  Motivated by the second author's construction of a classifying space for the
group of pure symmetric automorphisms of a free product, we introduce and study
a family of topological operads, the operads of based cacti, defined for every
pointed topological space $(Y,\bullet)$. These operads also admit linear
versions, which are defined for every augmented graded cocommutative coalgebra
$C$. We show that the homology of the topological operad of based $Y$-cacti is
the linear operad of based $H_*(Y)$-cacti. In addition, we show that for every
coalgebra $C$ the operad of based $C$-cacti is Koszul. To prove the latter
result, we use the criterion of Koszulness for operads due to the first author,
utilising the notion of a filtered distributive law between two quadratic
operads. We also present a new proof of that criterion which works over the
ground field of arbitrary characteristic.
