nLab enriched Lawvere theory

Idea

The notion of enriched Lawvere theory or enriched algebraic theory is a generalization of Lawvere theories to the setting of enriched categories.

Definitions and elementary properties

Definition

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.

Definition

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 $X$.

Proposition

For the case $A=Set$ the above definition recovers the usual notion of a Lawvere theory and an algebra over a Lawvere theory.

Proposition

The forgetful functor from algebras to $A$ is a finitary enriched monadic functor? (over $V$).

Proposition

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.

References

• Eduardo Dubuc, Enriched semantics-structure (meta) adjointness, Rev. Un. Mat. Argentina 25 (1970): 5-26.

• 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)

• Nathanael Arkor, Monadic and Higher-Order Structure, PhD thesis, Cambridge (2022) [doi:10.17863/CAM.86347, pdf, pdf]