normalized cacti $$\circ_i:\cacti^1(n)\otimes \cacti^1(m)\rightarrow \cacti^1(n+m-1)$$ are defined on elements $(c,c') \mapsto c\circ_i c'$ simply by identifying the realization of the cycle $c_0$ of $c'$ with the cycle $c_i$ of $c$ via the maps $\phi_0(c')$ and $\phi_i(c)$ again with the orientation on the second $S^1$ reversed.

These maps together with the $\Sn$ action permuting the labels turn the collection $\{\cacti^1(n)\}$ into a homotopy associative quasi-operad $\cacti^1$. The collection $\{\cact^1(n)\}$ forms a homotopy associative quasi-suboperad $\cact^1$ of $\cacti^1$ [@cact].

Relations among cacti
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\[cactthm\] [@cact] Normalized cacti are homotopy equivalent through quasi-operads to the cacti. The same holds for the (quasi)-suboperads of normalized spineless cacti and spineless cacti.

[@cact] Normalized cacti are quasi-isomorphic as quasi-operads to cacti and normalized spineless cacti are quasi-isomorphic as quasi-operads to spineless cacti. In particular in both cases the homology quasi-operads are operads and are isomorphic as operads.

### Remarks on the bi-crossed product

In this section we recall the construction of the bi-crossed product as it was given in [@cact] to which we refer the reader for more details.

First notice that there is an action of $S^1$ on $\Cact(n)$ given by rotating the base point [*clockwise*]{} (i.e. in the orientation opposite the usual one of 