2 Search Results for "Perrone, Paolo"

Document
DOI: 10.4230/LIPIcs.STACS.2026.65
Demystifying Codensity Monads via Duality

Authors: Fabian Lenke, Nico Wittrock, Stefan Milius, and Henning Urbat

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
Codensity monads provide a universal method to generate complex monads from simple functors. Recently, a wide range of important monads in logic, denotational semantics, and probabilistic computation, such as several incarnations of the ultrafilter monad, the Vietoris monad, and the Giry monad, have been presented as codensity monads, using complex arguments. We propose a unifying categorical approach to codensity presentations of monads, based on the idea of relating the presenting functor to a dense functor via a suitable duality between categories. We prove a general presentation result applying to every such situation and demonstrate that most codensity presentations known in the literature emerge from this strikingly simple duality-based setup, drastically alleviating the complexity of their proofs and in many cases completely reducing them to standard duality results. Additionally, we derive a number of novel codensity presentations using our framework, including the first non-trivial codensity presentations for the filter monads on sets and topological spaces, the lower Vietoris monad on topological spaces, and the expectation monad on sets.
Cite as

Fabian Lenke, Nico Wittrock, Stefan Milius, and Henning Urbat. Demystifying Codensity Monads via Duality. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 65:1-65:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{lenke_et_al:LIPIcs.STACS.2026.65,
  author =	{Lenke, Fabian and Wittrock, Nico and Milius, Stefan and Urbat, Henning},
  title =	{{Demystifying Codensity Monads via Duality}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{65:1--65:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.65},
  URN =		{urn:nbn:de:0030-drops-255549},
  doi =		{10.4230/LIPIcs.STACS.2026.65},
  annote =	{Keywords: Codensity, Monad, Duality}
}
Document
DOI: 10.4230/LIPIcs.CALCO.2023.16
Weakly Markov Categories and Weakly Affine Monads

Authors: Tobias Fritz, Fabio Gadducci, Paolo Perrone, and Davide Trotta

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

Abstract
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.
Cite as

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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