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   {"id":"aristoteMonotoneWeakDistributive2025d","abstract":"In both the category of sets and the category of compact Hausdorff spaces, there is a monotone weak distributive law that combines two layers of non-determinism. Noticing the similarity between these two laws, we study whether the latter can be obtained automatically as a weak lifting of the former. This holds partially, but does not generalize to other categories of algebras. We then characterize when exactly monotone weak distributive laws over powerset monads in categories of algebras exist, on the one hand exhibiting a law combining probabilities and non-determinism in compact Hausdorff spaces and showing on the other hand that such laws do not exist in a lot of other cases.","author":[{"family":"Aristote","given":"Quentin"}],"citation-key":"aristoteMonotoneWeakDistributive2025d","event-place":"Paris, France","event-title":"PPS days (IRIF)","genre":"seminar","issued":{"date-parts":[["2025",7,3]]},"language":"en","note":"slides: https://gitlab.math.univ-paris-diderot.fr/-/project/1062/uploads/5d4b2a49c1a59a680d60bffa2b3ee01b/monotone-wdl-algebras.pdf","title":"Monotone weak distributive laws over the lifted powerset monad in categories of algebras","type":"speech","URL":"https://www.irif.fr/rencontres/pps2025/index"},
   {"id":"aristoteMonotoneWeakDistributive2025e","abstract":"Within the study of the semantics of programming languages, computational effects may be modelled with monads, and weak distributive laws between monads are then a tool to combine two such effects. The first part of the talk will be dedicated to introducing (monotone) (weak) distributive laws.\n\nIn both the category of sets and the category of compact Hausdorff spaces, there is a monotone weak distributive law that combines two layers of non-determinism. Noticing the similarity between these two laws, we study in the second part of the talk whether the latter can be obtained automatically as some sort of lifting of the former.\n\nMore specifically, we show how a framework for constructing monotone weak distributive laws in regular categories lifts to categories of algebras, giving a full characterization for the existence of monotone weak distributive laws therein. We then exhibit such a law, combining probabilities and non-determinism, in compact Hausdorff spaces; but we also show how such laws do not exist in a lot of other cases.","author":[{"family":"Aristote","given":"Quentin"}],"citation-key":"aristoteMonotoneWeakDistributive2025e","event-place":"Chambéry, France","event-title":"LIMD seminar (LAMA)","genre":"seminar","issued":{"date-parts":[["2025",12,18]]},"language":"en","note":"slides: https://gitlab.math.univ-paris-diderot.fr/-/project/1062/uploads/524cfdd95d3bc0c75b4940fdd65e2275/monotone-wdl-algebras.pdf","title":"Monotone weak distributive laws over the lifted powerset monad in categories of algebras","type":"speech","URL":"https://www.lama.univ-smb.fr/seminars/limd"},
   {"id":"aristoteMonotoneWeakDistributive2025f","abstract":"Joint work with Umberto Tarantino. We provide a bicategorical generalization of Barr's landmark 1970 paper, in which he describes how to extend Set-monads to relations and uses this to characterize topological spaces as the relational algebras of the ultrafilter monad. With two-sided discrete fibrations playing the role of relations in a bicategory, we first characterize, in terms of exact squares, when pseudomonads on a bicategory extend to its bicategory of two-sided discrete fibrations. As a wide class of examples, we show that every Set-monad induces a pseudomonad on the 2-category of categories satisfying our criterion and thus extending to profunctors. Among these, we then focus on the ultracompletion pseudomonad, whose pseudoalgebras are ultracategories: we characterize the normalized lax algebras of its profunctorial extension as ultraconvergence spaces, a recently-introduced categorification of topological spaces.","author":[{"family":"Aristote","given":"Quentin"}],"citation-key":"aristoteMonotoneWeakDistributive2025f","event-place":"Paris, France","event-title":"PPS days (IRIF)","genre":"seminar","issued":{"date-parts":[["2026",5,21]]},"language":"en","note":"slides: https://gitlab.math.univ-paris-diderot.fr/-/project/1195/uploads/85d2421a661b6d2fc3c05f5a26a7785b/profunctorial-algebras.pdf","title":"Profunctorial algebras","type":"speech","URL":"https://www.irif.fr/rencontres/pps2026/index"},
