Cacti and filtered distributive laws

TL;DR

This paper introduces topological and linear operads called based cacti, explores their homology, and proves their Koszul property using filtered distributive laws, extending previous criteria with a new proof applicable in any characteristic.

Contribution

It defines and studies based cacti operads for topological spaces and coalgebras, proving their Koszulness via filtered distributive laws and extending Koszul criteria to arbitrary characteristic.

Findings

Homology of based Y-cacti operad is the linear operad of based H_*(Y)-cacti.

Operad of based C-cacti is Koszul for every coalgebra C.

New proof of Koszulness criterion applicable over any characteristic.

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 topological space $(Y,\bullet)$. 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 the ground field of arbitrary characteristic.