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Stream: learning: questions

Topic: Limits of diagrams with interesting composition structure

Asad Saeeduddin (Jan 31 2022 at 20:05):

Hello there folks. I was recently musing on the fact that all of the interesting limits I know of (or at least interesting enough to be named concepts in their own right) arise from diagrams that have no interesting composition structure. By this I mean that the only composition structure in the relevant indexing category is precisely what is necessitated by the strictures of categoryhood.

In other words, the limits I know about are really indexed by (the free categories of) graphs. These include terminal objects, products, pullbacks, and equalizers.

I was wondering if folks can point me to interesting examples of limits arising from diagrams with "prescribed" compositions in the indexing category (i.e. where the indexing category is not equivalent to the free category generated by some graph). The simpler the example the better. Thanks!

Ralph Sarkis (Jan 31 2022 at 20:13):

If you view a group action as a diagram $\mathbf{B}G \to \mathbf{Set}$, the limit of this diagram is the set of elements fixed by all of $G$ and the colimit is the set of orbits of the action.

Paolo Perrone (Jan 31 2022 at 20:26):

A filtered limit (such as the stalks of a sheaf) also uses composition nontrivially.

Mike Shulman (Jan 31 2022 at 22:16):

However, it is true that in a 1-category, the limit of a diagram $C \to D$ is isomorphic to the limit of the corresponding diagram $FUC\to D$, where $FUC$ is the free category on the underlying directed graph of $C$. So in that sense, as long as you remain in the world of 1-categories, the composition structure in the indexing category is "irrelevant". But this fails once you move up to $n$-categories for $n>1$ (although I think there is a somewhat-analogous fact that limits in an $n$-category are determined by simplicial $n$-skeleta of indexing categories).

Ian Coley (Feb 01 2022 at 00:02):

Mike Shulman said:

But this fails once you move up to $n$-categories for $n>1$ (although I think there is a somewhat-analogous fact that limits in an $n$-category are determined by simplicial $n$-skeleta of indexing categories).

I think this can be found in Section 3 of https://arxiv.org/pdf/1910.04117.pdf

Asad Saeeduddin (Feb 01 2022 at 03:55):

@Mike Shulman very interesting. but I'm having trouble identifying why the BG -> Set example does not count as "1-categorical"

Reid Barton (Feb 01 2022 at 03:59):

It is 1-categorical, and the composition law of G does not matter--the limit is the set of G-fixed points, and these only depend on the actions of all the elements of G.

Leopold Schlicht (Feb 01 2022 at 12:48):

This discussion reminds me of https://mathoverflow.net/questions/391627/shapes-for-category-theory.