##### The Voronov Product of Operads– Sep 22, 2016 #math #operads #swiss-cheese

My first real post in a while! It turns out that writing an actual paper (cf. previous blog post) takes a lot of time and effort. Who knew?

The Voronov product of operads is an operation introduced by Voronov in his paper The Swiss-cheese operad (he just called it “the product”). It combines an operad and a multiplicative operad to yield a new colored operad; the main example I know is the homology of the Swiss-cheese operad. This is a construction that I use in my preprint Swiss-Cheese operad and Drinfeld center, where as far as I know I coined the name “Voronov product” – I haven’t seen this operation at all outside of Voronov’s paper. I wanted to advertise it a bit because I find it quite interesting and I’m eager to see what people can do with it.

The purpose of this post is to record the definition of $$\infty$$-operads and explain why it works like that. For this I’m using Lurie’s definition of $$\infty$$-operads; there is also a definition by Cisinski–Moerdijk–Weiss using dendroidal sets, about which I might talk later.
Indeed, the definition on an $$\infty$$-operad is a bit mysterious taken “as-is” – see [HA, §2.1.1.10]. My goal is to explain how to reach this definition, mostly for my own sake. Most of what follows is taken either from the book Higher Algebra, the $$n$$Lab, or the semester-long workshop about hgiher category theory in Lille in 2015.
This post is about something somewhat weird I noticed about infinitesimal bimodules over operads and their relationships with some $$E_n$$ operads. I don’t know if it’s something significant, and I’d definitely be interested to hear more about it.