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

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

Main Result

The main result is a multi-sorted generalization of a theorem by Badzioch:

Theorem. Let 𝒯\mathcal{T} be an algebraic theory. Any homotopy 𝒯\mathcal{T}-algebra is weakly equivalent as a homotopy 𝒯\mathcal{T}-algebra to a strict 𝒯\mathcal{T}-algebra.

The main result is stated:

Theorem. Let 𝒯\mathcal{T} be a multi-sorted algebraic theory. Any homotopy 𝒯\mathcal{T}-algebra is weakly equivalent as a homotopy 𝒯\mathcal{T}-algebra to a strict 𝒯\mathcal{T}-algebra.

In this paper, they are considering simplicial algebras, models of the theories in simplicial sets.

Examples

Several examples of multi-sorted theories are given.

  • (Example 3.2) Pairs (G,X)(G,X) where GG is a group and XX is a set.

  • (Example 3.2) Pairs (G,X)(G,X) as above, and an action of GG on XX.

  • (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)