  An algebraic theory $T$ is a category with objects $t_0,t_2...$ such that for
each $n$ the object $t_n$ is an $n$-fold categorical product of $t_1$. A strict
$T$-algebra is a product preserving functor $A: T\to Spaces$. Lawvere showed
that for a suitable choice of T giving such an algebra amounts to providing the
space $A(t_1)$ with a familiar structure of a monoid group, ring, Lie
algebra...
  Given a functor $X: T\to Spaces$ which preserves products up to a weak
equivalence we show that $X$ is more or less canonically weakly equivalent to a
strict $T$-algebra $LX$. Thus any `homotopy' algebraic structure on the space
$X(t_1)$ can be rigidified to a strict algebraic structure on a space weakly
equivalent to $X(t_1)$. This fact can be interpreted as a generalization of the
results establishing equivalence of homotopy theories of loop spaces and
simplicial groups, products of Eilenberg-Mac Lane spaces and abelian monoids
etc.
