#### Volume 16, issue 2 (2016)

 Recent Issues
Author Index
 The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 To Appear Other MSP Journals
Rectification of weak product algebras over an operad in $\mathcal{C}\mathit{at}$ and $\mathcal{T}\mathit{op}$ and applications

### Zbigniew Fiedorowicz, Manfred Stelzer and Rainer M Vogt

Algebraic & Geometric Topology 16 (2016) 711–755
##### 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 $\mathsc{C}\mathit{at}$, whereas their construction gives rise to simplicial categories. As an application we show that a simplicial algebra over a $\Sigma$–free $\mathsc{C}\mathit{at}$ operad $\mathsc{O}$ is functorially weakly equivalent to a $\mathsc{C}\mathit{at}$ algebra over $\mathsc{O}$. When combined with the results of a previous paper, this allows us to conclude that, up to weak equivalences, the category of $\mathsc{O}$–categories is equivalent to the category of $B\mathsc{O}$–spaces, where $B:\phantom{\rule{0.3em}{0ex}}\mathsc{C}\mathit{at}\to \mathsc{T}\phantom{\rule{0.3em}{0ex}}\mathit{op}$ 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 $\mathsc{C}\mathit{at}$.

##### Keywords
operads, categories, loop space machines
Primary: 18D50
Secondary: 55P48
##### Publication
Received: 23 May 2014
Revised: 3 June 2015
Accepted: 23 June 2015
Published: 26 April 2016
##### Authors
 Zbigniew Fiedorowicz Department of Mathematics The Ohio State University 100 Mathematics Building 231 West 18th Avenue Columbus, OH 43210-1174 USA http://https://people.math.osu.edu/fiedorowicz.1/ Manfred Stelzer Fachbereich Mathematik/Informatik Universität Osnabrück Albrechtstrasse 28a D-49076 Osnabrück Germany Rainer M Vogt Fachbereich Mathematik/Informatik Universität Osnabrück Albrechtstrasse 28a D-49069 Osnabrück Germany