Definition. Given a set , an -sorted algebraic theory is a small category with objects where for and varying, and such that each is equipped with an isomorphism .
Definition. Given an –sorted theory , a (strict simplicial) -algebra is a product-preserving functor . Here, product-preserving means that the canonical map , induced by the projections for all , is an isomorphism of simplicial sets.
Definition. Given an –sorted theory , a homotopy -algebra is a functor which preserves products up to homotopy, i.e. for all the canonical map , induced by the projections for all , is a weak equivalence of simplicial sets.
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.
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) The theory for operads has a sort for each natural number, corresponding to the arity of the operation.
(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 author’s comment and a supplement in: Reprints in Theory and Applications of Categories 5 (2004) pp 1–121. (abstract)