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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeFeb 27th 2018

Stub for triple category.

• CommentRowNumber2.
• CommentAuthormaxsnew
• CommentTimeAug 12th 2019

• CommentRowNumber3.
• CommentAuthormaxsnew
• CommentTimeMay 26th 2022
• (edited May 26th 2022)
Are the morphisms between double categories in the category of double categories used in the definition here *strict* double functors?
• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeSep 27th 2023

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?

• CommentRowNumber5.
• CommentAuthorDavid_Corfield
• CommentTimeSep 28th 2023

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).

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeSep 28th 2023
• (edited Sep 28th 2023)

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

$\,$

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.

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeSep 28th 2023

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

• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeOct 20th 2023

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)