New York Journal of Mathematics
Volume 27 (2021), 1009-1059


Nick Gurski, Niles Johnson, and Angélica M. Osorno

2-categorical opfibrations, Quillen's Theorem B, and S-1S

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Published: July 13, 2021.
Keywords: opfibration, group completion, 2-categories.
Subject: 19D23; 18D30, 18M05, 18N45, 18N10, 19D06, 55N25, 55P48.

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 E2 page of a homology spectral sequence associated to an opfibration and apply this machinery to a 2-categorical construction of S-1S. We show that if S is a symmetric monoidal 2-groupoid with faithful translations then S-1S models the group completion of S.


The third author was partially supported by NSF Grant DMS-1709302.

Author information

Nick Gurski:
Department of Mathematics, Applied Mathematics, and Statistics
Case Western Reserve University
Cleveland, Ohio 44106, USA


Niles Johnson:
Department of Mathematics
The Ohio State University at Newark
Newark, OH 43055, USA


Angélica M. Osorno:
Department of Mathematics
Reed College
Portland, Oregon 97202, USA