# The Azimuth Project Rigidification of algebras over multi-sorted theories (Rev #1, changes)

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## 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.