an operad is an operad the statement follows. Alternatively one could just remark, that there is also an obvious functorial map induced by the diagonal for these cells.

The chains are a chain model for the framed little discs operad since $\cacti^1(n)$ and $\cacti(n)$ are homotopy equivalent and the latter is equivalent to the framed little discs operad.

Although the above chain model is the one one would expect to use for framed little discs, it does not have enough cells for our purposes. In order to translate the proofs in the arc complex given in [@KLP] into statements about the Hochschild complex, we will need a slightly finer cell structure then the one above. After having used the larger structure one can reduce to the cell model with less cells as they are obviously equivalent.

A spine decoration $\sdec$ for a planted planar bi–partite tree is a $\Zz$ decoration together with the marking of one angle at each vertex labelled by one and a flag at each vertex labelled by zero. We call the set of such trees which are $n$-labelled by $\swlbptree(n)$ and again use this notation as well for the free Abelian group and the $k$ vector 