superconvex space (Rev #35, changes) in nLab

The where full the subcategory last consisting equality follows from the definition of the single superconvex object space structure onΔ ℕℕ \Delta_{\mathbb{N}} \mathbb{N} . is \end{proof} dense inSCvx\mathbf{SCvx}, and hence we can employ the restricted Yoneda embedding to view superconvex spaces as the functors A^=SCvx(⋅,A)∈Set Δ ℕ op\widehat{A}=\mathbf{SCvx}(\cdot,A) \in \mathbf{Set}^{\Delta_{\mathbb{N}}^{op}}, where Δ ℕ\Delta_{\mathbb{N}} is viewed as a monoid.

An The ideal category in a superconvex spaceASCvx A \mathbf{SCvx} has all limits and colimits. Furthermore it is a subset 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.ℐ\mathcal{I} such that whenever a 0∈ℐa_0 \in \mathcal{I} and ∑ i∈ℕp ia i\sum_{i \in \mathbb{N}}p_i a_i is a countable affine sum with the coefficient of a 0a_0 nonzero then ∑ i∈ℕp ia i∈ℐ\sum_{i \in \mathbb{N}}p_i a_i \in \mathcal{I}. Ideals are useful for defining functors to or from SCvx\mathbf{SCvx}.

The full subcategory consisting of the single object Δ ℕ\Delta_{\mathbb{N}} is dense in SCvx\mathbf{SCvx}, and hence we can employ the restricted Yoneda embedding to view superconvex spaces as the functors A^=SCvx(⋅,A)∈Set Δ ℕ op\widehat{A}=\mathbf{SCvx}(\cdot,A) \in \mathbf{Set}^{\Delta_{\mathbb{N}}^{op}}, where Δ ℕ\Delta_{\mathbb{N}} is viewed as a monoid.

An ideal in a superconvex space AA is a subset ℐ\mathcal{I} such that whenever a 0∈ℐa_0 \in \mathcal{I} and ∑ i∈ℕp ia i\sum_{i \in \mathbb{N}}p_i a_i is a countable affine sum with the coefficient of a 0a_0 nonzero then ∑ i∈ℕp ia i∈ℐ\sum_{i \in \mathbb{N}}p_i a_i \in \mathcal{I}. Ideals are useful for defining functors to or from SCvx\mathbf{SCvx}.

Examples

\begin{example} A fundamental superconvex space is the set ℕ\mathbb{N} with the superconvex space structure defined, for every sequence s:ℕ→ℕ\mathbf{s}: \mathbb{N} \rightarrow \mathbb{N} by

(8)∫ ℕsdp=inf i{s i|p i>0}. \textstyle{\int_{\mathbb{N}}} \mathbf{s} d\mathbf{p} \;=\; \inf_i \{\mathbf{s}_i \, | \, p_i \gt 0 \} \,.

That structure on ℕ\mathbb{N} shows that the function ϵ:𝒢(ℕ)→ℕ\epsilon: \mathcal{G}(\mathbb{N}) \rightarrow \mathbb{N} defined by ∑ i∈ℕp iδ i↦inf i{i|p i>0}\sum_{i \in \mathbb{N}} p_i \delta_i \mapsto \inf_i \{i | p_i\gt 0\} is a countably affine map. \end{example}

\begin{example} As an example of the utility of ideals in a superconvex space we note that for 𝒢\mathcal{G} the Giry monad the measurable space 𝒢(X)\mathcal{G}(X), with the smallest σ\sigma-algebra such that the evaluation maps ev U:𝒢(X)→[0,1]ev_U: \mathcal{G}(X) \rightarrow [0,1] are measurable for every measurable set UU in XX, has a superconvex space structure defined on it pointwise.

Note that if XX is any measurable space the maximal proper ideals of 𝒢X\mathcal{G}{X} are of the form ev U −1((0,1])ev_{U}^{-1}( (0,1]) or ev U −1([0,1))ev_U^{-1}([0,1)) for U≠XU\ne X a nonempty measurable set in XX.

To prove this, note that both (0,1](0,1] and [0,1)[0,1) are ideals in the superconvex space [0,1][0,1], with the natural superconvex space structure, and it follows that ev U −1([0,1))ev_{U}^{-1}( [0,1)) and ev U −1((0,1])ev_U^{-1}((0,1]) are ideals in 𝒢X\mathcal{G}{X}. (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 ev U −1([0,1))ev_U^{-1}( [0,1)). To show that this is a maximal proper ideal suppose that ℐ\mathcal{I} is another ideal of 𝒢X\mathcal{G}{X} such that ev U −1([0,1))⫋︀ℐev_U^{-1}( [0,1)) \varsubsetneqq \mathcal{I}. Every element P∈ℐP \in \mathcal{I} which is not in ev U −1([0,1))ev_U^{-1}([0,1)) has the defining property that P(U c)=1P(U^c)=1. Now let Q∈𝒢XQ \in \mathcal{G}{X}. If Q∉𝒥Q \notin \mathcal{J} and Q∉ev U −1([0,1))Q \notin ev_U^{-1}([0,1)) then Q(U c)≠1Q(U^c) \ne 1 which implies Q∈ev U −1([0,1))⫋︀JQ \in ev_U^{-1}( [0,1)) \varsubsetneqq J which is self-contradictory. Thus 𝒥\mathcal{J} must be all of 𝒢X\mathcal{G}{X} which shows ev U −1([0,1))ev_U^{-1}([0,1)) is a maximal (proper) ideal. The argument that the ideal ev U −1((0,1])ev_U^{-1}( (0,1]) is a maximal ideal is similiar except we replace the condition P(U c)=1P(U^c)=1 in the above proof with P(U)=0P(U)=0.

