Paolo Perrone - University of Oxford

Paolo Perrone

Occupation: Mathematician and Computer Scientist
Position: Research Associate
Institution: University of Oxford
Group leader: Sam Staton
Some coworkers and coauthors: Carmen Constantin, Tobias Fritz, Tomáš Gonda, Sharwin Rezagholi, Eigil Rischel, Brandon Shapiro, Dario Stein, Walter Tholen

Research Interests

Category Theory
Probability and Information Theory
Complexity and Networks
Theoretical Computer Science
Geometry and Topology

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

Curriculum Vitae

Short CV:

Born in Italy in 1989
Bachelor degree in Physics, University of Milan-Bicocca, Italy, in 2011
Master's degree in Physics, University of Milan, Italy, in 2013
PhD in Mathematics, Max Planck Institute for Mathematics in the Sciences and University of Leipzig, Germany, in 2018
Currently researcher at the University of Oxford, England

Full academic CV:

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

Writings

Books:

P. Perrone, Starting Category Theory, World Scientific, 2024. Link here.

Published articles:

P. Perrone, Lifting couplings in Wasserstein spaces. Compositionality, 7(2), 2025. Available here.
P. Perrone, Markov Categories and Entropy. IEEE Transactions on Information Theory 70(3), 2024. Available here.
C. Constantin, T. Fritz, P. Perrone and B. Shapiro, Weak cartesian properties of simplicial sets. Journal of Homotopy and Related Structures, 2023. Available here.
T. Fritz, T. Gonda, N. Gauguin Houghton-Larsen, P. Perrone and D. Stein, Dilations and information flow axioms in categorical probability. Mathematical Structures in Computer Science, 2023. Available here.
T. Fritz, F. Gadducci, P. Perrone and D. Trotta, Weakly Markov categories and weakly affine monads, Proceedings of CALCO, LIPIcs 10, 2023. Available here.
T. Fritz, T. Gonda, P. Perrone and E. Rischel, Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability, Theoretical Computer Science 961, 2023. Available here.
C. Constantin, T. Fritz, P. Perrone and B. Shapiro, Partial Evaluations and the Compositional Structure of the Bar Construction, Theory and Applications of Categories 39, 2023. Available here.
S. Moss and P. Perrone, A category-theoretic proof of the ergodic decomposition theorem, Ergodic Theory and Dynamical Systems, 2023. Available here.
S. Moss and P. Perrone, Probability monads with submonads of deterministic states. Proceedings of LICS, 2022. Extended version available here.
P. Perrone and W. Tholen, Kan extensions are partial colimits, Applied Categorical Structures 30, 2022. Available here.
T. Fritz, P. Perrone, and S. Rezagholi, Probability, valuations, hyperspace: Three monads on Top and the support as a morphism, Mathematical Structures in Computer Science 31(8), 2021. Available here.
T. Fritz, T. Gonda and P. Perrone, De Finetti's Theorem in Categorical Probability, Journal of Stochastic Analysis 2(4), 2021. Available here.
T. Fritz and P. Perrone, Stochastic Order on Metric Spaces and the Ordered Kantorovich Monad, Advances in Mathematics 366, 2020. Available here.
T. Fritz and P. Perrone, Monads, partial evaluations, and rewriting. Proceedings of MFPS 36, ENTCS, 2020. Available here.
T. Fritz and P. Perrone, A Probability Monad as the Colimit of Spaces of Finite Samples, Theory and Applications of Categories 34, 2019. Available here.
T. Fritz and P. Perrone, Bimonoidal Structure of Probability Monads. Proceedings of MFPS 34, ENTCS, 2018. Available here.
P. Perrone and N. Ay, Hierarchical Quantification of Synergy in Channels, Front. Robot. AI, 2016. Available here.

