* [[Julia E. Bergner]], [[Rigidification of algebras over multi-sorted theories]], _Algebraic & Geometric Topology 6, pages 1925-1955, 2006. [arXiv:0508152](https://arxiv.org/abs/math/0508152) ## Main Result 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. ## Examples 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. ## References * [[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 author's comment and a supplement in: Reprints in Theory and Applications of Categories **5** (2004) pp 1--121. ([abstract](http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html)) category:mathematical methods