Re: A well kept secret

["Reinhard Boerger" <Reinhard.Boerger@FernUni-Hagen.de>]

Hello,


I still like to add some remarks. Category theory is one part of
mathematics, and it should be treated not better, but not worse than others.
It looks more important to me that categorical thinking becomes popular in
other areas in mathematics. Some years ago, a functional analyst needed half
an our to prove that homeomorphic Banach spaces have homemorphic duals, a
simple consequence of the fact that all functors preserve isomorphisms. 

Another example from my own experience: People, who worked about
orthomodular lattices noticed that they have no tensor product. So they
tried to weaken the notions and ended up with effect algebras, but
unfortunately they did not admit a tensor product either. Su people looked
for other notions. But they had already shown that a tensor product of
effect algebras exists if one admits 0=1; i.e. the tensor product my
collapse. But because they did not admit this, they had to formulate their
result more complicated.

Later I saw that tensor products of orthomodular posets exist if one admits
0=1; the easy proof uses the Adjoint Functor Theorem and does not give much
insight into the structure. It also seems to work for orthomodular lattices.

My preference for orthomodular posets rather than lattices is also inspired
by categorical thinking. The idempotents of an arbitrary ring with 1 form an
orthomodular poset, and this construction yields a functor. This is the
non-commutative analogue to the Boolean algebra of idempotents of an
arbitrary ring. But most people were inspired by quantum dynamics and were
looking for an abstraction for the set of projections of a Hilbert space.
Here joins and meets exist (somehow accidentially) because projections
correspond to closed subspaces. But they are not continuous and have no
physical meaning in general. I think it is often better to look for
functorial notions than to use ad-hoc-abtractions.


Greetings
Reinhard




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