The current organizer is Christopher Dean. Information about winter 2024 seminars can be found here.
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 $(\infty, 1)$-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 $\infty$-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 $(\infty, n)$-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 $(\infty, n)$-categories and a future syntax of directed homotopy type theory.
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.
Partially traced categories
I will recall Haghverdi and Scott's notion of partially traced category, and discuss several examples, including the Haghverdi-Scott trace on finite dimensional vector spaces with direct sum, and the kernel-image trace. We'll also look at the Kleene trace, which is not a partial trace. The main theorem about partially traced categories was proved in my student Octavio Malherbe's 2010 Ph.D. thesis: every partially traced category can be faithfully embedded in a totally trace category, and vice versa, every monoidal subcategory of a totally traced category is partially traced.
Dirichlet functors
In my seminar talk last year I introduced the difference operator, a discrete version of derivative, for taut endofunctors of Set. All the examples given then were variations on polynomial functors. In this talk I will introduce a new class of taut functors, essentially different from polynomials, which I call Dirichlet. I will develop their main properties and give some cool examples. No knowledge of last year's talk is required.
Diagrammatic (∞,n)-categories
I will give an overview of the diagrammatic model of (∞,n)-categories, whose theory I have been developing together with Clémence Chanavat. This is designed to share (or improve) the good features of strict higher categories—such as a strong pasting theorem enabling explicit diagrammatic reasoning, and an expressive language for cellular models—while provably satisfying the homotopy hypothesis, and hopes to function as a bridge between non-algebraic and algebraic models.
Traces in categories of contractions
See processes taking multiple inputs and yielding multiple outputs? You are probably dealing with a monoidal category. Is it possible to feed the outputs of a process back to its own inputs? You are probably dealing with a traced monoidal category. The monoidal category of finite-dimensional vector spaces with tensor product is traced, where feeding the output of a linear operator back to its own input means taking the usual trace from linear algebra. Less famous, but also interesting: the monoidal category of finite-dimensional vector spaces with direct sum is partially traced. ("Partially" means that the trace is only defined for certain maps.) Here the trace would give meaning to feeding the outputs (rows) of a matrix back to its own inputs (columns). For finite-dimensional Hilbert spaces in particular, it is a totally defined trace when restricted to the monoidal subcategory of isometries (Bartha 2014), or more generally contractions (Andrés-Martínez 2022). We prove this result in the most general abstract context where we can see it works: a dagger finite biproduct category with negatives and Moore-Penrose inverses. This is joint work with Peter Selinger.
Squares K-theory and 2-Segal spaces
It's a well-known result that a simplicial set satisfies the Segal condition precisely if it's the nerve of a category; then, categories represent all possible 1-Segal sets. But what about 2-Segal? This question was studied by Bergner, Osorno, Ozornova, Rovelli and Scheimbauer, who proved that 2-Segal sets are in turn related to a certain class of double categories via a "K-theory construction". In this talk, I will discuss recent work joint with Maxine Calle where we explore the connection between 2-Segal sets/spaces and a newly introduced K-theory framework based on double categories called "Squares K-theory".