Submitted articles:

T. Fritz, T. Gonda, A. Lorenzin, P. Perrone and A. Shah Mohammed, Empirical Measures and Strong Laws of Large Numbers in Categorical Probability, 2025. Available here.
M. Bohinen and P. Perrone, Categorical algebra of conditional probability, 2025. Available here.
P. Perrone, How to Represent Non-Representable Functors, 2024. Available here.
P. Perrone and R. Van Belle, Convergence of martingales via enriched dagger categories, 2024. Available here.
N. Ensarguet and P. Perrone, Categorical probability spaces, ergodic decompositions, and transitions to equilibrium, 2023. Available here.
T. Fritz, T. Gonda, A. Lorenzin, P. Perrone and D. Stein, Absolute continuity, supports and idempotent splitting in categorical probability, 2023. Available here.

Lecture notes, conference papers and other material:

P. Perrone, Categorical Information Geometry, 2023. Geometric Science of Information (GSI 2023), Lecture Notes in Computer Science 14071. Available here.
P. Perrone, Notes on Category Theory with examples from basic mathematics, 2019. Lecture notes. Available here. Extended to a book here.
P. Perrone, Categorical Probability and Stochastic Dominance in Metric Spaces, 2018. PhD thesis, University of Leipzig. Full text here, abstract here.
T. Fritz and P. Perrone, A Criterion for Kan Extensions of Lax Monoidal Functors, 2018. Available here.
P. Perrone and N. Ay, Iterative Scaling Algorithm for Channels, 2016. Available here.
P. Perrone and N. Ay, Decomposition of Markov Kernels. Proceedings of WUPES 10, 2015. Available here.
P. Perrone, Dual Connections and Holonomy, 2015. Available here.

Teaching

Current and upcoming:

Tutorials, Integration, at the University of Oxford, Mansfield College. Hilary term 2025.
Tutorials, Statistics, at the University of Oxford, Mansfield College. Hilary term 2025.
Course, Applied Category Theory, at the University of Oxford, Department of Mathematics. Trinity term 2025.
Project, ACT Adjoint School, together with Tomáš Gonda.

Past lectures and seminars:

Tutorials, Integration, at the University of Oxford, Mansfield College. Hilary term 2023.
Tutorials, Integral Transforms, at the University of Oxford, Mansfield College. Hilary term 2023.
Tutorials, Probability I, at the University of Oxford, Mansfield College. Michaelmas term 2022.
Tutorials, Probability II, at the University of Oxford, Mansfield College. Michaelmas term 2022.
Tutorials, Quantum Computing and Modelling, Cambridge Center for International Research. Summer 2022.
Online course, Category Theory (in Italian), together with the ItaCa project. Main page here, videos here.
Tutorials, Groups and Group Actions, at the University of Oxford, Mansfield College. Hilary term 2021-22.
Tutorials, Linear Algebra II, at the University of Oxford, Mansfield College. Hilary term 2021-22.
Tutorials, Integration, at the University of Oxford, Mansfield College. Hilary term 2021-22.
Tutorials, Integral Transforms, at the University of Oxford, Mansfield College. Hilary term 2021-22.
Course, Bayesian Statistical Probabilistic Programming, at the University of Oxford, Department of Computer Science. Michaelmas term 2021. Together with Hugo Paquet and Sam Staton.
Tutorials, Introduction to Complex Numbers, at the University of Oxford, Mansfield College. Michaelmas term 2021.
Tutorials, Linear Algebra I, at the University of Oxford, Mansfield College. Michaelmas term 2021.
Course, Applied Category Theory, at the University of Oxford, Department of Mathematics. Trinity term 2021.
Tutorials, Groups and Group Actions, at the University of Oxford, Mansfield College. Trinity term 2021.
Applied Category Theory Adjoint School 2020, based at MIT (moved online), together with Carmen Constantin and Eliana Lorch. More information here.
Reading seminar, Categories and gauge theory, at MIT.
Course, Applied Linear Algebra, at the York University of Toronto. Fall semester 2019. For notes and other teaching material see the Moodle page (students only).
Course, Category theory and applications, at the Max Planck Institute of Leipzig. Summer semester 2019. Lecture notes here (which were extended to this book).
Teaching assistant at the Applied Category Theory Adjoint School, based at the University of Oxford. March-July 2019. For notes and other teaching material see the shared folder (students only).
Reading seminar on Applied Category Theory at the Max Planck Institute of Leipzig. Summer semester 2018.
"LikBez" seminar at the Max Planck Institute of Leipzig. Years 2017 and 2018.

