  Multi-sorted algebraic theories provide a formalism for describing various
structures on spaces that are of interest in homotopy theory. The results of
Badzioch and Bergner showed that an interesting feature of this formalism is
the following rigidification property. Every multi-sorted algebraic theory
defines a category of homotopy algebras, i.e. a category of spaces equipped
with certain structure that is to some extent homotopy invariant. Each such
homotopy algebra can be replaced by a weakly equivalent strict algebra which is
a purely algebraic structure on a space. The equivalence between the homotopy
categories of loop spaces and topological groups is a special instance of this
result.
  In this paper we will introduce the notion of a finite product sketch which
is a useful generalization of a multi-sorted algebraic theory. We will show
that in the setting of finite product sketches we can still obtain results
paralleling these of Badzioch and Bergner, although a rigidification of a
homotopy algebra over a finite product sketch is given by a strict algebra over
an associated multi-sorted algebraic theory.
