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The main result is a multi-sorted generalization of a theorem by Badzioch:
Theorem. Let be an algebraic theory. Any homotopy -algebra is weakly equivalent as a homotopy -algebra to a strict -algebra.
The main result is stated:
Theorem. Let be a multi-sorted algebraic theory. Any homotopy -algebra is weakly equivalent as a homotopy -algebra to a strict -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 where is a group and is a set.
(Example 3.2) Pairs as above, and an action of on .
(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)