analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
β¦
β¦
The notion of βsuperconvex spacesβ generalizes the idea of convex spaces by replacing finite affine sums with countable affine sums.
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 that measurable space we often write to denote the underlying set of . That set is the countably infinite-dimensional simplex. The set (= ) is the prototype space from which the axioms are abstracted.
Given any set , a sequence , and any , we refer to the formal expression 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
We say a set has the structure of a superconvex space if it comes equipped with a function
such that the following two axioms are satisfied:
Axiom 1. For every sequence and every the property
holds.
Axiom 2. If and is a sequence of probability measures on then
The second 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
Note the second axiom alone is sufficient because by choosing the constant sequence with value it follows that the second axiom implies the first axiom.
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 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 ββ-normββ 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.
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 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 this endows with a superconvex space structure note that for all to obtain
To prove 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 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.
One important aspect of the free space adjunction is that it implies
Lemma Let be an arbitrary set and the free superconvex space on . Let the quotient map be the coequalizer of the parallel pair . Then the quotient of the free space is the free space of the quotient,
Proof The quotient map is, up to isomorphism, just . Because the quotient map is countably affine it is completely specified by where it sends the elements . Because we have
The function specifies an equivalence relation on , yielding the set . Because is surjective we can thus endow the set with the superconvex space structure of using the bijective map of sets . Hence the above equation extends (trivially) on the right by .
The map specified by specifies a -isomorphism. By the isomorphism we can write this as . That completes the proof.
Now let denote a standard measurable space, so is standard also. Forgetting the measurable structure on we can take the free space of that set, which consist of all countable affine sums for all and all . These countable affine sums are defined pointwise for all .
We observe that the free space can also be viewed as the image of the functor on , where is the Giry monad (functor) viewed as a functor into .
We would like to extend the free space adjunction with an adjunction such that the composite is the Giry monad on . The category denotes some subcategory of so that the adjunction can be constructed. If such an adjunction exists the counit of the adjunction says every space is the quotient of a free space, i.e., is the coequalizer of a parallel pair of maps into a free space. To illustrate this consider the following elementary example.
The coequalizer of the pair of points and , where . The coequalizer of those pair of points is the discrete space with the structure defined by for all , for all , and for all .
If the functor exists it should recognize that that superconvex space arises as a quotient space of , and hence should have the barycenter map given by the mapping , , and for all .
Let denote the full subcategory of consisting of the two objects and , and let be the inclusion functor.
If we look at the category there are two countably affine maps,
and
from which every other countably affine map can be obtained by composing either or with a monotonic function . Because is discrete there are no non-constant countably affine maps .
The coequalizer of those two maps yields the discrete space for which is the space we are trying to find - it is the smallest free space such that there is a countably affine map .
Whether probing a space with arrows to objects and products of objects in is sufficient is unknown but a clue is given in the fact that every standard space is -isomorphic to either or a countable discrete space. Hence should intuitively consists of quotients of (and smaller) - the countability condition being the critical condition. That condition rules out spaces such as and the pathological space of Example 5.4.
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 August 19, 2022 at 09:12:06. See the history of this page for a list of all contributions to it.