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Cacti and filtered distributive laws

Vladimir Dotsenko and James Griffin

Algebraic & Geometric Topology 14 (2014) 3185–3225

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.

based cactus products, Koszul operad, Gröbner basis, distributive law
Mathematical Subject Classification 2010
Primary: 18D50
Secondary: 20L05, 16S15
Received: 10 November 2011
Revised: 5 March 2014
Accepted: 23 March 2014
Published: 15 January 2015
Vladimir Dotsenko
School of Mathematics
Trinity College Dublin
College Green
Dublin 2
James Griffin
School of Mathematics and Statistics
University of Glasgow
University Gardens
Glasgow G12 8QW