Category Theory and So-Called Fundamental Results of Mathematics

Category Theory and So-Called Fundamental Results of Mathematics

[Ellis D. Cooper]

At http://en.wikipedia.org/wiki/Fundamental_theorem a "fundamental
theorem" in a field of mathematics is defined to be "a theorem [or
lemma] considered central to that field." The designation
"fundamental" is often a matter of tradition, belonging to the
history or sociology of mathematics.

Some results, for example Hilbert's Nullstellensatz in algebraic
geometry, are fundamental yet not generally designated "fundamental,"
although at
http://en.wikipedia.org/wiki/Timeline_of_category_theory_and_related_mathematics
it is indeed called that, along with a number of other results not
called "fundamental" elsewhere.

At http://en.wikipedia.org/wiki/Category:Fundamental_theorems there
are links to articles on fundamental theorems of finitely generated
abelian group, algebra, arithmetic, calculus, calculus of variations,
combinatorial enumeration, curves, cyclic groups, Galois theory,
homomorphisms, and linear algebra.

To these can be added three "fundamental theorems of functional
analysis" (Hahn-Banach, Open Mapping, Uniform Boundedness), the
"fundamental theorem of Lie groups," and even a "fundamental theorem
of fractal geometry" (Iterated Function System Convergence).

Some of these results have been or can be revealingly stated if not
yet proved in category theory terms (finitely generated abelian
group, Galois theory, homomorphisms,...). My question is, what if any
are the obstructions to extending this virtue to all of the so-called
fundamental theorems?

Ellis D. Cooper


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]