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Nathanael Arkor (Jul 28 2022 at 17:03):Given a double category with companions, a conjoint to a tight arrow is a loose arrow which is right adjoint to the companion . Is there a standard notion dual to conjoints, which could be defined in terms of a left adjoint to companions? A motivating example of a (pseudo) double category admitting companions and co-conjoints would be that of small categories, functors, and coprofunctors (defined in terms of free completion rather than free cocompletion).
Nathanael Arkor (Jul 28 2022 at 17:21):I think perhaps the concept I am looking for is a conjoint in the transpose of a double category, as least when the double category is strict. Does this have a name? (It could well be that I've forgotten to flip some arrows and this turns out to be the same as a conjoint or something similar.)
Mike Shulman (Jul 29 2022 at 03:47):I don't think there's a name or a nice characterization of that. It's related to how left adjoints of restriction in a fibration have a nice characterization in terms of opcartesian arrows, but right adjoints don't.
Matteo Capucci (he/him) (Aug 01 2022 at 11:57):Pardon the OT, but I suggest changing the name of coprofunctor as it literally means 'shit functor' :grinning_face_with_smiling_eyes:
Zhen Lin Low (Aug 01 2022 at 12:08):Indfunctor, antefunctor, counterfunctor, postfunctor, ...
Nathanael Arkor (Aug 01 2022 at 12:32):Matteo Capucci (he/him) said:
Pardon the OT, but I suggest changing the name of coprofunctor as it literally means 'shit functor' :grinning_face_with_smiling_eyes:
That's an interesting point :) Perhaps "codistributor" would be a better choice.
John van de Wetering (Aug 02 2022 at 12:45):Matteo Capucci (he/him) said:
Pardon the OT, but I suggest changing the name of coprofunctor as it literally means 'shit functor' :grinning_face_with_smiling_eyes:
In my first course on Category Theory by Bart Jacobs, he made this joke when introducing coproducts. That they are like products, but shittier.
John Baez (Aug 02 2022 at 20:54):Actually the pipe out of the toilet counts as a coproduct.
Mike Shulman (Aug 06 2022 at 00:22):I'm not sure I even know what a "co-profunctor" is supposed to be.
Nathanael Arkor (Aug 06 2022 at 03:16):A 1-cell in the Kleisli bicategory of the free completion relative pseudomonad.