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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

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 Σ–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 En spaces) are functorially weakly equivalent to classifying spaces of n–fold monoidal categories. Another application is a change of operads construction within Cat.

operads, categories, loop space machines
Mathematical Subject Classification 2010
Primary: 18D50
Secondary: 55P48
Received: 23 May 2014
Revised: 3 June 2015
Accepted: 23 June 2015
Published: 26 April 2016
Zbigniew Fiedorowicz
Department of Mathematics
The Ohio State University
100 Mathematics Building
231 West 18th Avenue
Columbus, OH 43210-1174
Manfred Stelzer
Fachbereich Mathematik/Informatik
Universität Osnabrück
Albrechtstrasse 28a
D-49076 Osnabrück
Rainer M Vogt
Fachbereich Mathematik/Informatik
Universität Osnabrück
Albrechtstrasse 28a
D-49069 Osnabrück