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Stream: deprecated: id my structure

Topic: A category of diagrams, but over which shape?

fosco (Feb 24 2023 at 09:46):

I am working with this kind of structure today: image.png

The ambient category is a 2-category $\mathcal K$, and $i,o$ are fixed 1-cells (and it is important that $i$ is an endo-1-cell). A "cone" (although I should not say cone for a non-conical diagram) is a triple $(e,\delta,\sigma)$ composed of a 1-cell $e : A\to B$ (so, parallel to $o$) and 2-cells $\delta,\sigma$ filling as above.

I am trying to find a "shape" $S[i,o]$ such that a diagram of that form is precisely a 2-functor $S[i,o]\to \mathcal K$. Or more precisely, I'd like to define the category of diagrams as above as a category of natural transformations in the fashion of weighted limits.
The reason why I believe this can be done is that I know the terminal such triple $(e,\delta,\sigma)$ is the right Kan extension of $o$ along the free monad on $i$ (provided all this exists, of course).
So, this must be some category of diagrams, the terminal object of which is a weighted limit, right?

I played a bit and I can stretch and turn the diagram into this more promising shape, image.png
which is evidently a certain element in the lax slice $\mathcal K//B$.
But how do I impose the two constraints on the shape of the diagram, namely that the domains of the objects in the slice are all the same, and the leftmost morphism is an identity? At this point, I'm not even sure this can be done!

Mike Shulman (Feb 24 2023 at 18:30):

I'm not quite sure what you are asking. There is a 2-category freely generated by two objects $a,b$, three morphisms $i:a\to a$, $o:a\to b$, $e:a\to b$, and two 2-cells $\delta : e\circ i \Rightarrow e$ and $\sigma : e \Rightarrow o$, and a 2-functor out of that 2-category will be a diagram of the above shape. Then you could fix $i$ and $o$ by saying that the restriction of your 2-functor to the subcategory generated by $a,b,i,o$ only has some fixed value.

fosco (Feb 24 2023 at 21:19):

In a similar way to what you do when you compute what is $X\mapsto Nat(W,\hom(X,F))$ in order to understand what is the $W$-weighted limit of $F$, I'd like to express the category whose

• objects are triples $(e,\delta,\sigma)$ as above
• morphisms are 2-cells $\phi : e \Rightarrow f$ compatible with everything

as a category of natural transformations. Is it possible, and how?

fosco (Feb 24 2023 at 21:21):

I understand the question might appear unclear: the dotted 1-cell and the 2-cells are not part of the diagram $F$, the diagram is just a span. But now I'd like to define a weight W so that a natural transformation W -> hom(X,F) corresponds to an arrangement of 2-cells like that

fosco (Feb 24 2023 at 21:23):

Similarly, a choice of weight and diagram like this image.png from the walking cospan (category at the left), is mapped by the diagram $F$ into the right cospan in $\mathcal K$, and by the weight $W$ to the cospan of categories at the center

fosco (Feb 24 2023 at 21:23):

where $\{0\to 1\}=\bf 2$ is the walking arrow

fosco (Feb 24 2023 at 21:24):

A natural transformation $W\to hom(X,F)$ now is a 2-cell filling this square image.png

and the W-weighted limit of F is the comma object of $f,g$

fosco (Feb 24 2023 at 21:25):

What is $W$ in my case, given that I know that the weighted limit is the right Kan extension of $o$ along the free monad on $i$?

fosco (Feb 24 2023 at 21:26):

I hope this is clearer. I never reverse-engineered a 2-dimensional shape in order to understand what category of diagrams it is generated by.

fosco (Feb 24 2023 at 22:59):

Something like this: for this choice of weight $W$ and diagram $G$, image.png

it seems to me that a natural transformation from $W$ to $\mathcal{K}(*,G)$ ($*$ a terminal object in K) is exactly a diagram of my shape

$Ga = Gb = i, Gc=o$

functoriality of $W$ implies $W(c.a)=Wa$ and $W(c.b)=Wb$

fosco (Feb 24 2023 at 23:00):

still not exactly what I was looking for, but at least now I think it's clear what I mean when I say "I want to describe the things above like a cat of diagrams"