The Azimuth Project
Rigidification of algebras over multi-sorted theories (Rev #2, changes)

Showing changes from revision #1 to #2: Added | Removed | Changed

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