Tensor product of operads and iterated loop spaces
We use the terminology of \textit{J. P. May} [The geometry of iterated loop spaces, Lect. Notes Math. 271 (1972; Zbl 0244.55009)]. Let \({\mathcal C}_ n\) denote the operad of little n-cubes, which describe n-fold loop structures on a topological space. The n canonical inclusions \({\mathcal C}_ 1\to {\mathcal C}_ n\) define an inclusion of the n-fold tensor product \({\mathcal C}_ 1^{\otimes n}\) of \({\mathcal C}_ 1\) in \({\mathcal C}_ n\). The main result of the present paper is that this inclusion is a local \(\Sigma\)-equivalence. Hence any \({\mathcal C}_ n\)-space is (weakly) homotopy equivalent to a \({\mathcal C}_ 1^{\otimes n}\)-space. This, in particular, implies that an n-fold loop structure is up to homotopy the same as n commuting one-fold loop structures. The author also extends the passage from loop structures to Moore loop structures (which are strictly associative) to \({\mathcal C}\)-spaces for an arbitrary operad \({\mathcal C}\) and analyses what happens to a \({\mathcal C}^{\otimes n}\)-space after this passage: It becomes a space with n strict monoid structures which commute with each other on suitable explicitly given subsets, but not globally (recall, two commuting monoid structures define the structure of an abelian monoid).
scientific article; zbMATH DE number 4100171
scientific article