Paolo Perrone

What I'm working on

Also: editor of Compositionality and of the International Journal of Mathematics; author of Starting Category Theory (World Scientific, 2024).

Research interests

• Probability & information theory
• Probabilistic ML & programming
• AI safety & alignment
• Bayesian inference & statistics
• Optimal transport
• Category theory

Here is an elementary introduction to some of the theoretical work I do.

Publications

(See also my arXiv and Google Scholar.)

Books

• P. Perrone, Starting Category Theory. World Scientific, ISBN 978-981-12-8600-1. (2024) · link

Published articles

• M. Bohinen, P. Perrone, Categorical algebra of conditional probability. Applied Categorical Structures, vol. 34, art. 56. (2026) · arXiv:2502.14941
• T. Fritz, T. Gonda, A. Lorenzin, P. Perrone, D. Stein, Absolute continuity, supports and idempotent splitting in categorical probability. Theory and Applications of Categories, vol. 45, no. 39. (2026) · arXiv:2308.00651
• T. Fritz, T. Gonda, A. Lorenzin, P. Perrone, A. Shah Mohammed, Empirical Measures and Strong Laws of Large Numbers in Categorical Probability. Logical Methods in Computer Science, vol. 22, issue 2. (2026) · arXiv:2503.21576
• P. Perrone, Lifting couplings in Wasserstein spaces. Compositionality, vol. 7, art. 2. (2025) · arXiv:2110.06591
• P. Perrone, Markov Categories and Entropy. IEEE Transactions on Information Theory, vol. 70, issue 3. (2024) · arXiv:2212.11719
• C. Constantin, T. Fritz, P. Perrone, B. Shapiro, Weak cartesian properties of simplicial sets. Journal of Homotopy and Related Structures. (2023) · arXiv:2105.04775
• T. Fritz, T. Gonda, N. Gauguin Houghton-Larsen, P. Perrone, D. Stein, Dilations and information flow axioms in categorical probability. Mathematical Structures in Computer Science. (2023) · arXiv:2211.02507
• T. Fritz, F. Gadducci, P. Perrone, D. Trotta, Weakly Markov categories and weakly affine monads. Proceedings of CALCO 10, LIPIcs. (2023) · arXiv:2303.14049
• T. Fritz, T. Gonda, P. Perrone, E. F. Rischel, Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability. Theoretical Computer Science, vol. 961. (2023) · arXiv:2010.07416
• C. Constantin, T. Fritz, P. Perrone, B. Shapiro, Partial evaluations and the compositional structure of the bar construction. Theory and Applications of Categories, vol. 39, no. 11. (2023) · arXiv:2009.07302
• S. Moss, P. Perrone, A category-theoretic proof of the ergodic decomposition theorem. Ergodic Theory and Dynamical Systems. (2023) · arXiv:2207.07353
• S. Moss, P. Perrone, Probability monads with submonads of deterministic states. Proceedings of LICS. (2022) · arXiv:2204.07003
• P. Perrone, W. Tholen, Kan extensions are partial colimits. Applied Categorical Structures, vol. 30. (2022) · arXiv:2101.04531
• T. Fritz, P. Perrone, S. Rezagholi, Probability, valuations, hyperspace: Three monads on Top and the support as a morphism. Mathematical Structures in Computer Science, vol. 31, no. 8. (2021) · arXiv:1910.03752
• T. Fritz, T. Gonda, P. Perrone, De Finetti's Theorem in Categorical Probability. Journal of Stochastic Analysis, vol. 2, no. 4, art. 6. (2021) · arXiv:2105.02639
• T. Fritz, P. Perrone, Stochastic order on metric spaces and the ordered Kantorovich monad. Advances in Mathematics, vol. 366. (2020) · arXiv:1808.09898
• T. Fritz, P. Perrone, Monads, partial evaluations, and rewriting. Proceedings of MFPS 36, ENTCS. (2020)
• T. Fritz, P. Perrone, A Probability Monad as the Colimit of Spaces of Finite Samples. Theory and Applications of Categories, vol. 34, no. 7. (2019) · arXiv:1712.05363
• T. Fritz, P. Perrone, Bimonoidal Structure of Probability Monads. Proceedings of MFPS 34, ENTCS. (2018)
• P. Perrone, N. Ay, Hierarchical Quantification of Synergy in Channels. Frontiers in Robotics and AI. (2016)

