symmetric monoidal (∞,1)-category of spectra
Let be the category defined as follows: * its set of objects is the free magma? on one generator, or equivalently the set of rooted binary tree?s. * the set of morphisms between two objects is given by the braid group whenever and are words of the same legnth , and is empty otherwise.
Then the collection of the ‘s is a braided operad?. The composition
is given by replacing the th strand of the first braid, by the second braid made very thin.
also have an obvious structure of a braided monoidal category. In fact:
let be the groupoid defined as follows: * it set objects are parenthesized permutations of , that is non-associative, non-commutative monomials on this set in which every letter appears exactly once. * morphisms between two objects are braids connecting each letter in to the same letter in . In other words, let be the canonical projection from the braid group to the symmetric group whose kernel is the pure braid group. Then, forgetting the parenthesization and viewing as permutations:
Then is an (ordinary) operad, the operadic structure being the same as for the non-colored version.
A topological interpretation of is as follows:
was originally defined in