For further material about current and past courses please contact me directly.

Events

Upcoming and current events:

Categories of relations which compose independently. Plenary talk, Category Theory 2025, Masaryk University, Brno (Czechia), 13-19th July 2025. Main page here.
Currently coordinating the OASIS Seminar at the University of Oxford.

Past events

2025:

Categories for probability theory. Invited talk, Logic, Categories and Applications Seminar, Università degli Studi di Bologna (Italy), June 2025. Main page here.
Probability and information theory with Markov categories. Invited talk, Computer Science Colloquium and Seminar, Centre of Data Innovation Research, University of Hertfordshire, Hatfield (United Kingdom), June 2025.
Empirical distributions and strong laws of large numbers in categorical probability. Applied Category Theory 2025, University of Florida, Gainsville FL (USA), June 2025. Speaker: Tobias Fritz (joint work). Main page here.
Categorical algebra of conditional probability. 13th Symposium on Compositional Structures, University College London (UK), April 2025. Speaker: Mika Bohinen (joint work). Main page here.
Safeguarded AI Creator Kick-Off Event. Invited event, ARIA, University of Nottingham (UK), April 2025.
Markov Categories. Invited talk, Quantitative Reasoning Workshop, Bellairs Research Institute (Mc Gill University), Holetown (Barbados), March 2025.
Categorical Probability with Additional Geometry. 10th Workshop on Information Geometry, Quantum Mechanics and Applications, Universidad Autónoma de Madrid (Spain), February 2025. Main page here.
Notions of expectation and probability monads. PUDDLE Seminar, Department of Computer Science, University of Oxford (UK), February 2025.
Safeguarded AI, TA1.1 Creator Cohort. Invited event, ARIA, University of York (UK), February 2025.
A Categorical Theory of Conditional Expectations. Invited talk, AMS Special Session, Joint Mathematics Meetings (JMM), Seattle, WA (USA), January 2025. Main page here.

2024:

The Formal Theory of Random Variables. ItaCa 2024, Università degli Studi di Padova (Italy), December 2024. Main page here.
Markov categories, symmetries, and generative AI. Invited lecture, Massachusetts Institute of Technology (MIT), Cambridge MA (USA), November 2024. Together with Rob Cornish. Held online. Video here.
Notions of expectation and probability monads. Distinguished Lecture, Peripathetic Seminar on Sheaves and Logic (PSSL), University of Leiden (Netherlands), November 2024. Main page here.
Mathematics and modelling are the keys we need to safely unlock transformative AI. Invited event, ARIA, Manchester (UK), October 2024. More information here.
When limits are limits: topological enrichment with an application to probability. Plenary talk, Category Theory 2024, Universidad de Santiago de Compostela (Spain), June 2024. Main page here, related video here.
Convergence of martingales via enriched dagger categories. Keynote talk, Applied Category Theory 2024, University of Oxford (UK), June 2024. Speaker: Ruben Van Belle (joint work). Main page here, video here.
Markov categories, Bayesian inversions, and statistical experiments. Tweag, Paris (France), June 2024. Video here.
Markov categories: towards user-friendly probability theory. Invited talk, Finance and Stochastics seminar, Imperial College, London (UK), May 2024. Link here.
Conditional expectation for category theorists. PUDDLE Seminar, Department of Computer Science, University of Oxford (UK), May 2024.
Monoidal preorders and enrichment. Guest lecture at the Applied Category Theory course, Department of Mathematics, University of Oxford (UK), May 2024.
Martingales and their convergence, categorically. Invited talk, Four decades of probabilistic semantics, Bellairs Research Institute (Mc Gill University), Holetown (Barbados), April 2024.
Higher abstraction for probability and statistics. Logic, Uncertainty, Computation and Information (LUCI) Lab, Università degli Studi di Milano (Italy), March 2024. Held online. Main page here.
Safeguarded AI: Constructing safety by design. Invited event, ARIA, Birmingham (UK), March 2024. More information here.

