Talks, preprints and publications

A(n hopefully) complete list of talks, preprints and publications I produced, in reverse chronological order.

To download my academic CV instead, click here.

2025

  • Talk: A Taste of Quantitative Logic
    Part 1 [slides, video], Part 2 [slides, video]
    Oxford Seminar (Topos UK)
    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 talk I will showcase these features by describing a sequent calculus for a quantitative version of linear logic.
    In the second part of the talk I will introduce p-means and argue they form a good quantitative analogue of first-order quantifiers. I will then sketch the construction of a hyperdoctrine valued in enriched graded preorders which forms the intended semantics of a first-order quantitative linear logic.
    This is work in progress with Atkey, Grellois, and Komendantskaya.
  • Preprint: Classifying strict discrete opfibrations with lax morphisms
    with David Jaz Myers
    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.
  • Talk: 2-classifiers for 2-algebras [slides, video]
    Oxford Seminar (Topos UK)
    In this talk I report on work in progress, joint with David Jaz Myers, about lifting discrete opfibration classifiers (2-classifiers, i.e. a ‘Set’-like object) from a 2-category 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.
  • Talk: Contextads, abridged [slides]
    PSSL110
    We introduce contextads and the Ctx construction, unifying various structures and constructions in category theory dealing with context and contextful arrows — comonads and their Kleisli construction, actegories and their Para construction, adequate triples and their Span 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: Syntax and Semantics of QPL [slides]
    Slides from the talk at the informal seminar for the ARIA ‘Safeguarded AI’ seminar. A follow up of last year’s preprint on quantifiers for quantitative reasoning.
  • Talk: An Elementary Account of the Internal Model Principle [slides]
    DIEP seminar
    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.
  • Preprint: A Bayesian Interpretation of the Internal Model Principle
    with Manuel Baltieri, Martin Biehl, and Nathaniel Virgo
    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.
  • Talk: Representable Behaviour in Double Categorical Systems Theory [slides, recording]
    Topos Colloquium talk
    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.

