# The Azimuth Project Enriched algebraic theories and monads for a system of arities (changes)

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## Idea

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

## Definitions

Definition. A system of arities in $\mathcal{V}$ is a fully faithful strong symmetric monoidal $\mathcal{V}$-functor $j \colon \mathcal{J} \to \mathcal{V}$.

Definition. A $\mathcal{V}$-enriched algebraic theory with arities $\mathcal{J} \hookrightarrow \mathcal{V}$ (briefly, a $\mathcal{J}$-theory) is a $\mathcal{V}$-category $\mathcal{T}$ equipped with a $\mathcal{J}$-cotensor-preserving identity-on-objects $\mathcal{V}$-functor $\tau \colon \mathcal{J}^{op} \to \mathcal{T}$.

Definition. Let $\mathcal{T}$ be a $\mathcal{J}$-theory. Given a $\mathcal{V}$-category $\mathcal{C}$, a $\mathcal{T}$-algebra in $\mathcal{C}$ is a $\mathcal{J}$-cotensor-preserving $\mathcal{V}$-functor $A \colon \mathcal{T} \to \mathcal{C}$. We shall often call $\mathcal{T}$-algebras in $\mathcal{V}$ simply $\mathcal{T}$-algebras.

Definition. Let $\mathcal{T}$ be a $\mathcal{J}$-theory. We call $\mathcal{V}$-natural transformations between $\mathcal{T}$-algebras $\mathcal{T}$-homomorphisms. If the object of $\mathcal{V}$-natural transformations [\mathcal{T} , \mathcal{C} ](A, B) = \int_{J \in \mathcal{J}} \mathcal{C}} (AJ, BJ)$[\mathcal{T} , \mathcal{C}](A, B) = \int_{J \in \mathcal{J}} \mathcal{C} (AJ, BJ)$ exists in  [\mathcal{T} \mathcal{V} , \mathcal{C} ](A, B) = \int_{J \in \mathcal{J}} \mathcal{C}}\mathcal{V} for all $\mathcal{T}$-algebras $A$, $B$ in $\mathcal{C}$, then $\mathcal{T}$ -algebras in $\mathcal{C}$ are the objects of an evident $\mathcal{V}$-category $\mathcal{T}-Alg_{\mathcal{C}}$. We denote $\mathcal{T}-Alg_{\mathcal{V}}$ by just \mathcal{T}}-Alg$\mathcal{T}-Alg$.

## Examples

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

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

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

## Random facts

This paper uses the term “eleutheric”. 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