Multi-sorted algebraic theory

A multi-sorted algebraic theory generalizes the notion of a Lawvere theory.

We give the definition found in Rigidification of algebras over multi-sorted theories. We don’t know where it is first given.

**Definition.** Given a set $S$, an $S$-sorted algebraic theory $\mathcal{T}$ is a small category with objects $T_{\underline{\alpha^n}}$ where $\underline{\alpha^n} = \langle \alpha_1, \dots, \alpha_n \rangle$ for $\alpha_i \in S$ and $n \geq 0$ varying, and such that each $T_{\underline{\alpha^n}}$ is equipped with an isomorphism $T_{\underline{\alpha^n}} \cong \prod_{i=1}^n T_{\alpha_i}$.

- Julia E. Bergner?, Rigidification of algebras over multi-sorted theories,
*Algebraic & Geometric Topology 6, pages 1925-1955, 2006. arXiv:0508152*

category: mathematical methods