Rectification of weak product algebras over an operad in $\mathcal{C}\mathit{at}$ and $\mathcal{T}\mathit{op}$ 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 $\mathcal{C}\mathit{at}$, whereas their construction gives rise to simplicial categories. As an application we show that a simplicial algebra over a $\Sigma$–free $\mathcal{C}\mathit{at}$ operad $\mathcal{O}$ is functorially weakly equivalent to a $\mathcal{C}\mathit{at}$ 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 $B:\mathcal{C}\mathit{at}\to \mathcal{T}\mathit{op}$ 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 $\mathcal{C}\mathit{at}$.