2023:

Conditional expectations as split idempotents. ItaCa 2023, Università degli Studi di Torino (Italy), December 2023. Main page here.
Mathematics and modelling are the keys we need to safely unlock transformative AI. Invited event, ARIA, Birmingham (UK), December 2023. More information here.
Probability theory with Markov categories. Invited talk, University College London (UK), October 2023.
Information theory with Markov categories. Invited talk, Geometry of Information Theory, Universidad Carlos III, Madrid (Spain), September 2023. Main page here.
Categorical Information Geometry. 6th International Conference on Geometric Science of Information (GSI), Saint-Malo (France), August-September 2023. Main page here.
Markov categories and entropy. Applied Category Theory 2023, University of Maryland, College Park, MD (USA), July-August 2023. Main page here, video here.
Absolute continuity, supports and idempotent splitting in Markov categories. Applied Category Theory 2023, University of Maryland, College Park, MD (USA), July-August 2023. Speaker: Tomáš Gonda (joint work). Main page here.
Markov categories, a tutorial. Tutorial session, Applied Category Theory 2023, University of Maryland, College Park, MD (USA), July-August 2023. Main page here, video here.
Universal properties in probability theory. Keynote talk, Category Theory 2023, Université Catholique de Louvain, Louvain-la-Neuve (Belgium), July 2023. Main page here, video here.
Weakly Markov categories and weakly affine monads. CALCO 2023, University of Indiana, Bloomington, IN (USA), June 2023. Main page here, slides here.
Compositional Robotics: Mathematics and Tools, ICRA 2023 Workshop. ExCeL London (UK), June 2023. Main page here.
Probability, Symmetry, and Entropy with Markov Categories. Invited talk, Computational Statistics and Machine Learning Seminar (OxCSML), Department of Statistics, University of Oxford (UK), May 2023.
Categorical Probability and Information Theory. Invited talk, Q-Math Seminar, Universidad Carlos III, Madrid (Spain), May 2023. Video here (in Spanish).
Probabilistic programming: its challenges and opportunities. Invited event, Alan Turing Institute, London (UK), April 2023.
Research week on Markov categories. Invited event, Universität Innsbruck (Austria), January 2023.

2022:

Dilations and information flow axioms in categorical probability. 10th Symposium on Compositional Structures, University of Edinburgh (Scotland), December 2022. Speaker: Tomáš Gonda (joint work). Main page here.
GS-monoidal structures and thunkability in Kleisli categories. Invited talk, Università di Pisa (Italy), December 2022.
Introduction to Markov categories. Sheaf Talk Seminar, University of Oxford (UK), November 2022. Link here, related video here.
When measurable spaces don't have enough points. Theoretical Computer Science Seminar, University of Birmingham (UK), November 2022. Main page here, related video here.
Lifting weights: enriched lenses between transport plans. 9th Symposium on Compositional Structures, Como (Italy), September 2022. Main page here.
Probability monads with submonads of deterministic states. Logic in Computer Science (LICS), Technion, Haifa (Israel), August 2022. Speaker: Sean Moss (joint work). Main page here.
When measurable spaces don't have enough points. Invited talk, Seminário de Teoria das Categorias, University of Coimbra (Portugal), July 2022. Held online. Related video here.
A category-theoretic proof of the ergodic decomposition theorem. Applied Category Theory 2022, University of Strathclyde, Glasgow (Scotland), July 2022. Main page here, video here.
Introduction to categorical probability. Guest lecture at the Applied Category Theory course, University of Oxford (UK), June 2022.
Compositional Robotics: Mathematics and Tools ICRA 2022 Workshop, Philadelphia, PA (USA), May 2022. Hybrid event. Main page here.

