Logic, Categories, and Applications Seminar

Given a measured groupoid G, together with Filippo Sarti, we defined a cohomology theory which generalizes the measurable bounded cohomology of a locally compact group. In the particular case of a groupoid associated to a measure preserving action, our cohomology boils down to the usual bounded cohomology of the group with twisted coefficients. We will discuss the possible applications of this result to orbit equivalence. 

Given a realcompact space X, we denote by Exp(X) the smallest infinite cardinal κ such that X is homeomorphic to a closed subspace of Rκ. In this talk, we analyze the realcompactness number of countable spaces. We will show that, for every cardinal κ, there exists a countable crowded space X such that Exp(X) = κ if and only if p ≤ κ ≤ c. On the other hand, we show that a scattered space of weight κ has pseudocharacter at most κ in any compactification. This will allow us to calculate Exp(X) for an arbitrary (that is, not necessarily crowded) countable space. This is a joint work with Andrea Medini and Lyubomyr Zdomskyy.

In this completely self-contained talk, I'll discuss various versions of the Garden of Eden theorem, the Gottschalk surjunctivity conjecture, and Kaplansky's conjecture on stable finiteness of group rings. This will include a quick review of the notions of amenability and soficity for groups, of the theory of cellular automata, and entropies of dynamical systems.

The Hochschild cohomology of a differential graded algebra or more generally of a differential graded category admits a natural map to the graded center of its derived category: the characteristic homomorphism. We interpret this map as an edge homomorphism in a spectral sequence, which allows to study the characteristic homomorphism systematically in many interesting examples from algebra, geometry, topology and physics. To illustrate this, we will discuss several concrete examples related to coherent sheaves on algebraic curves and cochains of classifying spaces of Lie groups. If time permits, I will also indicate a new extension of this framework to $A_\infty$-categories. Some of this is joint work with M. Szymik (Sheffield) other with A. Phimister (Leicester).

In this talk I will study generalized automata (in the sense of Adámek-Trnková) in Joyal’s category of combinatorial species; as an important preliminary step, I will provide examples of coalgebras for the "derivative" endofunctor ∂ and for the ‘Euler homogeneity operator’ L∂ arising from the adjunction L⊣∂⊣R. The theory is connected with, and in fact provides nontrivial examples of, differential 2-rigs—a concept I recently introduced by treating combinatorial species in the same way that a generic (differential) semiring (R,d) relates to the (differential) semiring N[[X]] of power series with natural coefficients. Joyal himself has long regarded species as categorified formal power series. This perspective aligns with a fundamental category-theoretic insight: free objects in the category of rings naturally acquire a canonical differential structure. At the heart of this phenomenon lies the representability of the prestack of derivations by an object of Kähler differentials. These ideas categorify elegantly within the 2-category of differential 2-rigs, revealing that species possess a universal property as differential 2-rigs. The desire to study categories of ‘state machines’ valued in an ambient monoidal category (K,⊗) gives a pretext to further develop the abstract theory of differential 2-rigs, proving lifting theorems of a differential 2-rig structure from (R,∂) to the category of ∂-algebras on objects of R, and to categories of Mealy automata valued in (R,⊗), as well as various constructions inspired by differential algebra such as jet spaces and modules of differential operators. This talk covers the content of the paper Automata and Coalgebras in Categories of Species (Proceedings of CMCS24, Luxembourg), as well as parts of an ongoing project with Todd Trimble.

TBA