Rigidification of Homotopy Algebras over Finite Product Sketches

Abstract

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.