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The notion of enriched Lawvere theory is a generalization of Lawvere theories to the setting of enriched categories.
An enriched Lawvere theory over a locally finitely presentable? $V$-enriched category $A$ is a small $V$-enriched category $L$ together with a $V$-enriched functor $J\colon A_f^{op}\to L$ that induces an identity map on the set of objects. Here $A_f$ denotes the full subcategory of finitely presentable objects in $A$. The category $V$ is a locally finitely presentable closed symmetric monoidal category.
An algebra over an enriched Lawvere theory is an object $X$ in $A$ equipped with a functor $M\colon L\to V$ whose precomposition with $J$ is isomorphic to the representable functor of $X$.
For the case $A=Set$ the above definition recovers the usual notion of a Lawvere theory and an algebra over a Lawvere theory.
The forgetful functor from algebras to $A$ is a finitary enriched monadic functor? (over $V$).
The category of enriched Lawvere theories over $A$ is equivalent to the category of finitary enriched monads? over $A$. The corresponding $V$-enriched categories of algebras are also equivalent.
John Power, Enriched Lawvere theories, tac
Koki Nishizawa?, John Power, Lawvere theories enriched over a general base. Journal of Pure and Applied Algebra 213, Issue 3, March 2009, Pages 377–386. (pdf),
MR2477057, Zbl:1158.18003, doi:10.1016/j.jpaa.2008.07.009.
Sam Staton, Freyd categories are
enriched Lawvere theories_, pdf
Rory B. B. Lucyshyn-Wright, Enriched algebraic theories and monads for a system of arities. arXiv:1511.02920.
Stephen Lack, John Power, Gabriel-Ulmer Duality and Lawvere Theories Enriched over a General Base, pdf
Richard Garner, John Power, An enriched view on the extended finitary monad–Lawvere theory correspondence, (arXiv:1707.08694)
Last revised on June 5, 2018 at 03:52:14. See the history of this page for a list of all contributions to it.