The Azimuth Project
Enriched algebraic theories and monads for a system of arities

Idea

The point of this paper is that various notions of algebraic theory arise as choices of β€œeleutheric” systems of arities in your enriching category.

Definitions

Definition. A system of arities in 𝒱\mathcal{V} is a fully faithful strong symmetric monoidal 𝒱\mathcal{V}-functor j:π’₯→𝒱j \colon \mathcal{J} \to \mathcal{V}.

Definition. A 𝒱\mathcal{V}-enriched algebraic theory with arities π’₯β†ͺ𝒱\mathcal{J} \hookrightarrow \mathcal{V} (briefly, a π’₯\mathcal{J}-theory) is a 𝒱\mathcal{V}-category 𝒯\mathcal{T} equipped with a π’₯\mathcal{J}-cotensor-preserving identity-on-objects 𝒱\mathcal{V}-functor Ο„:π’₯ op→𝒯\tau \colon \mathcal{J}^{op} \to \mathcal{T}.

Definition. Let 𝒯\mathcal{T} be a π’₯\mathcal{J}-theory. Given a 𝒱\mathcal{V}-category π’ž\mathcal{C}, a 𝒯\mathcal{T}-algebra in π’ž\mathcal{C} is a π’₯\mathcal{J}-cotensor-preserving 𝒱\mathcal{V}-functor A:π’―β†’π’žA \colon \mathcal{T} \to \mathcal{C}. We shall often call 𝒯\mathcal{T}-algebras in 𝒱\mathcal{V} simply 𝒯\mathcal{T}-algebras.

Definition. Let 𝒯\mathcal{T} be a π’₯\mathcal{J}-theory. We call 𝒱\mathcal{V}-natural transformations between 𝒯\mathcal{T}-algebras 𝒯\mathcal{T}-homomorphisms. If the object of 𝒱\mathcal{V}-natural transformations [𝒯,π’ž](A,B)=∫ J∈π’₯π’ž(AJ,BJ)[\mathcal{T} , \mathcal{C}](A, B) = \int_{J \in \mathcal{J}} \mathcal{C} (AJ, BJ) exists in 𝒱\mathcal{V} for all 𝒯\mathcal{T}-algebras AA, BB in π’ž\mathcal{C}, then 𝒯\mathcal{T} -algebras in π’ž\mathcal{C} are the objects of an evident 𝒱\mathcal{V}-category π’―βˆ’Alg π’ž\mathcal{T}-Alg_{\mathcal{C}}. We denote π’―βˆ’Alg 𝒱\mathcal{T}-Alg_{\mathcal{V}} by just π’―βˆ’Alg\mathcal{T}-Alg.

Examples

  • The system of arities given by FinCardβ†ͺSet\mathsf{FinCard} \hookrightarrow \mathsf{Set} corresponds to ordinary Lawvere theories.

  • For a closed monoidal 𝒱\mathcal{V}, the system of arities given by 𝒱 fpβ†ͺ𝒱\mathcal{V}_{fp} \hookrightarrow \mathcal{V} gives Power’s enriched Lawvere theories.

  • Borceux and Day’s enriched finite power theories come from β„• 𝒱→𝒱\mathbb{N}_\mathcal{V} \to \mathcal{V}.

Random facts

This paper uses the term β€œeleutheric”. Eleutheria is an ancient and modern Greek term for, and personification of, liberty.

References

  • Max Kelly?, Basic concepts of enriched category theory?, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press 1982, 245 pp.; remake: TAC reprints 10, tac

  • J. Power?, Enriched Lawvere theories?, Theory and Applications of Categories 6 (1999), 83–93. TAC