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 . These
operads also admit linear versions, which are defined for every augmented graded cocommutative
coalgebra .
We show that the homology of the topological operad of based
–cacti is the linear
operad of based
–cacti.
In addition, we show that for every coalgebra
the operad of
based
–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