$-category. Formal nonsense of lax products produces a bialgebra in $\Omega^2\mathcal{C}$ from a retract $X \rightleftarrows 1_{\mathcal{C}}$ with certain adjunctibility conditions. Surprisingly, this bialgebra is always a Hopf algebra. More surprisingly, it can be a Hopf algebra with noninvertible antipode. This is joint work with David Reutter.
February 4: Thomas Blom, Max Planck Institute for Mathematics
On the straightening of every functor
In this talk, I will explain a straightening equivalence for arbitrary functors between ∞-categories. Explicitly, this result states that for a given ∞-category C, there is a natural equivalence between the ∞-category of ∞-categories over C and the ∞-category of unital lax functors from C to a certain double ∞-category of correspondences. I will also describe how one can deduce various classical straightening equivalences for ∞-categories, such as those of Lurie and Ayala–Francis, from our result.
February 11: Max Demirdilek, University of Hamburg
Surface diagrams for Grothendieck-Verdier categories
A Grothendieck-Verdier category is a monoidal category with a duality structure that is more general than rigid duality. Motivating examples include finite-dimensional bimodules over finite-dimensional algebras and modules over vertex operator algebras.
In this talk, I will explain how to extend Joyal and Street's two-dimensional calculus for rigid monoidal categories to a three-dimensional calculus for Grothendieck-Verdier categories. Using this calculus, I will study Frobenius algebras in Grothendieck-Verdier categories and present a lifting theorem, which implies that the category of finite-dimensional modules over a full Hopf algebroid carries a Grothendieck-Verdier structure. I will also discuss Frobenius-Schur indicators for pivotal Grothendieck-Verdier categories if time permits.
Throughout the talk, I will share 3D printouts of surface diagrams created using the graphical proof assistant homotopy.io.
February 25: Lukas Mueller, Perimeter Institute for Theoretical Physics
Deformation Quantization via Categorical Factorization Homology
Factorization homology `integrates' (higher) categorical structures, such as representation categories of Hopf algebras, VOAs, or quantum groups, over manifolds. In my talk, I will discuss an approach for constructing local-to-global deformation quantizations of symplectic manifolds, such as moduli spaces of flat bundles (character varieties), based on factorization homology. In this approach, local quantizations are governed by $E_2$-deformations of symmetric monoidal categories, which, when integrated over 2-dimensional manifolds, lead to Poisson structures and deformation quantizations. Applying the general framework to the Drinfeld category reproduces deformations previously introduced by Li-Bland and Ševera. As a direct consequence, we can conclude a precise relation between their quantization and those introduced by Alekseev, Grosse, and Schomerus. No prior knowledge of factorization homology will be assumed. The talk is based on joint work with Eilind Karlsson, Corina Keller, and Ján Pulmann.
March 4: Scott Wesley, Dalhousie University
How to Build a Parameterized Map
Parameterized functions are ubiquitous in mathematics and physics. For example, the time evolution of a quantum mechanical system is a family of unitary functions parameterized by a time variable. Of course, this parameterization is continuous in the appropriate sense. Upon careful examination, it can be shown that these parameterized functions form a monoidal category, and that continuous reparameterizations induce endofunctors on this category. In this talk, we generalize this construction to show how to construct parameterized maps in arbitrary $\mathcal{V}$-enriched categories. The parameterized maps inherit the morphism structure of $\mathcal{V}$. We show that in order to recover all of the nice properties enjoyed by unitary time evolutions, $\mathcal{V}$ must be Cartesian monoidal.
