HAL : hal-00339331, version 1
 Free models of T-algebraic theories computed as Kan extensions
 (2008)
 One fundamental aspect of Lawvere's categorical semantics is that every algebraic theory (eg. of monoid, of Lie algebra) induces a free construction (eg. of free monoid, of free Lie algebra) computed as a Kan extension. Unfortunately, the principle fails when one shifts to linear variants of algebraic theories, like Adams and Mac Lane's PROPs, and similar PROs and PROBs. Here, we introduce the notion of T-algebraic theory for a pseudomonad T -- a mild generalization of equational doctrine -- in order to describe these various kinds of algebraic theories''. Then, we formulate two conditions (the first one combinatorial, the second one algebraic) which ensure that the free model of a T-algebraic theory exists and is computed as an Kan extension. The proof is based on Bénabou's theory of distributors, and of an axiomatization of the colimit computation in Wood's proarrow equipments.
 1 : Preuves, Programmes et Systèmes (PPS) CNRS : UMR7126 – Université Paris VII - Paris Diderot
 Domaine : Mathématiques/Catégories et ensemblesMathématiques/Algèbre commutative
 Mots Clés : Lawvere theories – PROs – PROPs – PROBs – operads – Kan extensions – distributors – enriched categories – free constructions – algebras – coalgebras
Liste des fichiers attachés à ce document :
 PDF
 free-models.pdf(415.2 KB)
 hal-00339331, version 1 http://hal.archives-ouvertes.fr/hal-00339331 oai:hal.archives-ouvertes.fr:hal-00339331 Contributeur : Nicolas Tabareau <> Soumis le : Lundi 17 Novembre 2008, 15:43:49 Dernière modification le : Lundi 17 Novembre 2008, 17:05:47