2-Segal spaces and K-theory constructions

Julie Bergner (University of Virginia)

Monday 14th October 16:00-17:00 Maths 311B

Abstract

Just as Segal spaces can be regarded as an up-to-homotopy version of topological categories, 2-Segal spaces encode a similar but weaker algebraic structure in which composition need not exist or be unique, but which is still associative.  A foundational result of both Dyckerhoff and Kapranov and of Galvez, Kock, and Tonks is that the result of applying Waldhausen's S-construction to an exact category is a 2-Segal space.  In joint work with Osorno, Ozornova, Rovelli, and Scheimbauer, we proved that every 2-Segal space can be obtained by applying an S-construction to a suitable general input.   However, Waldhausen categories do not necessarily fit into this framework, so a natural question is how much of the structure of a 2-Segal space is recovered in this case.   In recent work, Carawan has given a general answer to this question, but has also shown that many common examples of Waldhausen categories have more structure than we might expect.

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