Preprints

• M. Di Meglio, C. Heunen, J-S. P. Lemay, P. Perrone, D. Stein, Dagger categories of relations: the equivalence of dilatory dagger categories and epi-regular independence categories. (2025) · arXiv:2508.01146
• P. Perrone, How to Represent Non-Representable Functors. (2024) · arXiv:2410.08019
• P. Perrone, R. Van Belle, Convergence of martingales via enriched dagger categories. (2024) · arXiv:2404.15191
• N. Ensarguet, P. Perrone, Categorical probability spaces, ergodic decompositions, and transitions to equilibrium. (2023) · arXiv:2310.04267

Lecture notes & other material

• P. Perrone, Categorical Information Geometry. Geometric Science of Information (GSI 2023), LNCS 14071. (2023) · link
• P. Perrone, Notes on Category Theory with examples from basic mathematics. Lecture notes. (2019) · arXiv:1912.10642
• T. Fritz, P. Perrone, A Criterion for Kan Extensions of Lax Monoidal Functors. (2018) · arXiv:1809.10481
• P. Perrone, N. Ay, Iterative Scaling Algorithm for Channels. (2016) · arXiv:1603.07181
• P. Perrone, N. Ay, Decomposition of Markov Kernels. Proceedings of WUPES 10. (2016)
• P. Perrone, Dual Connections and Holonomy. (2015) · arXiv:1511.07737

PhD thesis

• P. Perrone, Categorical Probability and Stochastic Dominance in Metric Spaces. PhD thesis, University of Leipzig. (2018) · link

Teaching

Courses & lectures

• Lecturer, Applied Category Theory (graduate course), University of Oxford (U.K.). (2026)
• Lecturer, Applied Category Theory (graduate course), University of Oxford (U.K.). (2025)
• College Tutor, Integral Transforms (undergraduate course), Mansfield College, University of Oxford (U.K.). (2025)
• College Tutor, Integration (undergraduate course), Mansfield College, University of Oxford (U.K.). (2025)
• Mentor, Applied Category Theory (summer school and online seminars), University of Oxford (U.K.). (2024)
• College Tutor, Integral Transforms (undergraduate course), Mansfield College, University of Oxford (U.K.). (2023)
• College Tutor, Integration (undergraduate course), Mansfield College, University of Oxford (U.K.). (2023)
• College Tutor, Probability I (undergraduate course), Mansfield College, University of Oxford (U.K.). (2022)
• College Tutor, Probability II (undergraduate course), Mansfield College, University of Oxford (U.K.). (2022)
• Tutor, Quantum Computing and Modelling: an Applied Category Theory Perspective (undergraduate course), Cambridge Center for International Research (U.K.). (2022)
• College Tutor, Integral Transforms (undergraduate course), Mansfield College, University of Oxford (U.K.). (2022)
• College Tutor, Integration (undergraduate course), Mansfield College, University of Oxford (U.K.). (2022)
• College Tutor, Groups and Group Actions (undergraduate course), Mansfield College, University of Oxford (U.K.). (2022)
• College Tutor, Linear Algebra II (undergraduate course), Mansfield College, University of Oxford (U.K.). (2022)
• Lecturer, Teoria delle categorie (online, in Italian). (2022) — Collaboration with the ItaCa Project.
• Lecturer, Bayesian Statistical Probabilistic Programming (graduate course), University of Oxford (U.K.). (2021) — Together with Hugo Paquet and Sam Staton.
• College Tutor, Linear Algebra I (undergraduate course), Mansfield College, University of Oxford (U.K.). (2021)
• Lecturer, Applied Category Theory (graduate course), University of Oxford (U.K.). (2021)
• College Tutor, Groups and Group Actions (undergraduate course), Mansfield College, University of Oxford (U.K.). (2021)
• Teacher, Categories and gauge theory (reading seminar), Massachusetts Institute of Technology, Cambridge, MA (U.S.A.). (2020)
• Lecturer, Applied Linear Algebra (undergraduate course), University of York, Toronto, ON (Canada). (2019)
• Teaching assistant, Applied Category Theory (summer school and online seminars), University of Oxford (U.K.). (2019)
• Lecturer, Category theory and applications (graduate course), Max Planck Institute, Leipzig (Germany). (2019)

