Algebraic & Geometric Topology

Cacti and filtered distributive laws

Vladimir Dotsenko and James Griffin

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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.

Article information

Algebr. Geom. Topol., Volume 14, Number 6 (2014), 3185-3225.

Received: 10 November 2011
Revised: 5 March 2014
Accepted: 23 March 2014
First available in Project Euclid: 19 December 2017

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Zentralblatt MATH identifier

Primary: 18D50: Operads [See also 55P48]
Secondary: 20L05: Groupoids (i.e. small categories in which all morphisms are isomorphisms) {For sets with a single binary operation, see 20N02; for topological groupoids, see 22A22, 58H05} 16S15: Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)

based cactus products Koszul operad Gröbner basis distributive law


Dotsenko, Vladimir; Griffin, James. Cacti and filtered distributive laws. Algebr. Geom. Topol. 14 (2014), no. 6, 3185--3225. doi:10.2140/agt.2014.14.3185.

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