Cacti and filtered distributive laws

Vladimir Dotsenko, James Griffin

    Research output: Contribution to journalArticle

    3 Citations (Scopus)

    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 (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.

    Original languageEnglish
    Pages (from-to)3185-3225
    Number of pages41
    JournalAlgebraic and Geometric Topology
    Volume14
    Issue number6
    DOIs
    Publication statusPublished - 15 Jan 2015

    Keywords

    • Based cactus products
    • Distributive law
    • Gröbner basis
    • Koszul operad

    ASJC Scopus subject areas

    • Geometry and Topology

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