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Stream: theory: category theory

Topic: ✔ Left adjoint to quintet construction

Amar Hadzihasanovic (Mar 31 2023 at 11:19):

The [[quintet construction]] is one of the ways of constructing a double category from a 2-category, defining a functor $\mathbf{2Cat} \to \mathbf{DblCat}$.

It is mentioned in the nLab article that this functor has a right adjoint, picking out the companion pairs in the double category.

However I've convinced myself that it also has a left adjoint, given by “treating double-categorical pasting diagrams as if they were 2-categorical pasting diagrams”.
More explicitly, given a double category $\mathbb{D}$, this constructs the 2-category whose

• objects are those of $\mathbb{D}$,
• morphisms are generated by both the vertical and horizontal morphisms of $\mathbb{D}$, quotiented by the compositions that exist in $\mathbb{D}$ & by identifying the vertical and horizontal units -- hence can be seen as equivalence classes of “paths” of alternating horizontal and vertical morphisms;
• 2-morphisms are generated by the squares $(v_0, \alpha, v_1): h_0 \Rightarrow h_1$ of $\mathbb{D}$, seen as 2-cells $\alpha: (h_0, v_1) \Rightarrow (v_0, h_1)$, quotiented by the “pasting” compositions that exist in $\mathbb{D}$.

I haven't been able to find this fact from a quick search, but I'm not that well-acquainted with the literature on double categories, so: has this been noticed and/or applied before?

(This is related to my unanswered question from a few days ago -- I've realised that one way of getting that zig-zag construction is by first building a double category out of $\mathcal{B}^\mathrm{op}$ and $\mathcal{F}$, then applying this functor to get a 2-category, then truncating to a 1-category)

Bryce Clarke (Mar 31 2023 at 12:07):

https://categorytheory.zulipchat.com/#narrow/stream/229136-theory.3A-category-theory/topic/Adjoint.20to.20the.20quintet.20construction

Bryce Clarke (Mar 31 2023 at 12:11):

Amar Hadzihasanovic said:

I haven't been able to find this fact from a quick search, but I'm not that well-acquainted with the literature on double categories, so: has this been noticed and/or applied before?

See Proposition 8 of Multiple Functors IV

Bryce Clarke (Mar 31 2023 at 12:23):

I have added a small note to the nlab page [[quintet construction]].

Amar Hadzihasanovic (Mar 31 2023 at 12:54):

Oh, that's great, thank you! I didn't think of actually searching Zulip for this :flushed:

Notification Bot (Mar 31 2023 at 12:56):

Amar Hadzihasanovic has marked this topic as resolved.

Matteo Capucci (he/him) (Apr 02 2023 at 10:53):

We're starting to exhaust the space of possible questions

Matteo Capucci (he/him) (Apr 02 2023 at 10:53):

We should have a chatbot that answers with possible related sources :thinking:

Bruno Gavranović (Apr 02 2023 at 11:18):

Since the Zulip archive is public facing, it ought to be possible to create a ChatGPT plugin :)