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**Published in:** LIPIcs, Volume 270, 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023)

Introduced in the 1990s in the context of the algebraic approach to graph rewriting, gs-monoidal categories are symmetric monoidal categories where each object is equipped with the structure of a commutative comonoid. They arise for example as Kleisli categories of commutative monads on cartesian categories, and as such they provide a general framework for effectful computation. Recently proposed in the context of categorical probability, Markov categories are gs-monoidal categories where the monoidal unit is also terminal, and they arise for example as Kleisli categories of commutative affine monads, where affine means that the monad preserves the monoidal unit.
The aim of this paper is to study a new condition on the gs-monoidal structure, resulting in the concept of weakly Markov categories, which is intermediate between gs-monoidal categories and Markov ones. In a weakly Markov category, the morphisms to the monoidal unit are not necessarily unique, but form a group. As we show, these categories exhibit a rich theory of conditional independence for morphisms, generalising the known theory for Markov categories. We also introduce the corresponding notion for commutative monads, which we call weakly affine, and for which we give two equivalent characterisations.
The paper argues that these monads are relevant to the study of categorical probability. A case at hand is the monad of finite non-zero measures, which is weakly affine but not affine. Such structures allow to investigate probability without normalisation within an elegant categorical framework.

Tobias Fritz, Fabio Gadducci, Paolo Perrone, and Davide Trotta. Weakly Markov Categories and Weakly Affine Monads. In 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 270, pp. 16:1-16:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{fritz_et_al:LIPIcs.CALCO.2023.16, author = {Fritz, Tobias and Gadducci, Fabio and Perrone, Paolo and Trotta, Davide}, title = {{Weakly Markov Categories and Weakly Affine Monads}}, booktitle = {10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023)}, pages = {16:1--16:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-287-7}, ISSN = {1868-8969}, year = {2023}, volume = {270}, editor = {Baldan, Paolo and de Paiva, Valeria}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2023.16}, URN = {urn:nbn:de:0030-drops-188133}, doi = {10.4230/LIPIcs.CALCO.2023.16}, annote = {Keywords: String diagrams, gs-monoidal and Markov categories, categorical probability, affine monads} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

We introduce the notion of a Gödel fibration, which is a fibration categorically embodying both the logical principles of traditional Skolemization (we can exchange the order of quantifiers paying the price of a functional) and the existence of a prenex normal form presentation for every logical formula. Building up from Hofstra’s earlier fibrational characterization of de Paiva’s categorical Dialectica construction, we show that a fibration is an instance of the Dialectica construction if and only if it is a Gödel fibration. This result establishes an intrinsic presentation of the Dialectica fibration, contributing to the understanding of the Dialectica construction itself and of its properties from a logical perspective.

Davide Trotta, Matteo Spadetto, and Valeria de Paiva. The Gödel Fibration. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 87:1-87:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{trotta_et_al:LIPIcs.MFCS.2021.87, author = {Trotta, Davide and Spadetto, Matteo and de Paiva, Valeria}, title = {{The G\"{o}del Fibration}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {87:1--87:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.87}, URN = {urn:nbn:de:0030-drops-145272}, doi = {10.4230/LIPIcs.MFCS.2021.87}, annote = {Keywords: Dialectica category, G\"{o}del fibration, Pseudo-monad} }

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