I will discuss ongoing work aimed at constructing higher categories of (enriched) higher categories. This should give the appropriate targets for many interesting examples of extended topological quantum field theories, including extended versions of the classical examples of TQFTs due to Turaev-Viro, Reshetikhin-Turaev, etc.

# Past Topology Seminar

Klarrich showed that the Gromov boundary of the curve complex of a hyperbolic surface is homeomorphic to the space of ending laminations on that surface. Independent results of Bestvina-Reynolds and Hamenstädt give an analogous statement for the free factor graph of a free group, where the space of ending laminations is replaced with a space of equivalence classes of arational trees. I will give an introduction to these objects and describe some joint work with Bestvina and Horbez, where we show that the Gromov boundary of the free factor graph for a free group of rank N has topological dimension at most 2N-2.

Let \Gamma_{g,1} denote the mapping class group of a genus g surface with one parametrized boundary component. The group homology H_i(\Gamma_{g,1}) is independent of g, as long as g is large compared to i, by a famous theorem of Harer known as homological stability, now known to hold when 2g > 3i. Outside that range, the relative homology groups H_i(\Gamma_{g,1},\Gamma_{g-1,1}) contain interesting information about the failure of homological stability. In this talk, I will discuss a metastability result; the relative groups depend only on the number k = 2g-3i, as long as g is large compared to k. This is joint work with Alexander Kupers and Oscar Randal-Williams.

The smooth homotopy category is a simultaneous enlargement of the usual homotopy category and of the category of smooth manifolds. Its structure can be described very simply and explicitly by a version of van Est's theorem. It provides us with an interpolation between topology and geometry (and with a toy model of derived algebraic geometry and motivic homotopy theory, though I shall not pursue those directions). My talk will list some situations which the category seems to illuminate: one will be Kapranov's beautiful description of the Lie algebra of the 'group' of based loops in a manifold.

Boundaries of hyperbolic spaces have played a key role in low dimensional topology and geometric group theory. In 1993, Paulin showed that the topology of the boundary of a (Gromov) hyperbolic space, together with its quasi-mobius structure, determines the space up to quasi-isometry. One can define an analogous boundary, called the Morse boundary, for any proper geodesic metric space. I will discuss an analogue of Paulin’s theorem for Morse boundaries of CAT(0) spaces. (Joint work with Devin Murray.)

We will discuss a construction of cobordism maps on the full link complex for decorated link cobordisms. We will focus on some formal properties, such as grading change formulas and local relations. We will see how several expressions for mapping class group actions can be interpreted in terms of pictorial relations on decorated surfaces. Similarly, we will see how these pictorial relations give a "connected sum formula" for the involutive concordance invariants of Hendricks and Manolescu.

After giving an introduction to functorial field theories I will explain a natural generalization thereof, called "twisted" field theories by Stolz-Teichner. The definition uses the notion of lax or oplax natural transformations of strong functors of higher categories for which I will sketch a framework. I will discuss the fully extended case, which gives a comparison to Freed-Teleman's "relative" boundary field theories. Finally, I will explain some examples, one of which explicitly arises from factorization homology and whose target is the higher Morita category of E_n-algebras, bimodules, bimodules of bimodules etc.

An efficient way to descibe binary operations which are associative only up to coherent homotopy is via simplicial spaces. 2-Segal spaces were introduced independently by Dyckerhoff--Kapranov and G\'alvez-Carrillo--Kock--Tonks to encode spaces carrying multivalued, coherently associative products. For example, the Waldhausen S-construction of an abelian category is a 2-Segal space. It describes a multivalued product on the space of objects given in terms of short exact sequences.

The main motivation to study spaces carrying multivalued products is that they can be linearised, producing algebras in the usual sense of the word. For the preceding example, the linearisation yields the Hall algebra of the abelian category. One can also extract tensor categories using a categorical linearisation procedure.

In this talk I will discuss double 2-Segal spaces, that is, bisimplicial spaces which satisfy the 2-Segal condition in each variable. Such bisimplicial spaces give rise to multivalued bialgebras. The second iteration of the Waldhausen S-construction is a double 2-Segal space whose linearisation is the bialgebra structure given by Green's Theorem. The categorial linearisation produces categorifications of Zelevinsky's positive, self-adjoint Hopf algebras.