2001/10/09 by Bernard Badzioch, Badzioch, Bernard · 2 citations
Mathematics · #18G55 (Secondary) #55P48 (Primary) #Algebraic Topology (math.AT) #FOS: Mathematics #math.AT #msc:18G55 #msc:55P48
paper · pdf · doi:10.48550/arxiv.math/0110101
19 pages, published version
arxiv created 2004/11/22 · arxiv updated 2009/11/30
An algebraic theory T is a category with objects t0,t2... such that for each n the object tn is an n-fold categorical product of t1. A strict T-algebra is a product preserving functor A: T→ Spaces. Lawvere showed that for a suitable choice of T giving such an algebra amounts to providing the space A(t1) with a familiar structure of a monoid group, ring, Lie algebra... Given a functor X: T→ 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(t1) can be rigidified to a strict algebraic structure on a space weakly equivalent to X(t1). 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.