it's 12:04 am on a thursday. time for my inferiority complex to kick in

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today's question: what the Fuck is a pullback

or, rather, let me elaborate. Why The Fuck is a pullback

lewd

@hazel it's when y
it's when you pull out and
and y
you

@hazel I'm bad at telling if this is a serious question but in case it is: the name comes from geometry. One way to think of a map f: B -> Y is that you're taking the space Y, and at each point y you're attaching the space f^-1(y), to get the total space B. When certain properties are satisfied you call this a vector bundle, fiber bundle, etc. A pullback is when you have a bundle B over Y, and you "pull it back" over a map X -> Y, to get a new bundle over X, namely X ×_Y B.

In algebraic geometry this construction is nice because being categorical makes it clear how the algebra should work, even when your category is very different from Set. Also, a lot of nice properties of morphisms are preserved under pullbacks.

@hazel I guess also the important special case is that X is a single point. Then the pullback is just the fiber over its image in Y. This gives you a way to describe fibers algebraically.

It also means that if P is some property of morphisms that is preserved by pullbacks, then if f satisfies P, all its fibers also satisfy P. This helps with intuition about what properties of morphisms actually mean: e.g. if f is a finite morphism, that means at least that all its fibers are finite.

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