+  {"id":"aristoteMonotoneWeakDistributive2025g","abstract":"In 1970, Barr characterized which Set-functors and natural transformations extend to the locally posetal category of sets and relations. His goal was to then describe topological spaces as certain \"relational algebras\", but his result also proved relevant for the study of linear logic (through its weighted relational model), transition systems (through coalgebras) and effectful programming (through distributive laws).\n\nIn this talk I will review the main idea behind Barr's characterization --- seeing relations as spans --- and how it has been adapted to more general settings. I will focus on two of my own results: a characterization of extensions to the Kleisli categories of powerset-like monads, with the aim of building weak distributive laws involving non-determinism, and, as jww. Umberto Tarantino, a characterization of extensions to the bicategory of profunctors, with the aim of adapting Barr's description of topological spaces to the 2-dimensional setting.","author":[{"family":"Aristote","given":"Quentin"}],"citation-key":"aristoteMonotoneWeakDistributive2025g","event-place":"Villetaneuse, France","event-title":"LoCal seminar (LIPN)","genre":"seminar","issued":{"date-parts":[["2026",6,10]]},"language":"en","note":"slides: https://gitlab.math.univ-paris-diderot.fr/-/project/1274/uploads/3cdc0688535f912652f703e2c89bf9b4/relational-extensions.pdf","title":"Relational extensions","type":"speech","URL":"https://lipn.univ-paris13.fr/pages/fr/research/teams/local/seminars.html"},
   {"id":"aristoteMulticategoricalFrameworkMinimization2024","abstract":"Joint work with Daniela Petrişan. This paper provides a unifying category-theoretic framework for minimization and learning algorithms for bottom-up tree automata with effects. Our aim is two-fold: encompass existing algorithms for various forms of tree automata – deterministic bottom-up tree automata, residual finite tree automata, tree automata weighted over a field – and instantiate the abstract framework in order to obtain new results – tree automata weighted over principal ideal domains (PIDs).","author":[{"family":"Aristote","given":"Quentin"}],"citation-key":"aristoteMulticategoricalFrameworkMinimization2024","container-title":"Tree Automata","event-place":"Bordeaux, France","event-title":"Highlights of Logic, Games and Automata (Highlights 2024)","genre":"conference","issued":{"date-parts":[["2024",9,18]]},"language":"en","note":"slides: https://gitlab.math.univ-paris-diderot.fr/-/project/1003/uploads/c415dcc8a1e22c5d91cb91cfd8d6d55f/minimal-tree-automata.pdf","title":"Multicategorical framework for minimization and learning of bottom-up tree automata with effects","type":"speech","URL":"https://highlights-conference.org/2024/"},
   {"id":"aristoteNumberRingsAre2025","abstract":"We study automata weighted over number rings, that is, rings of integers in an algebraic number field.\n\nWe show that number rings are what we call \"almost strong Fatou\": if an n-state automaton weighted in a number field recognizes an integer-valued series, then it admits an equivalent n+1-state automaton weighted in the corresponding ring of integers.\n\nWe give a polynomial time procedure for computing such an n+1-state integer-weighted automaton, and show that deciding whether this automaton is minimal among all equivalent integer-weighted automata is as hard as PIP, a problem for which the best known algorithm is in quantum polynomial time.\n\nThis procedure provides a reduction between learning problems for automata weighted in number rings and in fields, and we show that it is an instance of a more general, abstract reduction procedure between learning problems for automata valued in two categories related by well-behaved functors.","author":[{"family":"Aristote","given":"Quentin"}],"citation-key":"aristoteNumberRingsAre2025","event-place":"Marseille, France","event-title":"Combinatorics, Automata and Number Theory 2025","genre":"conference","issued":{"date-parts":[["2025",9,29]]},"language":"en","note":"slides: https://gitlab.math.univ-paris-diderot.fr/-/project/996/uploads/73d3ba029f37d98b7090cce6fd81d8f1/number-rings-almost-strong-fatou.pdf\nposter: https://gitlab.math.univ-paris-diderot.fr/-/project/996/uploads/6c9a3e6d87931780518a540aabb432de/removing-fractions.pdf","title":"Number rings are almost strong Fatou","type":"speech","URL":"http://www.discmath.ulg.ac.be/CANT2025/"},
   {"id":"aristoteOpenPowerObjectsCategories2025","abstract":"https://conference.math.muni.cz/ct2025/data/uploads/abstracts/aristote.pdf","author":[{"family":"Aristote","given":"Quentin"}],"citation-key":"aristoteOpenPowerObjectsCategories2025","event-place":"Brno, Czech Republic","event-title":"International Category Theory Conference (CT 2025)","genre":"conference","issued":{"date-parts":[["2025",7,18]]},"language":"en","note":"slides: https://gitlab.math.univ-paris-diderot.fr/-/project/1062/uploads/620105dfc6c8ec7fef78737368585a25/open-powerobjects-algebras.pdf","title":"Open Power-Objects in Categories of Algebras","type":"speech","URL":"https://conference.math.muni.cz/ct2025/"},