Resumen de Rigidification of homotopy algebras over finite product sketches
Bruce R. Corrigan Salter
Multi-sorted algebraic theories provide a formalism for describing various structures on spaces that are of interest in homotopy theory. The results of Badzioch (2002) [1] and Bergner (2006) [4] 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 simplicial multi-sorted algebraic theory.
