The point of this paper is that various notions of algebraic theory arise as choices of βeleuthericβ systems of arities in your enriching category.
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)
exists in $[\mathcal{T} , \mathcal{C} ](A, B) = \int_{J \in \mathcal{J}} \mathcal{C}}\mathcal{V}$ 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
.
The system of arities given by $\mathcal{T}}\mathsf{FinCard} \hookrightarrow \mathsf{Set}$ corresponds to ordinary Lawvere theories.
For a closed monoidal $\mathcal{V}$, the system of arities given by $\mathcal{V}_{fp} \hookrightarrow \mathcal{V}$ gives Powerβs enriched Lawvere theories.
Borceux and Dayβs enriched finite power theories come from $\mathbb{N}_\mathcal{V} \to \mathcal{V}$.
This paper uses the term βeleuthericβ. Eleutheria is an ancient and modern Greek term for, and personification of, liberty.
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), 83β93. TAC