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The notion of “superconvex spaces” generalizes the idea of convex spaces by replacing finite affine sums with countable affine sums.
The category of superconvex spaces is the category of algebras for the countable distribution monad defined on by:
For a set let be the set whose elements are functions such that
for only countably many , and
. (The limit of the countable sum is one.)
Note that the sum above is countable if one excludes all the zero addenda.
The elements of are called countable distributions or countably-supported probability measures over .
Given a function , one defines the pushforward as follows. Given , then is the function
(Note that, up to zero addenda, the sum above is again countable. Moreover, once we have defined countably affine, note that the pushforward map is countably affine.)
Let denote the set of all probability measures on the set of natural numbers, hence every can be represented as where with each . Here is the Giry monad, but because we want to forget the -algebra associated with the measurable space we often write for its underlying set. That set may be regarded as the countably infinite-dimensional simplex, as such it is the prototypical example of a superconvex space.
Given any set , a sequence , and any , we refer to the formal sum as a countably affine sum of elements of , and for brevity we use the notation to refer to a countably affine sum dropping the explicit reference to the condition that the limit of partial sums converges to one. An alternative notation to the countable affine sum notation is to use the integral notation
(superconvex spaces)
We say a set has the structure of a superconvex space if it comes equipped with a function
such that the following axiom is satisfied:
Axiom. If and is a sequence of probability measures on then
The axiom uses the pushforward measure and the natural transformation of the Giry monad at component , , which yields the probability measure on the measurable space whose value at the measurable set is given by the composite of measurable maps
(category of superconvex spaces)
A morphism of superconvex spaces, called a countably affine map, is a set function such that
where the composite gives the sequence in with component . Composition of countably affine maps is the set-theoretical composition.
Superconvex spaces with morphisms the countably affine maps thus form a category denoted .
In physics, superconvex spaces have been referred to as strong convex spaces, and since probability amplitudes are employed there one makes use of the -norm instead of the tradition ‘’-norm’‘ which is used above. By using
where
and is the complex conjugate of , applied to the above axioms one obtains superconvex spaces useful for physics.
For every sequence and every the property
holds.
Choose the constant sequence with value applied to the axiom yields the result.
The most basic property of superconvex spaces, is
For any superconvex space every countably affine map is uniquely specified by a sequence in , hence we have .
Every element has a unique representation as a countable affine sum , and hence a countably affine map is uniquely determined by where it maps each Dirac measure . Thus specifies a sequence in .
A function is a countably affine map if and only if is monotone, implies .
Necessary condition. Suppose that is a countably affine map. Let . By the superconvex space structure on it follows, for all , that . If is not monotone then there exist a pair of elements such that with . This implies, for all , that
which contradicts our hypothesis that is a countably affine map.
Sufficient condition. Suppose is a monotone function, and that we are given an arbitrary countably affine sum in , so that for all we have . Since the condition defining the superconvex structure is conditioned on the property , the countably affine sum is not changed by removing any number of terms in the countable sum whose coefficient . Hence for all such that it follows that so that
where the last equality follows from the definition of the superconvex space structure on .
The standard free space construction can be applied to superconvex spaces to obtain an adjoint pair where consists of all formal countable affine sums, for and , modulo the relations
so that the sole necessary axiom of a superconvex space is satisfied.
The monad arising from this adjunction, with the usual unit and counit maps (for the free and forgetful functors), is the countable distribution monad defined in the Idea section, and the category of algebras of that monad is superconvex spaces.
Note that the finite distribution monad also arises from the free functor and forgetful functor adjunction. (The verification that the finite and countable distributions arise from the free functor and forgetful functor is straight forward.)
A direct method to prove that the category of algebras of , , is equivalent to the category of superconvex spaces is to show that the comparison functor has a two-sided inverse . This functor is defined on objects by taking as the set with the superconvex structure defined by where the countable sum on the right hand side is a formal countable sum, which is an element of . Given any map of algebras it follows, as shown by the (only) proof given on the Giry monad page is that the function , which we now view in , is necessarily a countably affine map. (Simply replace the terms with the term which is an element in .)
The category has all limits and colimits. Furthermore it is a symmetric monoidal closed category under the tensor product. The proof of the latter condition follows the proof by Meng, replacing finite sums with countable sums.
The full subcategory consisting of the single object is dense in , and hence we can employ the restricted Yoneda embedding to view superconvex spaces as the functors , where is viewed as a monoid.
An ideal in a superconvex space is a subset such that whenever and is a countable affine sum with the coefficient of nonzero then . Ideals are useful for defining functors to or from .
A fundamental superconvex space is the set with the superconvex space structure defined, for every sequence by
That structure on shows that the function defined by is a countably affine map.
As an example of the utility of ideals in a superconvex space we note that for the Giry monad the measurable space , with the smallest -algebra such that the evaluation maps are measurable for every measurable set in , has a superconvex space structure defined on it pointwise.
Note that if is any measurable space the maximal proper ideals of are of the form or for a nonempty measurable set in .
To prove this, note that both and are ideals in the superconvex space , with the natural superconvex space structure, and it follows that and are ideals in . (The preimage of an ideal under a countably affine map is an ideal in the domain space. The proof is the standard argument for ideals in any category.) Consider the ideal . To show that this is a maximal proper ideal suppose that is another ideal of such that . Every element which is not in has the defining property that . Now let . If and then which implies which is self-contradictory. Thus must be all of which shows is a maximal (proper) ideal. The argument that the ideal is a maximal ideal is similiar except we replace the condition in the above proof with .
If we restrict to the category of standard measurable spaces then every object has a countable generating basis and it is then clear that every ideal in is a countable intersection of maximal ideals, and hence measurable. This implies, for example, that every countably affine map is also a measurable function with having the powerset -algebra. (The only ideals of are the principal ideals .)
The one point compactification of the real line , with one point adjoined, denoted , which satisfies the property that any countably affine sum if either (1) and for any index , or (2) the sequence of partial sums does not converge, is a superconvex space. The real line is not a super convexspace since we could take and and the limit of the sequence does not exist in . Thus while is a convex space it is not a superconvex space.
The only nonconstant countably affine map is given by for all and (for the superconvex space structure on determined by ).
A pathological space, useful for counterexamples, is given by the closed unit interval with the superconvex space structure defined by the infimum function, .
Consider the probability monad on compact Hausdorff spaces, where the algebras are precisely the compact convex sets in locally convex topological vector spaces together with the barycenter maps .
Given such a space we can endow it with a superconvex space structure by defining, for all , countable affine sums by which, along with the pointwise superconvex space structure on makes the barycenter map a countably affine map.
To prove that this endows with a superconvex space structure note that for all to obtain
To prove that is countably affine on use the property that .
The method employed in this example is not restricted to locally convex compact Hausdorff spaces. It shows that the algebras of a probability monad are a superconvex space. That implication is the motivation for the next example.
The notion of superconvex spaces originates with:
The proof that has no cogenerator is due to:
The fact that is a symmetric monoidal argument can be proven the same way it is for convex spaces simply by replacing the finite affine sums with countable affine sums. That proof was given by
Proposition 1.2 in
is particularly useful for viewing superconvex spaces as positively convex spaces which are somewhat easier to work with because the condition is replaced by the inequality .
The fact that the functor is a codense functor can be found in
although several aspects, such as the construction with the right-Kan is incorrect.
For purposes of constructing models of complex systems using superconvex spaces the construction given in Example 6.1 of the following article applies equally well to superconvex spaces.
The term strong convex space was employed in:
Last revised on July 5, 2024 at 07:50:49. See the history of this page for a list of all contributions to it.