2021:

The rise of quantitative category theory. Topos Colloquium. Topos Institute, Berkeley, CA (USA), November 2021. Held online. Main page here, video here.
Kan extensions are partial colimits. Category Theory 2021, University of Genoa (Italy), August-September 2021. Main page here, video here.
De Finetti's theorem in categorical probability. Applied Category Theory 2021, University of Cambridge (UK), July 2021. Held online. Speaker: Tobias Fritz (joint work). Main page here.
An invitation to categorical probability and statistics with Markov categories. Invited talk, Categories and Companions. Macquarie University, Sydney (Australia), June 2021. Held online. Speaker: Tobias Fritz (joint work). Main page here.
Compositional structure of partial evaluations. Categories and Companions. Macquarie University, Sydney (Australia), June 2021. Held online. Speaker: Brandon Shapiro (joint work). Main page here.
Compositional Robotics: Mathematics and Tools, ICRA 2021 Workshop. ETH Zürich (Switzerland) May 2021. Held online. Main page here.
Weighted limits in metric geometry. Metric geometry seminar. Max Planck Institute for Mathematics in the Sciences, Leipzig (Germany), May 2021. Held online.
Markov categories: towards a syntax for probability. Logics and Semantics for Dummies seminar. University of Cambridge (UK), May 2021. Held online. Link here.
The law of large numbers in categorical probability. Topos Colloquium. Topos Institute, Berkeley, CA (USA), May 2021. Held online. Speaker: Tobias Fritz (joint work). Main page here, video here.
The de Finetti theorem in categorical probability. Seminar talk, Cohomology in algebra, geometry, physics and statistics, Institute of Mathematics, Czech Academy of Sciences (Czech Republic), April 2021. Held online. Speaker: Tobias Fritz (joint work).
Categorical probability, Markov categories and the De Finetti theorem. Seminar talk, New York City Category Theory Seminar, CUNY, New York, NY (USA), March 2021. Speaker: Tobias Fritz (joint work). Held online. Main page here, video here.
Categorical probability, Markov categories and the De Finetti theorem. Impromptu seminar talk, New York City Category Theory Seminar, CUNY, New York, NY (USA), March 2021. Held online. Main page here, video here.
Partial evaluations: the results so far. Invited talk, Seminario de categorías UNAM, Universidad Nacional Autónoma de México, Mexico City (Mexico), February 2021. Held online. Main page here, video here.
Partial evaluations for monads and 2-monads. Invited talk, Seminário de Teoria das Categorias, University of Coimbra (Portugal), February 2021. Held online.
Kan extensions are partial colimits. Invited talk, Masaryk University Algebra Seminar, Masaryk University, Brno (Czech Republic), February 2021. Held online. Main page here, video here.
An introduction to monads. Invited lecture, Applied Compositional Thinking for Engineers (ACT4E), ETH Zürich (Switzerland), January 2021. Held online. Main page here, video here.
Markov categories: randomness and information flow. Oxford Quantum Group Workshop 2021, University of Oxford (UK), January 2021. Held online.

2020 (all events held online):

