Seems relevant to #4, but I haven’t checked the details:

]]>There are three more-or-less known notions of morphism between lax double functor–natural transformations, protransformations, and modules–and they have all been shown to give useful notions of morphism between models, generalizing functors, cofunctors, and profunctors between categories. (Evan Paterson)

Thanks. I think Williams’ thesis (where #4 is developed) is coming out soon, so a chance to see how these pictures relate.

]]>[me typing the following comment overlapped with #5 appearing]

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Probably by internally iterating the canonical examples of double categories:

For instance, consider the 3-category of 3-vector spaces as that of algebras with bimodules between them internal to a 2-category of ordinary algebras with bimodules between them.

The homomorphisms between these 3-vector space can now in principle have underlying morphisms of three kinds, I suppose: (1.) Plain linear maps, (2.) ordinary bimodule structures on ordinary vector space, (3.) bimodule structures on 2-vector spaces.

]]>A response from Christian Williams

]]>Dbl cats are logics: dim V is process (function) and dim H is relation.

Trp cats are metalogics: dim V is “metaprocess” (V-profunctors containing hetero-processes) and dim H is “metarelation” (H-profunctors), while dim T is transformation of inference (double functors).

Asking for a friend:

If double categories find good use in situations where one needs both something function-like and something relation-like, when might triple categories similarly arise? Can there be a third kind of morphism?

]]>Add another example.

]]>Stub for triple category.

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