Cacti and filtered distributive laws

Dotsenko, Vladimir¹ ; Griffin, James²

¹School of Mathematics, Trinity College Dublin, College Green, Dublin 2, Ireland
²School of Mathematics and Statistics, University of Glasgow, University Gardens, Glasgow G12 8QW, UK

Algebraic and Geometric Topology, Tome 14 (2014) no. 6, pp. 3185-3225

Résumé

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.

Détail

DOI : 10.2140/agt.2014.14.3185

Classification : 18D50, 20L05, 16S15
Keywords: based cactus products, Koszul operad, Gröbner basis, distributive law

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