On Several Varieties of Cacti and Their Relations

We consider four operads which are different varieties of the cacti operad. These are cacti without local zeros (or spines) and cacti proper as well as both varieties with fixed constant size one of the constituting loops. We show that both types of cacti are a semi–direct products as operads of their normalized versions with a re–scaling operad based on R>0 which are homotopic to direct products. Furthermore, we introduce the notion of bi–crossed operads and show that the cacti proper are a bi–crossed product of the operad of cacti without spines and the operad based on the monoid given by the circle group S. We also prove that this particular bi–crossed operad product is homotopy equivalent to the semi–direct product of the spineless cacti with the group S. Lastly we recall how to realize the cacti operads as suboperads of the Arc operad and show that the cacti are generated as a suboperad by the cacti without spines and a S twist whose corresponding cycle represents the BV operator.