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The yoneda lemma gives us a characterization of $Psh(\mathcal{C})$ that seems very similar to the theory of distributions. That is, we have a notion of representable presheaves, similar to representable distributions. The ability to talk about presheaves as colimits of representables correlates to the more complicated notion of distributions as derivatives and limits of representables. The whole idea of "test objects" is exactly the same as the notion of "test functions" and so on. Is there a deep connection there or is it just another case of stretching the terminology?

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In what way is the whole idea of "test objects" exactly the same as the notion of "test functions"? –  Mariano Suárez-Alvarez Dec 31 '09 at 22:52
    
This is just a general theme in mathematics. Approximate complicated objects by simpler and more manageable ones by using some kind of "limiting" procedure. Distributions are also an instance of an object in mathematics that can be developed from several different perspectives and some of those methods will certainly have relations to other constructions. –  davidk01 Jan 1 '10 at 2:23
    
If you want more specific answers then you should probably narrow the focus of the question. I've seen at least 4 different presentations of distributions and 2 have absolutely no relation to any kind of categorical constructs. –  davidk01 Jan 1 '10 at 2:58
    
Your objection is useless for the following reason: two of the constructions you've seen are related to categorical constructions. –  Harry Gindi Jan 1 '10 at 22:12
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up vote 4 down vote accepted

While maybe not exactly what you were after, here is something that you might enjoy looking into, which relates presheaves and distributions.

There exists a category of sheaves on certain test objects, such that

  • this category is a smooth topos into which the category of smooth manifolds embeds full and faithfully.

  • in this topos, there exists not only a notion of infinitesimals, as in every smooth topos, but also of invertible infinitesimals, in fact, this topos provides a model for nonstandard analysis.

  • Accordingly, in this topos distributions on manifolds are given by actual functions - internally in the topos.

So in a way, this topos makes precise and manifest the intuition that distributions are "generalized functions". They are functions in this topos.

The topos that I am talking about is described in great detail in section VI of the textbook Models for Smooth Infinitesimal Analysis. The test objects in this case, i.e. the objects in the site that the topos is a category of sheaves over, are smooth loci.

Distributions are discussed in section VII,3

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You're wrong in one respect. This is exactly what I was looking for. –  Harry Gindi Jan 5 '10 at 14:37
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