Oberseminar WS2025/26

It is well known that all c-basic varieties, studied by Mardare, Panangaden and Plotkin, are monadic categories over Met, the category of extended metric spaces. This opens the question of which monads correspond to those varieties.

For c=ω, all enriched finitary monads have that property. But Urbat presented an example of an ω-basic variety yielding a non-finitary monad.

We prove that a monad corresponds to an ω-basic variety iff it is a weighted colimit of finitary monads (in the category Mnd(Met) of enriched monads). Further, a monad corresponds to a c-basic variety for an uncountable cardinal c iff it is enriched and c-accessible.

The DIREGA project aims to create a formal, computable model of German commercial register law. This talk will delve into the methodological and technical hurdles faced during this formalization process. We will present key challenges for computational representation of the relevant regulations. Furthermore, we will critically analyze the different formalization techniques and modeling languages tested and the fundamental problems that necessitated shifts in our approach.

Codensity monads provide a universal method to generate complex monads from simple functors. Recently, a wide range of important monads in logic, denotational semantics, and probabilistic computation, such as several incarnations of the ultrafilter monad, the Vietoris monad, and the Giry monad, have been presented as codensity monads, using complex arguments. We propose a unifying categorical approach to codensity presentations of monads, based on the idea of relating the presenting functor to a dense functor via a suitable duality between categories. We prove a general presentation result applying to every such situation and demonstrate that most codensity presentations known in the literature emerge from this strikingly simple duality-based setup, drastically alleviating the complexity of their proofs and in many cases completely reducing them to standard duality results. Additionally, we derive a number of novel codensity presentations using our framework, including the first non-trivial codensity presentations for the filter monads on sets and topological spaces, the lower Vietoris monad on topological spaces, and the expectation monad on sets.

Machine Learning, in particular the development of Large Language Models (LLMs) in the last half decade, has sculpted the landscape of computer science more than any other broader field of research in a mutual manner. Although our work might not immediately be automated, we already observe the effects of broadly available and fairly precise natural language reasoning in the exercises to our foundational lectures. In this talk, we give an overview on recent developments in automated processing of language, and suggest various options how to incorporate machine learning either as a subject to research or as a tool for auxiliary tasks in our work.