Last week I was at the Max Planck Institute for the Conference for Young researchers in homotopy theory and categorical structures (which was, by the way, a great conference – thanks to the organizers), and I gave yet another talk about the Lambrechts–Stanley model for configuration spaces. So maybe it’s time I write a little bit about it on this blog. I’ll write a first post about the model itself, and later I will explain how the Fulton–MacPherson operad is involved in all this.
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, §22.214.171.124]. 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 the Milnor–Moore theorem, a powerful tool describing the structure of (co)commutative Hopf algebras. Like the Eckmann–Hilton argument, it shows that having multiple compatible operations on the same object can lead to unexpected results about the object. Briefly, the theorem says that as soon as the Hopf algebra is cocommutative and connected, then it is isomorphic to the universal enveloping algebra of a Lie algebra (and a similar dual statement is true for commutative Hopf algebras).
The theorem(s) of acyclic models are a rather powerful technique for proving that some functors defined on truncated chain complexes can be extended in higher dimensions, and that two such functors are homotopic, by proving it on a small class of “model” objects.
For some reason I only discovered this last year, and I always find myself forgetting the precise hypotheses and conclusion… Hopefully writing this blog post will fix them in my mind. My main reference will be:
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.
As promised, this post is about the famous Eckmann–Hilton argument. This argument, on the surface, looks like a simple algebraic trick; but it has deep consequences, which I will now try to explain. This post is an expanded version of a math.SE answer I wrote some time ago.