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Reid Barton (Oct 29 2020 at 15:16):Do people who study bicategories (or double categories) of spans ever use models in which the target of a 2-cell is a string of 1-morphisms, rather than the source, as a way to sidestep coherence issues?
Reid Barton (Oct 29 2020 at 15:19):It seems to make sense because the composition of 1-morphisms is defined via a limit, so it's better to map something into it.
David Kern (Oct 30 2020 at 15:08):Yes, they were used in Dawson-Paré-Pronk's series on the universal property of Spans as freely adding right adjoints, in particular The span construction (section 4) and Paths in double categories (section 2), under the name "oplax double categories", in precisely this context.
Reid Barton (Oct 30 2020 at 15:31):Thanks! Section 4 is just the sort of thing I had in mind.