#### Volume 14, issue 6 (2014)

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Cacti and filtered distributive laws

### Vladimir Dotsenko and James Griffin

Algebraic & Geometric Topology 14 (2014) 3185–3225
##### Abstract

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 $\left(Y,p\right)$. 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 $Y\phantom{\rule{0.3em}{0ex}}$–cacti is the linear operad of based ${H}_{\ast }\left(Y\right)$–cacti. In addition, we show that for every coalgebra $C$ the operad of based $C\phantom{\rule{0.3em}{0ex}}$–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.

##### Keywords
based cactus products, Koszul operad, Gröbner basis, distributive law
##### Mathematical Subject Classification 2010
Primary: 18D50
Secondary: 20L05, 16S15