Rectification of weak product algebras over an operad in Cat and Top and applications

Abstract

We develop an alternative to the May-Thomason construction used to compare operad-based infinite loop machines to those of Segal, which rely on weak products. Our construction has the advantage that it can be carried out in Cat, whereas their construction gives rise to simplicial categories. As an application we show that a simplicial algebra over a Sigma-free Cat operad O is functorially weakly equivalent to a Cat algebra over O. When combined with the results of a previous paper, this allows us to conclude that, up to weak equivalences, the category of O-categories is equivalent to the category of BO-spaces, where B: Cat -> Top is the classifying space functor. In particular, n-fold loop spaces (and more generally E-n spaces) are functorially weakly equivalent to classifying spaces of n-fold monoidal categories. Another application is a change of operads construction within Cat.