Frobenius structures in star-autonomous categories

Luigi Santocanale (1, 2, 3) , Cédric de Lacroix (1, 2)

1 LIS - Laboratoire d'Informatique et des Systèmes (LIS) (Marseille, Toulon)
2 LIRICA - Logique, Interaction, Raisonnement et Inférence, Complexité, Algèbre
3 ACRO - Algorithmique, Combinatoire et Recherche Opérationnelle

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.

Mots clés: Quantale; Frobenius quantale; Girard quantale; associative algebra; star-autonomous category; nuclear object; adjoint

Domaines: Catégories et ensembles [math.CT]; Logique en informatique [cs.LO]; Logique [math.LO]

Luigi Santocanale, Cédric de Lacroix. Frobenius structures in star-autonomous categories. 2022. ⟨hal-03739197⟩