Crossed modules over operads and operadic cohomology
This article introduces and studies secondary operads. These are defined as operads in the symmetric monoidal category \(Pair(Mod)\) whose objects are maps \(f:A\to B\) between pairs of graded modules. The definition is made explicit by a full description of all the structure maps of a secondary operad and of all the unit, associativity and equivariance conditions they satisfy. Algebras over such operads are called crossed modules and prolongations of crossed modules are called crossed extensions (Definition 6.1). Examples of secondary operads may be obtained from operads in the category of chain complexes over a field, by applying the secondary homology functor. This functor associates to a chain complex \((C_*,d)\) the pair \((\partial:s^{-1} \text{coker}(d)\to\text{ker}(d))\), where \(\partial\) is induced by \(d\) and \(s^{-1}\) is a suspension. Examples of crossed modules arise from algebras over an operad in chain complexes, again by applying secondary homology. In particular, an interesting example is given by applying secondary homology to \(C^*(X)\), the cochains on a topological space \(X\), considered as a algebra over a suitable \(E_\infty\) operad. With this framework, operadic cohomology theories are defined, via equivalence classes of crossed extensions. Examples are discussed and operadic cohomology is related to other theories. In particular, the operadic cohomology for the pair \((0\to \text{Ass})\), where Ass is the associative operad, is Hochschild cohomology (at least working over a field). This case has been studied in more detail in \textit{H.-J. Baues} and \textit{E. G. Minian}, [Homology Homotopy Appl. 4, No. 2(1), 63--82, electronic only (2002; Zbl 1004.18012)]. The secondary cohomology associated to an \(E_\infty\) operad gives a cohomology theory for commutative algebras. The precise relationship to other theories for commutative algebras in not fully understood. The construction leads to the identification of a natural characteristic class associated to the cochains on a space \(X\), living in the third operadic cohomology group. This generalizes the notion of characteristic class in Hochschild cohomology for a differential graded algebra. In the final section, operadic cohomology is redefined using cofibrant resolutions. This construction is used to prove that operadic cohomology vanishes, in degrees greater than \(2\), for free algebras.
scientific article; zbMATH DE number 2081554