Colimits as algebraic operations. Invited talk, ItaCaFest. Online seminar, Italy, September 2020. Link and abstract here, video here.
Compositional structure of partial evaluations. MIT Categories Seminar. Massachusetts Institute of Technology, Cambridge, MA (USA), September 2020. Speaker: Brandon Shapiro (joint work). Main page here, video here.
Distribution functors, second-order stochastic dominance and the Blackwell-Sherman-Stein Theorem in Categorical Probability. Keynote talk at the Applied Category Theory Conference, based at MIT, Cambridge, MA (USA), July 2020. Speaker: Tomáš Gonda (joint work). Main page here, video here.
Monads and comonads. Tutorial talk, Applied Category Theory Conference, based at MIT, Cambridge, MA (USA), July 2020. Main page here, video here.
Applied Category Theory Adjoint School, Massachusetts Institute of Technology, Cambridge, MA (USA), June-July 2020. Link here.
Monads, partial evaluations, and rewriting. Principles of Programming and Verification seminar, Boston University, Boston, MA (USA), June 2020. Link here.
Kan extensions are partial colimits. MIT Categories Seminar. Massachusetts Institute of Technology, Cambridge, MA (USA), June 2020. Held online. Main page here, video here.
Probability monads and stochastic dominance, Categorical Probability and Statistics Workshop, University of Ottawa, ON (Canada), June 2020. Main page here, video here.
What is a probability monad? Tutorial talk, Categorical Probability and Statistics Workshop, University of Ottawa, ON (Canada), June 2020. Main page here, video here.
Monads, partial evaluations, and rewriting. Mathematical Foundations of Programming Semantics 36, Paris (France), June 2020. Main page here, video here.
The support as a morphism from probability to possibility. TallCat Seminar. Tallinn Institute of Technology (TalTech), Tallinn (Estonia), May 2020. Speaker: Sharwin Rezagholi (joint work).
Stochastic orders and Kantorovich duality. Metric geometry seminar. Max Planck Institute for Mathematics in the Sciences, Leipzig (Germany), April 2020.
Composing partial evaluations. MIT Categories Seminar. Massachusetts Institute of Technology, Cambridge, MA (USA), March 2020. Main page here, video here.

2019:

The compositional structure of partial evaluations. Invited talk, University of Ottawa, ON (Canada), November 2019.
Partial evaluations, the bar construction, and second-order stochastic dominance. Project at the Applied Category Theory Adjoint School, University of Oxford (UK), July 2019. Link here.
The support is a morphism of monads. Applied Category Theory 2019, University of Oxford (UK), July 2019. Link here.
Categorical Probability: Results and Challenges. 4th Symposium on Compositional Structures, Chapman University, CA (USA), May 2019. Speaker: Tobias Fritz (joint work). Link here.
The support is a morphism of monads. 3rd Symposium on Compositional Structures, University of Oxford (UK), March 2019. Speaker: Sharwin Rezagholi (joint work). Link here.
Towards probability theory without measure theory. Invited talk, MIT Categories Seminar. Massachusetts Institute of Technology, Cambridge, MA (USA), February 2019. Speaker: Tobias Fritz (joint work). Main page here, video here.

2018:

Monads, Partial Evaluations, and Martingales. Invited talk, MIT Categories Seminar. Massachusetts Institute of Technology, Cambridge, MA (USA), October 2018. Link and abstract here.
Monads, Partial Evaluations, and Rewriting. 1st Symposium on Compositional Structures, University of Birmingham (United Kingdom), September 2018. Link and slides here.
On the Operational Meaning of the Bar Construction. Category Theory 2018, University of Azores, Ponta Delgada (Portugal), July 2018. Abstract and slides here.
Bimonoidal Structure of Probability Monads. Mathematical Foundations of Programming Semantics 34, Halifax, NS (Canada), June 2018. Abstract and paper here.
A Kantorovich Monad for Ordered Spaces. 26th Foundational Methods in Computer Science Workshop, Mount Allison University, Sackville, NB (Canada), May-June 2018. Abstract and slides here.

Earlier:

The Wasserstein Monad in Categorical Probability. Category Theory 2017, Vancouver, BC (Canada), July 2017. Abstract and slides here.
A Conceptual Viewpoint on Information Decomposition. Invited talk, Perimeter Institute for Theoretical Physics, Waterloo, ON (Canada), November 2016. Abstract and video here.
Decomposition of Markov Kernels. 10th Workshop on Uncertainty Processing, Monínec (Czechia), September 2015. Link and abstract here.
Synergy as a Linear Operator. Guided Self Organization 2014, Freiburg (Germany), December 2014. Link here.

Contact

Email: paolo.perrone at cs.ox.ac.uk
Address:
University of Oxford
Department of Computer Science
Wolfson Building, Parks Road
Oxford, OX1 3QD, U.K.
Office: Wolfson Building 331