Group operads as crossed interval groups

The goal of the paper is to establish and to investigate a fully faithful embedding of the category of group operads into that of crossed interval groups. For this, we introduce a monoidal structure on the slice of the category of operads over the operad of symmetric groups. Comparing with the monoidal structure on the category of interval sets discussed in the author's previous work, we obtain a monoidal functor connecting these two categories. It will be shown that this actually induces a fully faithful functor on monoid objects and does not change the underlying sets, so we obtain a required embedding. The conditions for crossed interval groups to belong to the essential image will be proposed; namely in terms of commutativity of certain elements. As a result, it will turn out that the group operads form a reflective subcategory of the category of crossed interval groups. Finally, we will discuss monoid objects in symmetric monoidal category and Hochschild homologies on them.