If I associate a copy of an algebra with every point on a manifold - such that one could specify a connection between algebras associated with neighboring points - have I specified an algebra bundle? If the definition of an algebra bundle is more restricted than that, is there a recognized term for what I've described?
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What are neighboring points on a manifold? – Matt Samuel Oct 31 '19 at 17:36
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By a neighboring point of p in a topological space, I mean a point that is in every neighboorhood of p (every contiguous open set that includes p). – rossng Oct 31 '19 at 17:52
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In a manifold, the only point in every neighborhood of $p$ is $p$. – Matt Samuel Oct 31 '19 at 17:53
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I thought a neighborhood is an open set for which one can move in any direction away from $p$ without leaving the set. Wouldn't any neighborhood of $p$ need to include other points (you can draw a path through the set that does not terminate at $p$)? – rossng Oct 31 '19 at 18:08
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Yes, it includes other points. But given any point $q$ in the neighborhood other than $p$, there is a neighborhood of $p$ contained in the original neighborhood that doesn't contain $q$. – Matt Samuel Oct 31 '19 at 18:43
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Ah ok, that makes sense. Rather than a connection between neighboring points, a connection defining how elements of the algebra at $p$ map onto the algebra at another point given a path between the points, which converges to the identity for points and paths within a neighborhood of $p$ as we shrink the neighborhood. Is that better defined? – rossng Oct 31 '19 at 19:37
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That makes more sense, but usually if we're attaching algebras to a space it would be a sheaf of algebras, which assigns an algebra to each open set. – Matt Samuel Oct 31 '19 at 19:47
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Let us continue this discussion in chat. – rossng Oct 31 '19 at 19:53
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I won't be a very good chat companion, but you can look at https://en.wikipedia.org/wiki/Sheaf_%28mathematics%29 – Matt Samuel Oct 31 '19 at 19:57