Student supervision

• Master thesis: Jón Hákon Garðarsson , Mathematical Foundations of Computer Science (MFoCS), University of Oxford (U.K.). (2025)
• Master thesis: Mika Bohinen , Mathematical Foundations of Computer Science (MFoCS), University of Oxford (U.K.). (2024)
• Master thesis: Noé Ensarguet , Mathematical Foundations of Computer Science (MFoCS), University of Oxford (U.K.). (2023)

For material from current and past courses, please contact me directly.

Talks & events

Invited talks, conference talks, and seminars.

2026

• Presheaves on Markov Categories and Expectation Values. Invited talk, Bilkent Üniversitesi, Ankara (Turkey). (Aug. 2026)
• Introduction to Markov categories. Invited lecture, Bilkent Üniversitesi, Ankara (Turkey). (Aug. 2026)
• Presheaves on Markov Categories and Expectation Values. Category Theory 2026, Johns Hopkins University, Baltimore MD (U.S.A.). (July 2026) · page
• Expectations in Markov Categories. Invited talk, Workshop on Compositional Foundations Seminar, University of Warwick (UK). (July 2026) · page
• Markov Categories and Entropy. Invited talk, FoSS Seminar, University of Sussex (UK). (June 2026)
• Markov Categories and Expectations. Invited talk, Quantitative Reasoning Workshop, Bellairs Research Institute (Barbados). (June 2026) · page
• Markov Categories and Entropy. Invited lecture, Seminar on Compositional Foundations Seminar, University of Warwick (UK). (Feb. 2026)
• Independent States Are Orthogonal: a categorical framework to treat probability geometrically. Invited talk, 11th Workshop on Information Geometry, Quantum Mechanics and Applications, Universidad Autónoma de Madrid (Spain). (Feb. 2026) · page

2025

• The Higher Structure of Martingales. ItaCa Live, Università degli Studi di Milano (Italy). (Dec. 2025) · page
• Descent in Probability Theory: The First Steps Downward. Invited talk, Logic, Algebra and Categories Seminar, Università degli Studi di Milano (Italy). (Nov. 2025) · page
• Independent States Are Orthogonal: a categorical framework to treat probability geometrically. Geometric Science of Information 2025, Saint-Malo (France). (Oct. 2025) (Held online.) · page · video
• Descent in Probability Theory: The First Steps Downward. Invited talk, Topos Institute, Oxford (UK). (Oct. 2025) · video
• Towards probabilistic regular logic. Invited talk, Southern and Midlands logic Seminar, University College London (UK). (Sept. 2025) · page
• Categories of Relations which Compose Independently. Category Theory 2025, Masaryk University, Brno (Czechia). (July 2025) · page · video
• Categories for Probability Theory. Invited talk, Logic, Categories and Applications Seminar, Università degli Studi di Bologna (Italy). (June 2025)
• Probability and information theory with Markov categories. Invited talk, Computer Science colloquium and seminar, University of Hertfordshire, Hatfield (United Kingdom). (June 2025) · page
• Empirical distributions and strong laws of large numbers in cateogrical probability. ACT 2025, University of Florida, Gainsville FL (U.S.A.). (June 2025) Speaker: Tobias Fritz (joint work). · page
• Categorical algebra of conditional probability. SYCO 13, University College London (United Kingdom). (Apr. 2025) Speaker: Mika Bohinen (joint work).
• Markov Categories. Invited talk, Quantitative Reasoning Workshop, Bellairs Research Institute (Barbados). (Mar. 2025) · page
• Categorical probability with additional geometry. Invited talk, 10th Workshop on Information Geometry, Quantum Mechanics and Applications, Universidad Autónoma de Madrid (Spain). (Feb. 2025) · page
• Notions of expectations and probability monads. Seminar talk, PUDDLE Seminar, University of Oxford (United Kingdom). (Feb. 2025)
• A Categorical Theory of Conditional Expectations. Invited talk, AMS Special Session, Joint Mathematics Meetings, Seattle, WA (U.S.A.). (Jan. 2025) · page

