Detecting Mapping Spaces and Derived Equivalence of Algebraic Theories
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We show that for spaces A which satisfy a certain smallness condition, there is a Lawvere theory T A so that a space X has the structure of a T A -algebra if and only if X is weakly equivalent to a mapping space out of A. In particular, spheres localized at a set of primes satisfy this condition. This problem motivates studying the Morita Theory of algebras in the derived setting. We give conditions which determine when a model category is Quillen equivalent to the model category of algebras over a simpicial algebraic theory, and when two simplicial algebraic theories have Quillen equivalent model categories of algebras.