make the obvious diagram commute:
The product of these objects is given by the fiber product or pullback:
X ---h--> X'
along with the map .
See that for consistenty, we could also have defined this as .
Since our condition is that , it all works out.
Said differently, we consider the product of fibers over the same base-point.
§ Fiber products of arbitrary bundle over a single-point base space
If , then the projections are always , and the fiber
product is the usual product.
§ Fiber products of singleton bundle over arbitrary base space
If is arbitrary while (for Point),
then this bundle will lie over some point in , given
by , where the special point
is chosen by . If we now consider some other bundle over ,
Then will pick the element .
That is , which is the fibre of over
the special point . This explains the name.
§ Fiber products of vector bundles
Consider a fiber bundle . Now consider a new base space
with a map . So we have the data:
Given this, we would like to pullback the bundle along to get a new
bundle over .This is defined by:
This is equipped with the subspace topology. We have the projection map
, . The projection into
the second factor gives a map , .
This makes the obvious diagram commute:
Any section of induces a section of
, by producing the function (given as a relation):
E' -h-> E
B' -f-> B
This has codomain . To check, if is in ,
we need . But this is true since
is a section, and thus .
Moreover, we need to check that is indeed a section of . For this,
we need to check that . Chasing definitions, we find
that this is:
Hence we are done, we have indeed produced a legitimate section.
§ Fiber products of Spec of affine scheme
Let be rings. consider . What is , in terms of
, , and whatever data you like about ? (Say I give you both and ).
The answer is that apparently, it's exactly .