Let D be a double category with an initial object. Any cotabulator Gamma(v) of a vertical morphism v:X-->Y gives rise to an extension (i.e., short exact sequence) X-->Gamma(v)-->Y in the vertical bicategory VD. If D has "open cokernels" then every extension in VD is equivalent to one of this form. Examples include the double categories Loc, Topos, Pos, and Cat, whose objects are locales, toposes, posets, and small categories, respectively; and Gamma(v) is given by Artin-Wraith glueing along v in the first two cases, and by the collage of v in the others.
Keywords: double categories, extensions, glueing, locales, toposes, posets, profunctors
2020 MSC: 18N10, 20E22, 18B25, 06D22, 18F70, 18D60, 18B35
Theory and Applications of Categories, Vol. 36, 2021, No. 13, pp 348-367.