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Stream: learning: questions

Topic: Sheaves and Bundles

Sahil Imtiyaz (Sep 24 2023 at 13:47):

There is an adjunction between category of sheaves over a topological space X and category of fiber bundles over X. I have read your post on it [https://johncarlosbaez.wordpress.com/2020/01/07/topos-theory-part-2/], but may be I am not purely into mathematical imagination so was bit hard for me.

I start thinking on this as; Let's say there is a fiber or a sheaf associated with X and we try to see what (say) a set over an open set U looks like; as per sheaves it would be locally homeomorphic to U in X and as per fiber it would be locally homeomorphic to the Cartesian product of U times F (F is standard fiber). It means the section in the set in fiber has a kind of dual characterisation: as a fiber and as a point or set in the total space which seems to me that sheaves don't consider total space at least naturally in their definition. Am I correct? And if it is the case then the adjunction between these categories would be to associate the total space structure to sheaves and in other sense to delete this notion from fiber bundles!! I am sorry for this mathematical abuse. Moreover, are fibers general than sheaves? @John Baez

John Baez (Sep 24 2023 at 14:08):

Let's say there is a fiber or a sheaf associated with X

Do you really mean 'fiber' here, or maybe 'fiber bundle'? I actually hope you meant to say 'bundle'.

A bundle, in a very general sense, is any map of topological spaces $p: E \to X$ where $E$ is called the total space and $X$ is called the base space. Then the fiber of $p: E \to X$ over the point $x \in X$ is defined to be the inverse image $p^{-1}(x)$.

John Baez (Sep 24 2023 at 14:09):

In the notes you referred to I described an adjunction between the category of sheaves over $X$ and the category of bundles over $X$.

John Baez (Sep 24 2023 at 14:14):

Note I was not discussing 'fiber bundles' in those notes! A fiber bundle is a very special sort of bundle: one that is 'locally trivializable', meaning roughly that for each point $x \in X$ there is a neighborhood $U$ for which the bundle is isomorphic to the trivial bundle $U \times F \to U$. Here $F$ is a topological space called the standard fiber $F$.

For a precise definition click the link. I am mentioning fiber bundles only to emphasize that my notes were not talking about those.

John Baez (Sep 24 2023 at 14:15):

I mention all this because you seem to be using the words 'bundle', 'fiber bundle' and 'fiber' in ways that make it hard for me to understand your question. So the first step might be to become very clear about their definitions, with examples of each.

John Baez (Sep 24 2023 at 14:15):

The adjunction I was discussing is between sheaves and bundles.

John Baez (Sep 24 2023 at 14:19):

Moreover, are fibers general than sheaves?

That doesn't make sense. It would make sense to ask if bundles are more general than sheaves.

But even then, "more general" is a funny concept when you have an adjunction: in this case our adjunction lets us turn any bundle into a sheaf and also turn any sheaf into a bundle.

Graham Manuell (Sep 24 2023 at 18:44):

At least if all the spaces involved are sober, the category of sheaves over X is a full subcategory of the category of bundles over X, so I think it's fair to say bundles are more general than sheaves in a fairly obvious sense. I don't know what happens in the non-sober case.

John Baez (Sep 24 2023 at 19:06):

Thanks!

Mike Shulman (Sep 25 2023 at 01:02):

More generally, if you have an adjunction in which one of the functors is fully faithful, it usually makes sense to think of the objects of the codomain of that functor as more general than those of its domain.