Follow up to A Taste of Quantitative Logic.

In this talk I introduce \(p\)-means and argue they form a good quantitative analogue of first-order quantifiers. I then sketch the construction of a hyperdoctrine valued in enriched graded preorders which forms the intended semantics of a first-order quantitative linear logic.

Quantitative logic, after Lawvere, is one whose judgments are valued in real numbers, rather than merely being a logic about real numbers. By doing so we can guarantee good structural properties of the logic, such as being able to treat addition as an additive connective in the sense of Girard. Moreover, by employing the full spectrum of sums multiplication distributes over, we are able to approximate 'hard' connectives with 'soft' ones, with application in machine learning. In the first part of the talk I will showcase these features by describing a sequent calculus for a quantitative version of linear logic.

This is work in progress with Atkey, Grellois, and Komendantskaya.

We study how discrete opfibration classifiers in a(n enhanced) 2-category can be endowed with the structure of a \(T\)-algebra and thereby lift to the enhanced 2-category of 2-algebras and lax morphisms. To support this study, we give a definition of discrete opfibration classifier in the enhanced setting in which tight (e.g. strict) discrete opfibrations are classified by loose (e.g. lax) maps.

We then single out conditions on the 2-monad \(T\) and the classifier that make this possible, and observe these hold in a wide range of examples: double categories (recovering the results of Parè and Lambert), (symmetric) monoidal categories, and all structures encoded by familial 2-monads. We also prove the properties needed on such 2-monads are stable under replacement by pseudo-algebra coclassifiers (when sufficient exactness conditions hold), allowing us to replace a pseudo-algebra structure on the classifier by a strict one.

To get to our main theorem, we introduce the concepts of cartesian maps and cartesian objects of a 2-algebra, which generalize various other notions in category theory such as cartesian monoidal categories, extensive categories, categories with descent, and more. As a corollary, we characterize when representable copresheaves are pseudo rather than lax in terms of the cartesianity at their representing object.

In this talk I report on work in progress, joint with David Jaz Myers, about lifting discrete opfibration classifiers (2-classifiers, i.e. a '\(\mathsf {Set}\)'-like object) from a 2-category \(\mathcal {K}\) to the 2-category of algebras of a 2-monad \(T\).

In the setting of DOTS, we often construct behaviour functors as 'representables', but without a 2-classifier one can't really call these 'representables'. Moreover, there is a strong connection between compositionality of such functors, the properties of the algebra they map out of, and the properties of the object(s) that represents them.

These phenomena are in fact completely general, so we set out to better understand the situation and found some frankly interesting notions and results, chiefly a tight result on the existence of 2-classifiers for 2-algebras.

We introduce contextads and the \(\mathbb {C}\mathsf {tx}\) construction, unifying various structures and constructions in category theory dealing with context and contextful arrows -- comonads and their Kleisli construction, actegories and their \(\mathcal {P}\mathsf {ara}\) construction, adequate triples and their \(\mathbb {S}\mathsf {pan}\) construction. Contextads are defined in terms of Lack-Street wreaths, suitably categorified for pseudomonads in a tricategory of spans. This abstract approach can be daunting, so in this talk we will work with a lower-dimensional version of contextads which is relevant to capture dependently graded comonads arising in functional monadic programming. In fact we show that many side-effects monads can be dually captured by discrete contextads, seen as dependently graded comonads, and gesture towards a general result on the 'transposability' of parametric right adjoint monads to dependently graded comonads.

Talk at the internal seminar for the ARIA 'Safeguarded AI' programme. A follow up of last year's preprint on quantifiers for quantitative reasoning.

The talk concerns the recent work with Baltieri, Biehl and Virgo on a categorical account of the classical 'internal model principle' from control theory and cybernetics in a broader sense. The aim is to distill the mathematical content of such an informal principle, following previous work of Wonham and Hepburn. In the talk I only use elementary mathematical notions and thus should be accessible to an audience acquainted with the basic vocabulary of sets and dynamical systems.

The internal model principle, originally proposed in the theory of control of linear systems, nowadays represents a more general class of results in control theory and cybernetics. The central claim of these results is that, under suitable assumptions, if a system (a controller) can regulate against a class of external inputs (from the environment), it is because the system contains a model of the system causing these inputs, which can be used to generate signals counteracting them. Similar claims on the role of internal models appear also in cognitive science, especially in modern Bayesian treatments of cognitive agents, often suggesting that a system (a human subject, or some other agent) models its environment to adapt against disturbances and perform goal-directed behaviour. It is however unclear whether the Bayesian internal models discussed in cognitive science bear any formal relation to the internal models invoked in standard treatments of control theory. Here, we first review the internal model principle and present a precise formulation of it using concepts inspired by categorical systems theory. This leads to a formal definition of 'model' generalising its use in the internal model principle. Although this notion of model is not a priori related to the notion of Bayesian reasoning, we show that it can be seen as a special case of possibilistic Bayesian filtering. This result is based on a recent line of work formalising, using Markov categories, a notion of 'model' generalising its use in the internal model principle. Although this notion of model is not a priori related to the notion of Bayesian reasoning, we show that it can be seen as a special case of possibilistic Bayesian filtering. This result is based on a recent line of work formalising, using Markov categories, a notion of 'interpretation', describing when a system can be interpreted as performing Bayesian filtering on an outside world in a consistent way.

Category theory has a long history of being applied to the study of general systems. Double Categorical Systems Theory (DCST) condenses many lessons learned along the way regarding compositional structures for the representation of systems, their behaviour and the interaction of these two aspects. In this talk I'll revisit old and new wisdom regarding functorial behaviour of systems represented by a category of timepieces, and prove old and new compositionality theorems for them.