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Stream: deprecated: id my structure

Topic: Cell in a 2 category

Patrick Nicodemus (Aug 01 2023 at 21:30):

I am reading a paper by Grandis and Pare, trying to understand their Conduche condition - "Lax Kan extensions between Double Categories".
I noticed the following generalization of their condition.
I want to know if there's a good way to think about this, or if anyone recognizes it. It's pretty simple so I am hoping someone recognizes it.

I am working in the 2 category Cat in what follows.
If $(q:a\to b, s:a\to c)$ is a span in Cat, by its 2-limit I mean a category $e$ with functors $x:e\to b, y:e\to a, z:e\to c$ and natural transformations $\alpha : x\to qy$, $\beta :sy\to z$, which is terminal among such cones.

Now say I have a square of functors connected by a 2 cell $\alpha : pq\Rightarrow rs$.

Then there is a canonical functor from the 2-limit of the span $(q, s)$ to the comma category of the cospan $(p, r)$.

The Conduche condition I am interested in says that this induced functor between limit categories is surjective and has connected fibers. Perhaps it would be more natural to strengthen this to an equivalence so the limit of the span is equivalent to the limit of the cospan in a sense or an adjunction.

Anyway, this got me thinking. Is anything known about like, if you make some assumptions on the 2-cell $\alpha$, what can we conclude about this induced functor?

Patrick Nicodemus (Aug 01 2023 at 21:47):

this kind of reminds me of dusko 's paper about viewing the beck Chevalley condition as a kind of interpolation condition. I feel like his paper involves a condition very similar to this, this is related to uniform interpolation

Patrick Nicodemus (Aug 02 2023 at 14:56):

@dusko Do you have any comments? Have you run into other situations where this interpolation condition comes up?

Patrick Nicodemus (Aug 02 2023 at 21:39):

Yooooo this might be what i need - https://ncatlab.org/nlab/show/exact+square

Patrick Nicodemus (Aug 03 2023 at 02:00):

I have added this realization to the page on the Conduche condition. Cool

Notification Bot (Aug 03 2023 at 02:00):

Patrick Nicodemus has marked this topic as resolved.

John Baez (Aug 03 2023 at 08:01):

What follows is some people testing what they can do after a topic has been "resolved":

Tsantilas Theophilos (Aug 03 2023 at 10:24):

posting a comment to the request of @John Baez . I hope this helps.

John Baez (Aug 03 2023 at 10:26):

Thanks!

Notification Bot (Aug 03 2023 at 15:56):

Todd Trimble has marked this topic as unresolved.

Todd Trimble (Aug 03 2023 at 15:57):

From what I thought I understood from before, some time has to elapse before the inability of ordinary users to unresolve kicks in.

John Baez (Aug 03 2023 at 16:29):

It turns out that anyone can post to a resolved comment no matter how long it's been resolved.