On mapping spaces of differential graded operads with the commutative operad as target

TL;DR

This paper investigates the homotopy types of mapping spaces from cofibrant E_n-operads to the commutative operad, revealing their classification by ground ring elements and their contractibility.

Contribution

It characterizes the homotopy classes of operad morphisms and proves the contractibility of mapping space components for E_n-operads, extending to E-infinity cases.

Findings

Homotopy class of morphisms determined by a multiplicative constant.

Connected components of Map(E_n,C) correspond to the ground ring.

Each component is contractible, indicating trivial homotopy types.

Abstract

The category of differential graded operads is a cofibrantly generated model category and as such inherits simplicial mapping spaces. The vertices of an operad mapping space are just operad morphisms. The 1-simplices represent homotopies between morphisms in the category of operads. The goal of this paper is to determine the homotopy of the operadic mapping spaces Map(E_n,C) with a cofibrant E_n-operad on the source and the commutative operad on the target. First, we prove that the homotopy class of a morphism phi: E_n -> C is uniquely determined by a multiplicative constant which gives the action of phi on generating operations in homology. From this result, we deduce that the connected components of Map(E_n,C) are in bijection with the ground ring. Then we prove that each of these connected components is contractible. In the case where n is infinite, we deduce from our results that the space of homotopy self-equivalences of an E-infinity-operad in differential graded modules has contractible connected components indexed by invertible elements of the ground ring.