We usually meet on Tuesdays at 2:30pm in room 319 of the Chase Building. The current organizers are Christopher Dean and Luuk Stehouwer. Information about fall 2024 seminars can be found here.
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.
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 $\Omega\mathcal{C} := End_{\mathcal{C}}(1_{\mathcal{C}})$. I will explain a "square" of this construction. Suppose that $(\mathcal{C}, 1_{\mathcal{C}})$ be a pointed $(\infty,3)$-category. Formal nonsense of lax products produces a bialgebra in $\Omega^2 \mathcal{C}$ from a retract $X \leftrightarrows 1_{\mathcal{C}}$ 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.
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.
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.
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.
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 V-enriched categories. The parameterized maps inherit the morphism structure of V. We show that in order to recover all of the nice properties enjoyed by unitary time evolutions, V must be Cartesian monoidal.
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
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.