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AG Seminar: Free Decomposition Spaces

Author: Lona

Speaker: Philip Hackney (U of Louisiana at Lafayette)


Simplicial objects are a fundamental tool in modern homotopy theory, (higher) category theory, and algebra. We will begin with a brief review / introduction, with an emphasis on the connection to categories, generalizing to the 2-Segal spaces of Dyckerhoff–Kapranov (also called decomposition spaces by Gálvez-Carillo–Kock–Tonks).

Any simplicial set has an underlying “outer face complex” obtained by forgetting about the degeneracy maps and the inner face maps. Our primary subject is the left adjoint to this forgetful functor, which freely adjoins inner faces and degeneracies to any outer face complex. A surprising fact is that this free functor takes any outer face complex to a decomposition space. Several previously-known examples of decomposition spaces, each of combinatorial origin and expressing a deconcatenation-type comultiplication, turn out to arise as such “free decomposition spaces.” We identify the category of outer face complexes with a certain category of decomposition spaces over the monoid of natural numbers. This talk is based on joint work with Joachim Kock.