The question is based on the following problem from Vakil's notes in Algebraic Geometry:
2.2.C. The identity and gluability axioms (of sheaves) may be interpreted as saying that $\mathcal{F}(\cup_i U_i)$ is a certain limit. What is that limit?
The precise solution / proof / answer to this question isn't as interesting to me as the intuitive approach we take to get there, because while thinking about this problem, I've realized that I am quite confused about the intuition behind limits and colimits.
Before going into my thoughts, here are some stack exchange questions and links I found in looking into ths:
- In (relatively) simple words: What is an inverse limit?
- Inverse vs direct limits
- intuition for limits
- Limits wiki page
(I have looked at more answers, but these seemed the most relevant to what I want to ask)
Here are two trains of thought I've had for each (these will be quite hand-wavey. As I said, I'm not looking at the proof just yet. I'm more interested in the intuition):
For viewing the axioms as limits: The motivating example of limits in my head are p-adic numbers. Ravi brings them up as his primary example as well (1.4.1 - 1.4.B). It seems like they key idea here is that the limit is some 'global' object that we demand that it behaves well locally. Based on the stack exchange posts linked above, I got the sense of something similar - inverse limits are 'zoomed out' objects, and we construct them from their local projections, just like p-adic numbers. Now I could argue that this is the spirit of the two axioms, one is defining a global object (the unique global section you get after gluing) and making sure it plays well locally.
For viewing the axioms as colimits: The motivating example of colimits is fractions, according to Ravi. I don't have a very good sense for this, even though I can formally work out proofs of exercises such as 1.4.C, which asks to prove that $\mathbb{Q} = \varinjlim \frac{1}{n}\mathbb{Z}$. I don't see any obvious way (even within the stack exchange answers) to view the gluing axioms in this sense. But I do intuitively feel comfortable with viewing colimits as disjoint unions modulo the relevant relations (one could ask me why this makes sense intuitively and not the fraction view, and one would not get a satisfying answer since I don't intuitively see their connection). With the disjoint union view, viewing colimits as unions makes intuitive sense, and this seems to be exactly the spirit of the gluing axioms - where we are literally gluing ('union-ing') the different sections to get a bigger section.
Both interpretations seem reasonable to me and so my question are 1) are these intuitive views accurate/correct or are there any glaring issues with these views? 2) (The more important question) How would you argue for the correct answer and against the incorrect answer? Where do the above views work and where do they fail in context of the correct and incorrect answers? And/or how would you approach approaching the problem intuitively. I.e. what are you looking for when you are trying to intuit whether something is a limit or colimit.
(Note, I am not looking for a proof unless the proof provides an insight to the second question above)
Thanks a lot!
