[Inquiry] Re: Differential And Riemannian Manifolds

[Jon Awbrey]

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DARM.  Note 15

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| 2.2.  Submanifolds, Immersions, Submersions (cont.)
|
| We use transversality as a sufficient condition under which the fiber product
| of two morphisms exists.  We recall that in any category, the "fiber product"
| of two morphisms f : X -> Z and g : Y -> Z over Z consists of an object P
| and two morphisms:
|
| g_1 : P -> X   and   g_2 : P -> Y
|
| such that f o g_1  =  g o g_2, and satisfying the universal mapping property:
|
| Given an object S and two morphisms:
|
| u_1 : S -> X   and   u_2 : S -> Y
|
| such that f o u_1  =  g o u_2, there exists a unique morphism u : S -> P
| making the following diagram commutative:
|
|             S
|             o
|            /|\
|           / | \
|          /  |  \
|     u_1 /   u   \ u_2
|        /    |    \
|       /     |     \
|      v      v      v
|   X o<------P------>o Y
|      \  g_1   g_2  /
|       \           /
|        \         /
|      f  \       /  g
|          \     /
|           \   /
|            v v
|             o
|             Z
|
| The triple (P, g_1, g_2) is uniquely determined,
| up to a unique isomorphism (in the obvious sense),
| and P is also denoted by X x_Z Y.
|
| Lang, DARM, p. 29.
|
| Serge Lang,
|'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.

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