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.

Keywords

based cactus products, Koszul operad, Gröbner basis, distributive law

Mathematical Subject Classification 2010

Primary: 18D50

Secondary: 20L05, 16S15

References

Publication

Received: 10 November 2011

Revised: 5 March 2014

Accepted: 23 March 2014

Published: 15 January 2015

Authors

Vladimir Dotsenko
School of Mathematics Trinity College Dublin College Green Dublin 2 Ireland

James Griffin
School of Mathematics and Statistics University of Glasgow University Gardens Glasgow G12 8QW UK

Volume 14, issue 6 (2014)

Vladimir Dotsenko and James Griffin

Abstract

Keywords

Mathematical Subject Classification 2010

References

Publication

Authors