ELEMENTS OF NON-LINEAR FUNCTIONAL ANALYSIS
by R. A. Graff
1. SECTION FUNCTORS
Let M be a smooth compact manifold, possibly with boundary.
1.1 Definition. FB(M) is the category whose objects are
smooth finite-dimensional fiber bundles over M (the fibers are mani-
folds without boundaries) and whose morphisms are defined as follows:
if Ew E
2
are objects of FB (M) , then
Map(E-,/E2) = {C fiber-preserving maps from E-. to E2}
1.2 Definition. A section functor W\ on FB(M) is a functor
from FB(M) to the category of topological spaces and continuous
maps such that:
(1) For each bundle E over M, the points of ^(E) are
continuous sections of E, and the natural inclusion
77{(E) + C (E) is continuous.
(2) For bundles E-. , E2 over M, and a morphism
f e Map(E1,E2), 771(f) (s) = fos for all s e ^(E-^ .
Note that we have (by definition) a natural transformation
n— .
Our first goal is to show that the local structure of 771(E)
depends only on the local structure of E. In order to do this, it
will be necessary to introduce a category which is related to FB(M).
1.3 Definition. FVB(M) is the category whose objects are
smooth vector bundles over M, and whose morphisms are defined as
follows: if §/ " H are smooth vector bundles over M, then
Map(|,r]) = {C fiber-preserving maps from £ to r\] .
Note that FVB(M) is a full subcategory of FB(M).
Received by the editor June 29, 1977.
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