Free models of T -algebraic theories computed as Kan extensions
نویسندگان
چکیده
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.
منابع مشابه
MPhil in Advanced Computer Science Advanced
SYLLABUS A range of topics for the course follows. 1. Algebraic theories: universal algebra; equational logic; soundness and completeness; theory translations and constructions. 2. Algebras: initial algebras; induction principle; recursive domain equations; free algebras. 3. Presheaves: cartesian closure; essential geometric morphisms; free cocompletions; Kan extensions; coends. 4. Simply typed...
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