quasi-operads $$\cacti^1 \sim \cacti \cong \cact \bowtie {\mathcal S}^1
\sim\cact^1 \bowtie {\mathcal S}^1\sim  \cact^1 \rtimes {\mathcal
S}^1$$

The proof of the first statement is given by verifying that the two operad structures coincide. For the second statement one notices that the homotopy diagonal is homotopy equivalent to the usual one and that one can find homotopies to the diagonal which continuously depend on the cactus. The third statement follows from contracting the factors $\mathbb{R}^n_{>0}$ and using Theorem \[cactthm\].

The homology operad of $\cacti$ is the semi-direct product of $\cacti$ and the homology of the operad $\mathcal{S}^1$ built on the monoid $S^1$.

Relation to (framed) little discs
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[@cact] The operad $\cact$ is equivalent to the little discs operad and the operad $\cacti$ is equivalent to the framed little discs operad.

The latter result has been first stated by Voronov in [@Vor].

A CW decomposition for $\cacti^1$ and a chain model for the framed little discs
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A $\Zz$ decoration for a black and white bipartite tree is a map $\Zdec: V_w \rightarrow \Zz$.

\[firstcells\] The quasi–operad of normalized cacti $\cacti^1$ has a CW–decomposition which is given by cells indexed by planar planted bi–partite trees with a $\Zz$ decoration. The $k$ 