Let K be a convex. Some subsets are the image of an affine map g of K into itself. What should such subsets be called? The affine map g may be considered to be idempotent. I guess that there exist a concept from category theory that describe objects L with the property that there exists morphisms f:L->K and g:K->L such that gf is the identity on L.
I am not well-versed in the generalities of this question, but when I ran into the basic concept while studying the structure of discriminating information, I ended up giving it the name "self-scaling" (for lack of a better term); namely, every "true" instantaneous discriminating information statistic (i.e., LLR) possesses the property that lambda=ln(p1(lambda)/p0(lambda)) for every allowed value of lambda. This property is extremely useful; for example, it provides a strong test of authenticity that any purported statistic is indeed a calibrated instantaneous measure of discriminating information.
@ John My question is actually related to discriminating information. What are the functions f and g in your "self-scaling" setup?
I will have to think about it. I never actually formulated functions mapping separate L to K and then K to L, only a unified mapping L to L (which is obtained by invoking symmetries on the more general forms, the same symmetries that lead to the symmetric form of the KL divergence). My distinct impression in working through all this is that the "symmetric structure" is an important piece of the story, but I haven't studied what happens if you remove that symmetry. The first paper I referred you to earlier (on the LLR and generalized SNR), among other things, develops self-scaling and some of its implications, but starting from a fully symmetric viewpoint.
Hi Peter--- In category theory, the morphisms you speak of (that is, f: L --> K and g: K--> L such that gf = identity of L) are called a retraction-section pair. g can be said to be a section of f, f a retraction of g. Then fg: K ---> K (what you initially called g in your question, which is of course different from the g appearing in the retraction-section pair) is indeed idempotent. I'm not sure whether there is a term for the object L (but see discussion below about images)... perhaps in keeping with the focus on morphisms in category theory, there is no special name for the object. (Just as sometimes, for an object B, the term "subobject of B" is used to refer to, say, a monomorphism from some object A into B, rather than to refer to A...) Anyway, if there's a name for an object relating to K like L does, namely the domain of a retraction (equivalently, codomain of a section), that would be the category-theoretic version of the concept you describe. There is also a category-theoretic notion of "image", but I don't recall enough detail at the moment to say whether we could call the "image of an idempotent affine map" that you refer to.., an example of an "image", although I suspect that indeed, L is in category theoretic terms the image of the idempotent fg.
Hi Howard Thank you. That makes sense. It is natural to say that an interval is a section of a disc in the category of convex sets and affine maps. The interval can be embedded as a section of the disc in several ways which leads to different sections. For a specific object the sections may be considered as objects of a category, which leads to a partial ordering of the domains of these sections.
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