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— c° .

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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