Cacti and filtered distributive laws

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 simplicial set .Y;p/ . 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 a ground field of arbitrary characteristic.