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Stream: theory: category theory

Topic: quintets construction

Matteo Capucci (he/him) (Sep 26 2022 at 16:01):

I'd like to understand the quintet construction a bit better, from an abstract nonsense standpoint. So far I've always thought of it as just a nice way to get a double category from a 2-category, no more than a toy, but recently I stumbled upon a construction which spits out $\mathbb Q(\cal K)$ for some reason and I want to understand if that's a bug or a feature.
On the nLab it says
image.png

Matteo Capucci (he/him) (Sep 26 2022 at 16:02):

which seems interesting, but I can't find a citation or anything. Eresmann paper about quintets is quite obscure to me, since he uses notation and terminology I'm very unfamiliar with (bonus: it's in French) so I can't really understand what's going on.

Matteo Capucci (he/him) (Sep 26 2022 at 16:02):

Can y'all recommend some references or pointers to learn more about this?

Mike Shulman (Sep 26 2022 at 17:03):

Possibly it is in Marco Grandis and Robert Pare, Adjoints for double categories?

Matteo Capucci (he/him) (Sep 26 2022 at 22:15):

Ha, thanks!

Tim Campion (Sep 28 2022 at 19:13):

There it is -- Thm 1.7!

Tim Campion (Sep 28 2022 at 19:13):

I've been confused about this for awhile too.

Tim Campion (Sep 28 2022 at 19:19):

I've updated the nlab page to include this reference.
That "morally" line on the page has been bothering me for maybe close to a year :upside_down:

Tim Campion (Sep 28 2022 at 19:20):

https://ncatlab.org/nlab/show/quintet+construction

Tim Campion (Sep 28 2022 at 19:25):

By the way, I'm pretty sure that the "quintets" construction also has a left adjoint, which is the left Kan extension of the Gray tensor product Cat x Cat --> 2Cat along the inclusion Cat x Cat --> DblCat

Matteo Capucci (he/him) (Sep 29 2022 at 08:43):

That's very cool!