The extended rational homotopy theory of operads

TL;DR

This paper develops a rational homotopy theory for operads in simplicial sets, extending previous work by allowing non-reduced arity one terms, and shows how the rational homotopy type is determined by a cochain dg-cooperad under certain conditions.

Contribution

It extends the rational homotopy theory of operads to include non-reduced arity one terms, connecting it with cochain dg-cooperads.

Findings

Rational homotopy type is determined by a cochain dg-cooperad.

Extension of rational homotopy theory to non-reduced arity one operads.

Applicable when underlying spaces are connected, nilpotent, with finite type rational cohomology.

Abstract

In this paper, we set up a rational homotopy theory for operads in simplicial sets whose term of arity one is not necessarily reduced to an operadic unit, extending results obtained by the author in the book "Homotopy of operads and Grothendieck-Teichm\"uller groups". In short, we prove that the rational homotopy type of such an operad is determined by a cooperad in cochain differential graded algebras (a cochain Hopf dg-cooperad for short) as soon as the Sullivan rational homotopy theory works for the spaces underlying our operad (e.g. when these spaces are connected, nilpotent, and have finite type rational cohomology groups).