March 11: Daniel Teixeira, Dalhousie University
Monoidal objects via fibrations
The bar construction relates monoids in a monoidal category $\mathcal{V}$ to simplicial objects $M\colon\Delta^\text{op}\to\mathcal{V}$ satisfying certain properties. However, when $\mathcal{V}$ is a higher category, one must beg $M$ for coherence, and this usually gets nasty. A solution in higher category theory is to turn $M$ into a fibration via some Grothendieck construction, turning coherence (data) into the (property of) existence of Cartesian lifts. This strategy recovers classical monoids as discrete fibrations, monoidal categories as Grothendieck fibrations, and monoidal ∞-categories as Cartesian fibrations. We will show that, similarly, monoidal bicategories correspond to certain Buckley fibrations over $\Delta^\text{op}$ (seen as a discrete bicategory) by unstraightening trifunctors to the tricategory $\text{Bicat}_\text{ps}$. With this, we will spell out a definition to the underlying (symmetric) monoidal bicategory to a (symmetric) monoidal (∞,2)-category. We will also explain a change of base result for discrete internal fibrations, which runs into complications when dealing with weak enrichment.
March 18: Sarah Meng Li, University of Amsterdam
A Complete and Natural Rule Set for Multi-Qutrit Clifford Circuits
We present a complete set of rewrite rules for n-qutrit Clifford circuits. This is the first completeness result for any fragment of quantum circuits in odd prime dimensions. We first generalize Selinger’s normal form for n-qubit Clifford circuits to the qutrit setting. Then, we present a rewrite system by which any Clifford circuit can be reduced to this normal form. We then simplify the rewrites in this procedure to a small natural set of rules, giving a clean presentation of the group of qutrit Clifford unitaries in terms of generators and relations.
Joint-work with: Michele Mosca, Neil J. Ross, John van de Wetering, and Yuming Zhao
March 25: Paula Verdugo, Max Planck Institute for Mathematics
A new strategy to construct model structures
Model structures have proven to be a powerful framework to do homotopy theory in a variety of contexts. It is often hard to verify by hand that given data does define a model structure, which has resulted in different strategies to build model structures or prove their existence. In this talk, we present a new strategy that focuses on knowing the desired cofibrations and desired fibrant objects together with classes of weak equivalences and fibrations between these. If time allows, we will present different examples of this, especially in the category of double categories and double functors. This is based on joint work with Léonard Guetta, Lyne Moser, and Maru Sarazola.
April 1: Lory Aintablian, Max Planck Institute for Mathematics
Differentiation of groupoid objects in tangent categories
The infinitesimal counterpart of a Lie group(oid) is its Lie algebra(oid). I will show that the differentiation procedure works in any category with an abstract tangent structure in the sense of Rosicky, which was later rediscovered by Cockett and Cruttwell. Mainly, I will construct the abstract Lie algebroid of a differentiable groupoid in a cartesian tangent category C with a scalar R-multiplication, where R is a ring object of C. Examples include differentiation of infinite-dimensional Lie groups, elastic diffeological groupoids, etc. This is joint work with Christian Blohmann.
September 17: Christopher Dean, Dalhousie University
Abstract Grothendieck Fibrations and Directed Path Objects
Structures from abstract homotopy theory, such as Brown's categories of fibrant objects and Quillen's model categories, provide a convenient 1-categorical middle ground between the very weak world of (∞,1)-categories and the strict syntax of homotopy type theory. A central ingredient of these structures is a distinguished class of arrows called fibrations. Fibrations satisfy a path-lifting property, and thus can be thought of as abstract isofibrations. These path-lifting properties allow one to reason about the internal ∞-groupoid structure of objects in a category of fibrant objects.
I will propose a directed analogue of a category with fibrant objects that I call a dipath category. Dipath categories have abstract classes of covariant and contravariant fibrations which behave much like Grothendieck (op)fibrations in the category of categories. These fibrations satisfy directed path lifting properties, which can be used to reason about the internal (∞,n)-category structure of objects in a dipath category. In this way dipath categories should serve as a convenient 1-categorical middle ground between the world of (∞,n)-categories and a future syntax of directed homotopy type theory.
September 24: Luuk Stehouwer, Dalhousie University
What is a 2-Hilbert space?
Baez introduced 2-Hilbert spaces as a categorification of Hilbert spaces. In this talk, I will explore 2-Hilbert spaces from a new perspective, deriving them from abstract categorical principles. The main tool is the recently developed theory of higher dagger categories. I hope these insights may contribute to the development of a notion of n-Hilbert space for larger n.