# Free models of T-algebraic theories computed as Kan extensions

Abstract : 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.
Keywords :
Type de document :
Pré-publication, Document de travail
2008
Domaine :

https://hal.archives-ouvertes.fr/hal-00339331
Contributeur : Nicolas Tabareau <>
Soumis le : lundi 17 novembre 2008 - 15:43:49
Dernière modification le : mardi 11 octobre 2016 - 13:55:39
Document(s) archivé(s) le : mardi 9 octobre 2012 - 15:30:50

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free-models.pdf
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### Identifiants

• HAL Id : hal-00339331, version 1

### Citation

Paul-André Melliès, Nicolas Tabareau. Free models of T-algebraic theories computed as Kan extensions. 2008. <hal-00339331>

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