Can metric properties can be expressed in category theoretical terms?

A simple example: If you are given the category of Hilbert spaces with the bounded linear mappings as morphism sets, then dualization is a contravariant endofunctor.

So we can talk about "qualitative" properties, or in other words, things which one might label as "soft analysis".

However, in contrast to this, "hard analysis" would not only ask for the dual morphism, but also would like to compare the norms of the morphism and the dual morphism. I have no clue whether category theoretical concepts are powerful enough to talk about such relations reasonably.

More generally, while I do not expect that estimates can be explicitly stated in these algebraic terms, I would like to express that many algebraic constructions are metrically well-behaved.

Suppose I am given some objects and morphism sets in the Hilbert space category, and build new objects and morphism sets from these, e.g. direct sums, tensor products, apply certain well-known functors. The morphism that are constructed are either with norm $1$ - e.g. inclusions and projections for the direct sum - or they are constructed through a functor, like dualization, such that their norms can be easily estimated in terms of the norms of their 'preimages'.

What does this tell me about the reach of category theory, and can we describe the metric behaviour of categorial constructions in categorial terms?

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Don't we have $\lVert f^*\rVert =\lVert f\rVert$? –  Rasmus Nov 11 '11 at 7:48
I don't really understand what you're asking, but note that there's no need to confine yourself to one category of Hilbert spaces. Another reasonable one would be the category of Hilbert spaces and morphisms of norm $\leq 1$. This has a few properties that the one you mention lacks and is the adequate setting for isometric considerations and the proper definition of the tensor product, the adjoint, etc. The category of Hilbert spaces and bounded linear maps is a localization of it and the functors you mention descend nicely. –  t.b. Nov 11 '11 at 11:17
@Rasmus: We indeed have this equivalence. But besides the algebraic statement, that the dualization functor is naturally equivalent to an endofunctor, I also would like to express in categorial terms that this equivalence holds. In contrast, the norm of the dual morphism could have no relation at all to the original morphism. Such a functor would be useless for analytic purposes. –  shuhalo Nov 11 '11 at 15:06

Forgive the brevity of this answer but I don't have enough time, I hope the following may help you.

It's indeed possible express lots of analysis in categorical term, for instance you can define a metric space as an enriched category over the category/linear order of real numbers, for more details you can take a look to this link. For more information about application of category theory in context of analysis John Baez web site is a good place where to start, there you can find a lot of material, links and references.

If I find some other time I'll add other stuff.

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I know you were in a hurry, but just in case: please write analysis in the future, not analisys :) –  t.b. Nov 11 '11 at 11:09
Sorry, thanks to have provided the corrections. –  Giorgio Mossa Nov 11 '11 at 11:55

You might be interested in learning the basics of Gel'fand duality, $C^*$-algebras (from a categorical viewpoint) (depending on what you know about CT you may find it boring or too difficult) and so on.

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