2024

  • Preprint: Contextads as Wreaths; Kleisli, Para, and Span Constructions as Wreath Products
    with David Jaz Myers
    We introduce contextads and the Ctx construction, unifying various structures and constructions in category theory dealing with context and contextful arrows — comonads and their Kleisli construction, actegories and their Para construction, adequate triples and their Span construction. Contextads are defined in terms of Lack–Street wreaths, suitably categorified for pseudomonads in a tricategory of spans in a 2-category with display maps. The associated wreath product provides the Ctx construction, and by its universal property we conclude trifunctoriality. This abstract approach lets us work up to structure, and thus swiftly prove that, under very mild assumptions, a contextad equipped colaxly with a 2-algebraic structure produces a similarly structured double category of contextful arrows. We also explore the role contextads might play qua dependently graded comonads in organizing contextful computation in functional programming. We show that many side-effects monads can be dually captured by dependently graded comonads, and gesture towards a general result on the `transposability’ of parametric right adjoint monads to dependently graded comonads.
  • Paper: A Fibrational Theory of First Order Differential Structures
    with Geoffrey Cruttwell, Neil Ghani and Fabio Zanasi
    We develop a categorical framework for reasoning about abstract properties of differentiation, based on the theory of fibrations. Our work encompasses the first-order fragments of several existing categorical structures for differentiation, including cartesian differential categories, generalised cartesian differential categories, tangent categories, as well as the versions of these categories axiomatising reverse derivatives. We explain uniformly and concisely the requirements expressed by these structures, using sections of suitable fibrations as unifying concept. Our perspective sheds light on their similarities and differences, as well as simplifying certain constructions from the literature.
  • Paper: Organizing Physics with Open Energy-driven Systems
    with Owen Lynch and David Spivak
    Organizing physics has been a long-standing preoccupation of applied category theory, going back at least to Lawvere. We contribute to this research thread by noticing that Hamiltonian mechanics and gradient descent depend crucially on a consistent choice of transformation — which we call a reaction structure — from the cotangent bundle to the tangent bundle. We then construct a compositional theory of reaction structures. Reaction-based systems offer a different perspective on composition in physics than port-Hamiltonian systems or open classical mechanics, in that reaction-based composition does not create any new constraints that must be solved for algebraically.
    The technical contributions of this paper are the development of symmetric monoidal categories of open energy-driven systems and open differential equations, and a functor between them, functioning as a “functorial semantics” for reaction structures. This approach echoes what has previously been done for open games and open gradient-based learners, and in fact subsumes the latter. We then illustrate our theory by constructing an n-fold pendulum as a composite of n-many pendula.
  • Paper: On a fibrational construction for optics, lenses, and Dialectica categories
    with Bruno Gavranović, Abdullah Malik, Francisco Rios, Jonathan Weinberger
    Categories of lenses/optics and Dialectica categories are both comprised of bidirectional morphisms of basically the same form. In this work we show how they can be considered a special case of an overarching fibrational construction, generalizing Hofstra’s construction of Dialectica fibrations and Spivak’s construction of generalized lenses. This construction turns a tower of Grothendieck fibrations into another tower of fibrations by iteratively twisting each of the components, using the opposite fibration construction.
  • Preprint: On Quantifiers for Quantitative Reasoning
    We explore a kind of first-order predicate logic with intended semantics in the reals. Compared to other approaches in the literature, we work predominantly in the multiplicative reals [0,\infty], showing they support three generations of connectives, that we call non-linear, linear additive, and linear multiplicative. Means and harmonic means emerge as natural candidates for bounded existential and universal quantifiers, and in fact we see they behave as expected in relation to the other logical connectives. We explain this fact through the well-known fact that min/max and arithmetic mean/harmonic mean sit at opposite ends of a spectrum, that of p-means. We give syntax and semantics for this quantitative predicate logic, and as example applications, we show how softmax is the quantitative semantics of argmax, and Rényi entropy/Hill numbers are additive/multiplicative semantics of the same formula. Indeed, the additive reals also fit into the story by exploiting the Napierian duality −\log \dashv 1/\mathrm{exp}, which highlights a formal distinction between ‘additive’ and ‘multiplicative’ quantities. Finally, we describe two attempts at a categorical semantics via enriched hyperdoctrines. We discuss why hyperdoctrines are in fact probably inadequate for this kind of logic.
  • Talk: Softmax is Argmax, and the Logic of the Reals [slides, video]
    MSP101 talk
    I will report some work in progress on the semiotics of softmax. This is an operator used in machine learning (but familiar to physicists way before that) to normalize a log-distribution, turning a vector of (thus, a function valued in) logits (i.e. additive reals) into a probability distribution. Its name is due to the fact it acts as a ‘probabilistic argmax’, since the modes of a softmax distribution reflects the minima (by an accident of duality) of the function. I will show an attempt to make this statement precise, by exhibiting the semantics of a ‘very linear logic’ on the *-autonomous quantale of extended multiplicative reals. In this logic, additive connectives are also linear, but are still in the same algebraic relation with the multiplicative ones. I will show how to define quantifiers, and thus softmax. If time permits, I’ll show a construction of an enriched equipment of relations in which softmax should be characterizable as a Kan lift, in the same way argmax is characterized as a Kan lift in relations.

2023

  • Talk: Constructing triple categories of cybernetic systems [slides]
    Talk at the Sixth International Conference on Applied Category Theory
    We illustrate a generalized version of the Para construction which allows to systematically construct triple categories of cybernetic processes, as well as further extensions thereof to cybernetic systems. While Para works for actions in categories, our generalization works for any suitably complete 2-category and for more general notions of action (what we call ‘oplax dependent actegories’). To exemplify the construction, we show how applying our generalized Para to the self-action of a monoidal double category of lenses and charts produces a triple category of parametric lenses, lenses and charts which improves on Spivak and Shapiro’s Org.
  • Talk: On a fibrational construction for lenses, optics and Dialectica categories [slides]
    Talk at the Joint Mathematics Meeting 2023 for the AMS Special Session on Applied Category Theory
    Categories of lenses/optics and Dialectica categories are both comprised of bidirectional morphisms of basically the same form. In this talk I’m going to introduce both and show how they can be considered a special case of an overarching fibrational construction, generalizing Hofstra’s construction of Dialectica fibrations. At its highest level of generality, it’s a construction that turns a tower of fibrations into another tower of fibrations by twisting each of the components using the opposite fibration construction.

