the usual maps $$\begin{gathered}
 \circ_i:(S^1)^{\times n} \times (S^1)^{\times
m} \stackrel{(id \times \pi_i)(\Delta) \times id}
{\longrightarrow} (S^1)^{\times n} \times
S^1\times (S^1)^{\times m}\\
\stackrel{id \times \rho}{\longrightarrow} (S^1)^{\times n} \times
(S^1)^{\times m}
\stackrel{\bar\circ_i}{\longrightarrow}(S^1)^{\times n+m-1}\end{gathered}$$ where now $\rho$ is the diagonal action of $S^1$ on $(S^1)^{\times
n}$. The maps $\circ_i$ and the permutation action on the factors give the collection $\{\mathcal{S}^1(n)\}=(S^1)^{\times n}$ the structure of an operad. In fact this is exactly the usual construction of an operad built on a monoid.

\[cactbicross\] [@cact] The operad of cacti is the bi–crossed product of the operad $\cact$ of spineless cacti with the operad $\mathcal {S}^1$ based on $S^1$. Furthermore this bi–crossed product is homotopic to the semi–direct product of the operad of cacti without spines with the circle group $S^1$. $$\cacti \cong \cact \bowtie {\mathcal S}^1 \simeq \cact \rtimes
{\mathcal S}^1$$ The multiplication in the bi-crossed product is given by $$(C,\theta) \circ_i (C',\theta') = (C\circ_i^{\theta_i} C',
\theta\circ_{i}^{C'}\theta')$$ The multiplication in the semi-direct product is given by $$(C,\theta) \circ_i (C',\theta') = (C\circ_i^{\theta_i} C',
\theta\circ_{i}\theta')$$ Also, normalized cacti are homotopy equivalent to cacti which are homotopy equivalent to the bi-crossed product of normalized cacti with $\mathcal{S}^1$ and the semi-direct product with $\mathcal{S}^1$, where all equivalences are as 