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A model structure for weakly horizontally invariant double categories

Lyne Moser, Maru Sarazola and Paula Verdugo

Algebraic & Geometric Topology 23 (2023) 1725–1786

We construct a model structure on the category DblCat of double categories and double functors, whose trivial fibrations are the double functors that are surjective on objects, full on horizontal and vertical morphisms, and fully faithful on squares; and whose fibrant objects are the weakly horizontally invariant double categories.

We show that the functor : 2Cat DblCat , a more homotopical version of the usual horizontal embedding , is right Quillen and homotopically fully faithful when considering Lack’s model structure on 2Cat . In particular, exhibits a levelwise fibrant replacement of . Moreover, Lack’s model structure on 2Cat is right-induced along from the model structure for weakly horizontally invariant double categories.

We also show that this model structure is monoidal with respect to Böhm’s Gray tensor product. Finally, we prove a Whitehead theorem characterizing the weak equivalences with fibrant source as the double functors which admit a pseudoinverse up to horizontal pseudonatural equivalence.

model structure, double categories, 2–categories, monoidal model structure, Whitehead theorem
Mathematical Subject Classification
Primary: 18D20, 18N10, 18N40
Received: 27 August 2020
Revised: 16 November 2021
Accepted: 14 December 2021
Published: 14 June 2023
Lyne Moser
Max Planck Institute for Mathematics
Maru Sarazola
Department of Mathematics
Johns Hopkins University
Baltimore, MD
United States
Paula Verdugo
Department of Mathematics and Statistics
Macquarie University
Sydney, NSW

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