2026-08-07T22:00:46Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:17479 2024-03-06T11:00:05Z ddc:004 open_access

Frobenius Structures in Star-Autonomous Categories de Lacroix, Cédric Santocanale, Luigi Quantale Frobenius quantale Girard quantale associative algebra star-autonomous category nuclear object adjoint It is known that the quantale of sup-preserving maps from a complete lattice to itself is a Frobenius quantale if and only if the lattice is completely distributive. Since completely distributive lattices are the nuclear objects in the autonomous category of complete lattices and sup-preserving maps, we study the above statement in a categorical setting. We introduce the notion of Frobenius structure in an arbitrary autonomous category, generalizing that of Frobenius quantale. We prove that the monoid of endomorphisms of a nuclear object has a Frobenius structure. If the environment category is star-autonomous and has epi-mono factorizations, a variant of this theorem allows to develop an abstract phase semantics and to generalise the previous statement. Conversely, we argue that, in a star-autonomous category where the monoidal unit is a dualizing object, if the monoid of endomorphisms of an object has a Frobenius structure and the monoidal unit embeds into this object as a retract, then the object is nuclear. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Cédric de Lacroix and Luigi Santocanale 2023 Is Part Of LIPIcs, Volume 252, 31st EACSL Annual Conference on Computer Science Logic (CSL 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CSL.2023.18 urn:nbn:de:0030-drops-174798 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.18 eng https://creativecommons.org/licenses/by/4.0/legalcode