62 research outputs found
On mapping spaces of differential graded operads with the commutative operad as target
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.Comment: 16 page
The bar complex of an E-infinity algebra
The standard reduced bar complex B(A) of a differential graded algebra A
inherits a natural commutative algebra structure if A is a commutative algebra.
We address an extension of this construction in the context of E-infinity
algebras. We prove that the bar complex of any E-infinity algebra can be
equipped with the structure of an E-infinity algebra so that the bar
construction defines a functor from E-infinity algebras to E-infinity algebras.
We prove the homotopy uniqueness of such natural E-infinity structures on the
bar construction.
We apply our construction to cochain complexes of topological spaces, which
are instances of E-infinity algebras. We prove that the n-th iterated bar
complexes of the cochain algebra of a space X is equivalent to the cochain
complex of the n-fold iterated loop space of X, under reasonable connectedness,
completeness and finiteness assumptions on X.Comment: 51 pages. Preprint put in Elsevier format. Minor additional writing
correction
Combinatorial operad actions on cochains
A classical E-infinity operad is formed by the bar construction of the
symmetric groups. Such an operad has been introduced by M. Barratt and P.
Eccles in the context of simplicial sets in order to have an analogue of the
Milnor FK-construction for infinite loop spaces. The purpose of this article is
to prove that the associative algebra structure on the normalized cochain
complex of a simplicial set extends to the structure of an algebra over the
Barratt-Eccles operad. We also prove that differential graded algebras over the
Barratt-Eccles operad form a closed model category. Similar results hold for
the normalized Hochschild cochain complex of an associative algebra. More
precisely, the Hochschild cochain complex is acted on by a suboperad of the
Barratt-Eccles operad which is equivalent to the classical little squares
operad.Comment: 46 pages. Final versio
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