2022

  • Talk: From categorical systems theory to categorical cybernetics [video, slides]
    Invited talk at the Virtual Double Categories Workshop
    Myers’ categorical system theory is a double categorical yoga for describing the compositional structure of open dynamical systems. It unifies and builds on previous work on operadic notions of system theory, and provides a strong conceptual scaffolding for behavioral system theory. However, some of the most interesting systems out have a richer compositional structure than that of dynamical systems. These are cybernetic systems, or in other words, interactive control systems. Notable and motivating examples are strategic games and machine learning models. In this talk I’m going to introduce the tools and language of categorical system theory and outline how categorical cybernetics theory might look like. At the end, we will briefly venture into the triple dimension.
  • Talk: Triple categories of open cybernetic systems [video, slides]
    Invited talk at ItaCa Fest 2022.
    Abstract: Categorical system theory (in the sense of Myers) is a double categorical yoga for describing the compositional structure of open dynamical systems. It unifies and improves on previous work on operadic notions of system theory, and provides a strong conceptual scaffolding for behavioral system theory. However, some of the most interesting systems out there escape the simple model of dynamical systems. They are instead cybernetic systems, or in other words, controllable dynamical systems. Notable and motivating examples are strategic games and machine learning models. In this talk I’m going to outline an upgrade of categorical system theory to deal with such systems by resorting to triple categories.
  • Paper: Diegetic representation of feedback in open games
    Talk at the Fifth International Conference on Applied Category Theory
    A game-changing (pun intended) paper about the status of open games as cybernetic systems. I show that by a process of generalized reverse-mode differentiation (of parametric lenses) we can get a cybernetic system representing accurately the feedback dynamics of a non-cooperative game. The resulting system can then be equipped with systems behaving as players, given by (parametric) selection functions and looking surprisingly alike learners. In a sense, in this paper I show agent in strategic games can be understood as experience backpropagation of feedback and doing gradient descent, for a given meaning of these words. Accepted at ACT22.
    Twitter thread, video, slides.
  • Talk: Dependent lenses are dependent optics
    An Intercats seminar about the recent developments on dependent optics. A recording is available on YouTube.
  • Preprint: Seeing double through dependent optics
    Some developments on dependent optics, prompted by recent advances by Vertechi and Milewski. I obtained the same definition they proposed from a ‘dependent’ Tambara theory based on actions of double categories, but then shunned it away because I couldn’t prove dependent lenses were an example. Vertechi found a way, Milewski found more examples, and thus I released my notes after some updating.
    Twitter thread
  • Preprint: Actegories for the working amthematician
    with Bruno Gavranović
    A long theory paper on actions of monoidal categories and their properties. We describe distributive laws between monoidal and actegorical structures and provide examples of their use in the theory of optics and parametric morphisms.
    Twitter thread
  • Preprint: Lenses for composable servers
    Using parametric dependent lenses for writing web servers.
    Twitter thread

2021

2020

  • Talk: Fantastic sheaves and where to find them
    An MSP101 talk about sheaf theory, covering cohomology and logical applications. There’s also an hint of a system-theoretic interpretation I’m working on.
  • Master thesis: Internal mathematics for stochastic calculus: a tripos theoretic approach
    My master thesis work. I explored how stochastic calculus could be simplified by a suitable internalization in a topos of ‘stochastic sets’. The results are encouraging, though much remains to be done.
  • Talk: Internal methods in algebraic geometry
    In the context of a series of short talks ending a course in homological algebra for topology, I spoke about topos-theoretic techniques applied to algebraic geometry, namely the methods of proving theorems by switching to the internal language of Zariski toposes. Since the audience wasn’t much acquainted with logic, I tried to skim over the most technical parts and keep it light, conceptual.