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 that of Segal, which relies 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 $Σ$-free $Cat$ operad $\mathcal{O}$ is functorially weakly equivalent to a $Cat$ algebra over $\mathcal{O}$. When combined with the results of a previous paper, this allows us to conclude that up to weak equivalences the category of $\mathcal{O}$-categories is equivalent to the category of $B\mathcal{O}$-spaces, where $B:Cat\to Top$ 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 $Cat$.