2024

• The Formal Theory of Random Variables. ItaCa Live, Università degli Studi di Padova (Italy). (Dec. 2024) · page
• Markov categories, symmetries, and generative AI. Invited lecture, Applied Category Theory for Engineering Design, Massachusetts Institute of Technology (MIT), Cambridge, MA (U.S.A.). (Nov. 2024) Together with Rob Cornish. (Held online.) · video
• Notions of Expectation and Probability Monads. Distinguished lecture, Peripatetic Seminar on Sheaves and Logic, Universiteit Leiden (Netherlands). (Nov. 2024) · page
• When limits are limits: topological enrichment with and application to probability. Category Theory 2024, Universidade de Santiago de Compostela (Spain). (June 2024) · page · video
• Convergence of martingales via enriched dagger categories. Keynote talk, Applied Category Theory 2024, University of Oxford (United Kingdom). (June 2024) Speaker: Ruben Van Belle (joint work). · page · video
• Markov categories, Bayesian inversions, and statistical experiments. Invited talk, Tweag, Paris (France). (June 2024) · video
• Markov categories: Towards user-friendly probability theory. Invited talk, Finance and Stochastics Seminar, Imperial College London (United Kingdom). (May 2024) · page
• Conditional expectation for category theorists. Seminar talk, PUDDLE Seminar, University of Oxford (United Kingdom). (May 2024)
• Monoidal preorders and enrichment. Guest lecture, Applied Category Theory course, University of Oxford (United Kingdom). (May 2024) (Held online.)
• Martingales and their convergence, categorically. Invited talk, Four Decades of Probabilistic Semantics, Bellairs Research Institute (Barbados). (Apr. 2024) · page
• Higher abstraction for probability and statistics. Invited talk, Logic, Uncertainty, Computation and Information (LUCI) Lab, Università degli Studi di Milano (Italy). (Jan. 2024) (Held online.) · page

2023

• Conditional expectations as split idempotents. ItaCa Live, Università degli Studi di Torino (Italy). (Dec. 2023) · page
• Probability Theory with Markov categories. Invited talk, University College London (UCL), London (United Kingdom). (Oct. 2023)
• Information Theory with Markov categories. Invited talk, Geometry of Information Theory workshop, Universidad Carlos III, Madrid (Spain). (Sept. 2023) · page
• Categorical information geometry. Geometric Science of Information 2023, Saint-Malo (France). (Aug. 2023) · page
• Markov categories and entropy. Applied Category Theory 2023, University of Maryland, College Park, MD (U.S.A.). (Aug. 2023) · page · video
• Absolute continuity, supports and idempotent splitting in Markov categories. Applied Category Theory 2023, University of Maryland, College Park, MD (U.S.A.). (Aug. 2023) Speaker: Tomáš Gonda (joint work). · page
• Markov categories: a tutorial. Tutorial talk for ACT 2023, University of Maryland, College Park, MD (U.S.A.). (July 2023) · page · video
• Universal properties in probability theory. Keynote talk, Category Theory 2023, Université Catholique de Louvain, Louvain-la-Neuve (Belgium). (July 2023) · page · video
• Weakly Markov categories and weakly affine monads. CALCO 2023, University of Indiana, Bloomington, IN (U.S.A.). (June 2023) (Held online.) · page · slides
• Probability, Symmetry, and Entropy with Markov Categories. Invited talk, Computational Statistics and Machine Learning Seminar (OxCSML), University of Oxford (United Kingdom). (May 2023) · page
• Categorical probability and information theory. Invited talk, Q-Math Seminar, Universidad Carlos III, Madrid (Spain). (May 2023) · page · video
• Probabilistic programming: its challenges and opportunities. Invited event, Alan Turing Institute, London (United Kingdom). (Apr. 2023)
• Research week on Markov categories. Invited event, Universität Innsbruck (Austria). (Jan. 2023)

