Pullback bundle as pullback example (category theory)

Sometimes I think I’m in an  odd variation of the game Twister except instead of colored circles, the circles are labeled with topics.  Currently I have at least one limb on TQFTs, another limb on Lie theory/representation theory, another limb on quantum mechanics, and the last limb moving back and forth across topics that come up.  Fiber bundles are an example. I know these turn up in physics (though I don’t know why yet).  I introduced them in Spheres-2.  To continue the analogy, I only moved a limb to the circle labled “Fiber Bundles” for a short while, enough to digest some information.  I’m glad I did though because they provide a nice illustration of pullbacks.

The term pullback is loaded.  I think of pullbacks of functions and differential forms in differential geometry for example.  It’s entirely possible that the category theory definition is consistent with that but I can’t see it yet.  I learned about pullbacks (and their dual: pushouts) recently watching the Catsters again on Youtube.  They have two very good videos: Pullbacks and Pushouts -1 and Pullbacks and Pushouts – 2.

The initial setup is this: you have three objects A,B, and C in your category and a pair of maps, f:A\to C and g:B\to C.  As a diagram,

CodeCogsEqn (2)

From here we can talk about the pullback of g along f or the pullback of f along g.  I think for the moment I’ll talk about the pullback of the diagram which, though I haven’t heard or read, seems appropriate.  It turns out that this construction is an example of a limit of a diagram (something incredibly cool in its own right).  The pullback of this diagram is another object D and another pair of maps s:D\to A and t:D\to B such that the following square commutes,

\begin{matrix}  D & \overset{t}{\longrightarrow} & B \\  \downarrow_s & & \downarrow_g \\  A & \overset{f}{\longrightarrow} & C  \end{matrix}

This construction, like many I’ve been learning about, satisfies a universal property.  This means that for any other object E and morphisms m:E\to A and n:E\to B there exists a unique morphism (dotted arrow) such that the resulting triangles commute,


As a verb, pullback seems appropriate at least in the sense that the maps g and f are pulled back to a common domain.  But it’s not quite the same as when you have a bunch of manifolds and functions M\overset{f}{\longrightarrow} N\overset{g}{\longrightarrow} S and you think of pulling back g to M.

This is where the pullback bundle comes in.  Suppose we have a fiber bundle E over a base space B with projection \pi : E\to B.  Suppose we have another topological space B’ and continuous map f:B'\to B then we can pull back the bundle over B to become a bundle over B’, call it E’.  We construct E’ by taking a subset of B'\times E composed of all those pairs (b', e)\in B'\times E such that f(b')=\pi(e).  We’ll need a projection map from E’ to B’ and we define it this way \pi'(b',e)=b'  We also have another projection onto the other coordinate, f':E'\to E.  The resulting diagram,

\begin{matrix}  E' & \overset{f'}{\longrightarrow} & E \\  \downarrow_{\pi'} & & \downarrow_{\pi} \\  B' & \overset{f}{\longrightarrow} & B  \end{matrix}

commutes by construction (details are from wikipedia’s page on the pullback bundle, but note the similarities between how Cheng (Catsters) constructs pullbacks).  So what about the fibers?  Roughly, if b\in B then \pi^{-1}(b) is homeomorphic to the fibers, F, of E.   What about B’?  Let’s take b'\in B' and consider \pi'^{-1}(b')\subset E'.  What is this?  It’s all the ordered pairs with first coordinate b’ and second coordinate any e\in E such that \pi(e) is projected to the same point as b’ is sent to under f. Those e\in E collectively form a fiber (or a subset homeomorphic to the fiber F). This means that \pi^{'-1}(b')={b'}\times\pi^{-1}(f(b')) which is homeomorphic (pretty sure anyways) to {b'}\times F which is certainly homeomorphic to F.  In this way we pulled back the space E which fibers over B to a space E’ that fibers over B’. The following  illustration might help,



About because0fbeauty

Fascinated by the way mathematics and physics interact, captivated by visual and tactile mathematics and hoping to become a better expositor of these things is why I blog...occasionally...when I remember.
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One Response to Pullback bundle as pullback example (category theory)

  1. Pingback: Lie groups and Lie algebras – 2 | because0fbeauty

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