Algebra/Topology Seminar

Speaker: Jan Steinebrunner

Title: From bordism categories to ∞-properads

Abstract: The bordism category Bord_d has a curious property: its space of morphisms Ar(Bord_d)^≈, which is an 𝐸_∞-algebra under disjoint union, is in fact a free 𝐸_∞-algebra. (Namely, it is freely generated by the connected bordisms). Abstracting this property of symmetric monoidal ∞-categories one arrives at the definition of an ∞-properad, of which Bord_d is an example. I will explain the basic theory of ∞-properads (developed in joint work with Shaul Barkan) and how this relates to the more classical notion of a properad, which is a generalisation of operads where each operation can have not only multiple inputs, but also multiple outputs. We also show that modular ∞-operads embed fully faithfully into ∞-properads as those where every object is dualisable. This implies a 1-dimensional cobordism hypothesis "with singularities" and I will sketch how it may be used to study various stable
moduli spaces of graphs or low-dimensional manifolds.