The main result is a multi-sorted generalization of a theorem by Badzioch:
Theorem. Let $\mathcal{T}$ be an algebraic theory. Any homotopy $\mathcal{T}$-algebra is weakly equivalent as a homotopy $\mathcal{T}$-algebra to a strict $\mathcal{T}$-algebra.
The main result is stated:
Theorem. Let $\mathcal{T}$ be a multi-sorted algebraic theory. Any homotopy $\mathcal{T}$-algebra is weakly equivalent as a homotopy $\mathcal{T}$-algebra to a strict $\mathcal{T}$-algebra.
In this paper, they are considering simplicial algebras, models of the theories in simplicial sets.
Several examples of multi-sorted theories are given.
(Example 3.2) Pairs $(G,X)$ where $G$ is a group and $X$ is a set.
(Example 3.2) Pairs $(G,X)$ as above, and an action of $G$ on $X$.
(Example 3.3) Ring-module pairs.
(Example 3.4) Operads.
(Example 3.5) Categories with a fixed object set.
B. Badzioch?, Algebraic theories in homotopy theory?, Ann. of Math. (2) 155, pages 895-913, 2002.
William Lawvere?, Functorial Semantics of Algebraic Theories? , Ph.D. thesis Columbia University (1963). Published with an authorβs comment and a supplement in: Reprints in Theory and Applications of Categories 5 (2004) pp 1β121. (abstract)