2022

• Dilations and information flow axioms in categorical probability. SYCO 10, University of Edinburgh (Scotland). (Dec. 2022) Speaker: Tomáš Gonda (joint work). · page
• GS-monoidal structures and thunkability in Kleisli categories. Invited talk, Università di Pisa (Italy). (Dec. 2022)
• Introduction to Markov categories. Invited talk, Sheaf Talk Seminar, University of Oxford (United Kingdom). (Nov. 2022) · page · video
• When measurable spaces don't have enough points. Invited talk, Theoretical Computer Science Seminar, University of Birmingham (United Kingdom). (Nov. 2022) · page · video
• Lifting weights: enriched lenses between transport plans. SYCO 9, Università dell'Insubria, Como (Italy). (Sept. 2022) · page
• Probability monads with submonads of deterministic states. Logic in Computer Science (LICS), Technion, Haifa (Israel). (Aug. 2022) Speaker: Sean Moss (joint work). · page
• When measurable spaces don't have enough points. Invited talk, Seminário de Teoria das Categorias, Universidade de Coimbra (Portugal). (July 2022) (Held online.) · video
• A category-theoretic prood of the ergodic decomposition theorem. Applied Category Theory 2022, University of Strathclyde (Scotland). (July 2022) · page · video
• Introduction to categorical probability. Guest lecture, Applied Category Theory course, University of Oxford (United Kingdom). (June 2022) (Held online.)

2021

• The rise of quantitative category theory. Invited talk, Topos Colloquium, Topos Institute, Berkeley. CA (U.S.A.). (Nov. 2021) · page · video
• Kan extensions are partial colimits. Category Theory 2021, Università degli Studi di Genova (Italy). (Aug. 2021) (Held in person.) · page
• De Finetti's theorem in categorical probability. Applied Category Theory 2021, University of Cambridge (U.K.). (July 2021) Speaker: Tobias Fritz (joint work). · page
• An invitation to categorical probability and statistics with Markov categories. Keynote talk, Categories and Companions 2021, Macquarie University, Sydney (Australia). (June 2021) Speaker: Tobias Fritz (joint work). · page
• Compositional structure of partial evaluations. Categories and Companions 2021, Macquarie University, Sydney (Australia). (June 2021) Speaker: Brandon Shapiro (joint work). · page
• Weighted limits in metric geometry. Seminar talk, Metric Geometry seminar, Max Planck Institute for Mathematics in the Sciences, Leipzig (Germany). (May 2021)
• Markov categories: towards a syntax for probability. Seminar talk, Logic and Semantics for Dummies, University of Cambridge (U.K.). (May 2021) · page
• The law of large numbers in categorical probability. Invited talk, Topos Colloquium, Topos Institute, Berkeley, CA (U.S.A.). (May 2021) Speaker: Tobias Fritz (joint work). · page · video
• The de Finetti theorem in categorical probability. Invited talk, Cohomology in algebra, geometry, physics and statistics, Institute of Mathematics, Czech Academy of Sciences (Czech Republic). (Apr. 2021) Speaker: Tobias Fritz (joint work). · page
• Categorical probability, Markov categories and the de Finetti theorem. Invited talk, New York City Category Theory Seminar, CUNY, New York, NY (U.S.A.). (Mar. 2021) Speaker: Tobias Fritz (joint work). · page · video
• Categorical probability, Markov categories and the de Finetti theorem. Impromptu seminar talk, New York City Category Theory Seminar, CUNY, New York, NY (U.S.A.). (Mar. 2021) · page · video
• Partial evaluations: the results so far. Invited talk, Seminario de categorías UNAM, Mexico City (Mexico). (Feb. 2021) · page · video
• Partial evaluations for monads and 2-monads. Invited talk, Seminário de Teoria das Categorias, Universidade de Coimbra (Portugal). (Feb. 2021) · page
• Kan extensions are partial colimits. Invited talk, Masaryk University Algebra Seminar, Brno (Czech Republic). (Feb. 2021) · video
• An introduction to monads. Invited lecture, Applied Compositional Thinking for Engineers, ETH, Zürich (Switzerland). (Jan. 2021) · page · video
• Markov categories: randomness and information flow. Oxford Quantum Group Workshop, University of Oxford (United Kingdom). (Jan. 2021) · page

