2-categorical opfibrations, Quillen's Theorem B, and $S^{-1}S$

zbMATH Open1519.18004arXiv2010.11173MaRDI QIDQ6351907

Angélica M. Osorno, Nick Gurski, Niles Johnson

Publication date: 21 October 2020

Abstract: In this paper we show that the strict and lax pullbacks of a 2-categorical opfibration along an arbitrary 2-functor are homotopy equivalent. We give two applications. First, we show that the strict fibers of an opfibration model the homotopy fibers. This is a version of Quillen's Theorem B amenable to applications. Second, we compute the $E^2$ page of a homology spectral sequence associated to an opfibration and apply this machinery to a 2-categorical construction of $S^{-1}S$. We show that if $S$ is a symmetric monoidal 2-groupoid with faithful translations then $S^{-1}S$ models the group completion of $S$.

Mathematics Subject Classification ID

Fibered categories (18D30) (Q)- and plus-constructions (19D06) Symmetric monoidal categories (19D23) Loop space machines and operads in algebraic topology (55P48) Homology with local coefficients, equivariant cohomology (55N25) 2-categories, bicategories, double categories (18N10) Monoidal categories, symmetric monoidal categories (18M05) Categories of fibrations, relations to (K)-theory, relations to type theory (18N45)

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