We discuss a W-construction that is inspired by Boardman-Vogt in the setting of cubical Feynman categories (joint work with B. Ward). Homotopically this yields a cofibrant replacement in a category of functors to topological spaces. Concretely this is given by a cubical complex which in special cases have a combinatorial description. It turns out that particular combinatorics of graphs appear. As a first consequence this allows to construct moduli spaces of curves and various compactifications in an ontologically small setting (joint with C. Berger). An interesting double categorical version of graphs underlies the specific computations. If time permits, we will comment on relations to bi-algebras (joint with I. Gálvez-Carrillo and A. Tonks) and Koszulness (joint with Ward).

I will explain that an infinity-operad in Lurie’s sense can be recovered from its monad for free algebras in spaces, and that this (almost) gives an equivalence between infinity-operads and a certain class of “analytic” monads. This builds on previous joint work with David Gepner and Joachim Kock where we developed the infinity-categorical theory of analytic monads and showed these are equivalent to dendroidal Segal spaces.

The coset poset of a finite group is the partially ordered set of cosets of all proper subgroups. Similarly, for a prime p, the p-coset poset is the partially ordered set of all proper p-subgroups. In both cases, the order relation is simply set inclusion. These posets are related to subgroup posets, but they are maybe less well-known. I will review what is known about coset posets and also report on some new results on the Euler characteristic of coset posets. There are fairly explicit results about Euler characteristic of p-coset posets of finite groups of Lie type in characteristic p. In other cases, there are more combinatorial, and much less explicit, expressions for the Euler characteristic.

Nielsen and Schreier proved in the 1920s that subgroups of free groups are free. Inspired by that result, we say that an operad P has the Nielsen-Schreier property if in the category of P-algebras subalgebras of free algebras are free. If one considers an operad generated by a single binary operation, only six among them were known to be Nielsen-Schreier. Using Gröbner bases for operads, we propose an effective combinatorial criterion for the Nielsen-Schreier property, and exhibit infinitely many new examples of Nielsen-Schreier operads. In particular, quite surprisingly to us, the operad of pre-Lie algebras is Nielsen-Schreier. This is joint work with with Ualbai Umirbaev.