2020

• Colimits as algebraic operations. Invited talk, ItaCa Fest III, ItaCa (Italy). (Sept. 2020) · page · video
• Compositional structure of partial evaluations. Invited talk, Massachusetts Institute of Technology (MIT), Cambridge, MA (U.S.A.). (Sept. 2020) Speaker: Brandon Shapiro (joint work). · page · video
• Distribution functors, second-order stochastic dominance and the Blackwell--Sherman--Stein Theorem in Categorical Probability. Applied Category Theory 2020, Massachusetts Institute of Technology (MIT), Cambridge, MA (U.S.A.). (July 2020) Speaker: Tomáš Gonda (joint work). · page · video
• Monads and Comonads. Tutorial talk for ACT 2020, Massachusetts Institute of Technology (MIT), Cambridge, MA (U.S.A.). (July 2020) · page · video
• Monads, Partial Evaluations, and Rewriting. Invited talk, POPV Seminar, University of Boston, Boston, MA (U.S.A.). (June 2020) · page
• Kan extensions are partial colimits. Invited talk, Massachusetts Institute of Technology (MIT), Cambridge, MA (U.S.A.). (June 2020) · page · video
• Probability monads and stochastic dominance. Categorical Probability and Statistics workshop, University of Ottawa, ON (Canada). (June 2020) · page · video
• What is a probability monad?. Tutorial talk for the CPS workshop, University of Ottawa, ON (Canada). (June 2020) · page · video
• Monads, Partial Evaluations, and Rewriting. MFPS 36, Université Paris VII, Paris (France). (June 2020)
• The support as a morphism from probability to possibility. Seminar talk, Tallinn Institute of Technology (TalTech), Tallinn (Estonia). (May 2020) Speaker: Sharwin Rezagholi (joint work). · page
• Stochastic orders and Kantorovich duality. Seminar talk, Metric Geometry seminar, Max Planck Institute for Mathematics in the Sciences, Leipzig (Germany). (April 2020)
• Composing partial evaluations. Seminar talk, Massachusetts Institute of Technology (MIT), Cambridge, MA (U.S.A.). (March 2020) · page · video

2019

• The compositional structure of partial evaluations. Invited talk, University of Ottawa, ON (Canada). (Nov. 2019)
• The support is a morphism of monads. Applied Category Theory 2019, University of Oxford (United Kingdom). (July 2019) · page
• Categorical Probability: Results and Challenges. SYCO 4, Chapman University, CA (U.S.A.). (May 2019) Speaker: Tobias Fritz (joint work). · page
• The support is a morphism of monads. SYCO 3, University of Oxford (United Kingdom). (March 2019) Speaker: Sharwin Rezagholi (joint work).
• Towards probability theory without measure theory. Invited talk, Massachusetts Institute of Technology (MIT), Cambridge, MA (U.S.A.). (Feb. 2019) Speaker: Tobias Fritz (joint work). · page · video

2018

• Monads, Partial Evaluations, and Martingales. Invited talk, Massachusetts Institute of Technology (MIT), Cambridge, MA (U.S.A.). (Oct. 2018) · page
• Monads, Partial Evaluations, and Rewriting. SYCO 1, University of Birmingham (United Kingdom). (Sept. 2018) · slides
• On the Operational Meaning of the Bar Construction. Category Theory 2018, Universidade dos Açores, Ponta Delgada, Azores (Portugal). (July 2018) · slides
• Bimonoidal Structure of Probability Monads. MFPS 34, Dalhousie University, Halifax, NS (Canada). (June 2018) · page
• A Kantorovich Monad for Ordered Spaces. FMCS 26, Mount Allison University, Sackville, NB (Canada). (May-June 2018) · slides

Earlier

• The Wasserstein Monad in Categorical Probability. Category Theory 2017, University of British Columbia, Vancouver, BC (Canada). (July 2017)
• A Conceptual Viewpoint on Information Decomposition. Invited talk, Perimeter Institute for Theoretical Physics, Waterloo, ON (Canada). (Nov. 2016)
• Decomposition of Markov Kernels. WUPES 10, Monínec (Czechia). (Sept. 2015)
• Synergy as a Linear Operator. GSO 2014, Albert-Ludwig-Universität, Freiburg (Germany). (Dec. 2014)

Curriculum Vitae

• Industry / research-scientist CV (pdf)
• Academic CV — English (pdf)
• Academic CV — German (pdf)
• Academic CV — Italian (pdf)

For a DORA/R4RI-style narrative CV, please contact me directly.