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 𝒢(X)\mathcal{G}(X) is a countable intersection of maximal ideals, and hence measurable. This implies, for example, that every countably affine map k:𝒢(X)→ℕk: \mathcal{G}(X) \rightarrow \mathbb{N} is also a measurable function with ℕ\mathbb{N} having the powerset σ\sigma-algebra. (The only ideals of ℕ\mathbb{N} are the principal ideals ↓0⊂↓1⊂…\downarrow \! 0 \subset \downarrow \! 1 \subset \ldots.) \end{example}

\begin{example} The one point compactification of the real line ℝ ∞\mathbb{R}_{\infty}, with one point adjoined, denoted ∞\infty, which satisfies the property that any countably affine sum ∑ i∈ℕp ir i=∞\sum_{i \in \mathbb{N}} p_i r_i = \infty if either (1) r j=∞r_j = \infty and p j>0p_j \gt 0 for any index jj, or (2) the sequence of partial sums does not converge, is a superconvex space. The real line ℝ\mathbb{R} is not a super convexspace since we could take p i=12 ip_i = \frac{1}{2^i} and r i=2 i+1r_i = 2^{i+1} and the limit of the sequence does not exist in ℝ\mathbb{R}. Thus while ℝ\mathbb{R} is a convex space it is not a superconvex space.

The only nonconstant countably affine map j:ℝ ∞→𝟚j: \mathbb{R}_{\infty} \rightarrow \mathbb{2} is given by j(u)=1j(u)=1 for all u∈ℝu \in \mathbb{R} and j(∞)=0j(\infty)=0 (for the superconvex space structure on 2\mathbf{2} determined by 120̲+121̲=0̲\frac{1}{2} \underline{0} + \frac{1}{2} \underline{1} = \underline{0}).
\end{example}

\begin{example} A pathological space, useful for counterexamples, is given by the closed unit interval with the superconvex space structure defined by the infimum function, ∑ i∈ℕp iu i:=inf i{u i|p i>0}\sum_{i \in \mathbb{N}} p_i u_i := inf_i \{ u_i | p_i \gt 0\}. \end{example}

\begin{example} Consider the probability monad on compact Hausdorff spaces, where the algebras are precisely the compact convex sets KK in locally convex topological vector spaces together with the barycenter maps β K:𝒫K→K\beta_K:\mathcal{P}K\rightarrow K.

Given such a space KK we can endow it with a superconvex space structure by defining, for all p∈𝒢ℕ\mathbf{p} \in \mathcal{G}\mathbb{N}, countable affine sums by ∑ i∈ℕp ik i:=β K(∑ i∈ℕp iδ k i)\sum_{i \in \mathbb{N}} p_i k_i := \beta_K( \sum_{i \in \mathbb{N}} p_i \delta_{k_i}) which, along with the pointwise superconvex space structure on 𝒫K\mathcal{P}K makes the barycenter map β K\beta_K a countably affine map.

To prove that this endows KK with a superconvex space structure note that β K(δ k i)=k i\beta_K(\delta_{k_i})= k_i for all k i∈Kk_i \in K to obtain

∑ i∈ℕp iβ K(δ k i)=∑ i∈ℕp ik i≔β K(∑ i∈ℕp iδ k i). \textstyle{\sum_{i \in \mathbb{N}}} p_i \beta_K(\delta_{k_i}) \;=\; \textstyle{\sum_{i \in \mathbb{N}}} p_i k_i \,\coloneqq\, \beta_K\big( \textstyle{\sum_{i \in \mathbb{N}}} p_i \delta_{k_i} \big) \,.

To prove that β K\beta_K is countably affine on 𝒫K\mathcal{P}K use the property that β K∘μ K=β K∘𝒫β K\beta_K \circ \mu_K = \beta_K \circ \mathcal{P}\beta_K.

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. \end{example}

\begin{example} The standard free space construction can be applied to superconvex spaces to obtain an adjoint pair ℱ:Set⇆SCvx:𝒰\mathcal{F}:\mathbf{Set} \leftrightarrows \mathbf{SCvx}: \mathcal{U} where ℱ(A)\mathcal{F}(A) consists of all formal countable affine sums, ∫ ℕadp:=∑ i∈ℕp ia i\int_{\mathbb{N}} \mathbf{a} \, d\mathbf{p} := \sum_{i \in \mathbb{N}}p_i a_i for p∈𝒢ℕ\mathbf{p} \in \mathcal{G}{\mathbb{N}} and a∈Set(ℕ,A)\mathbf{a} \in \mathbf{Set}(\mathbb{N},A), modulo the relations

(9)∫ j∈ℕ(∫ i∈ℕadQ i)dp≅∫ j∈ℕad(∫ i∈ℕQ •dp)∀Q∈Set(ℕ,𝒢ℕ), \textstyle{\int_{j \in \mathbb{N}}} \big( \textstyle{\int_{i \in \mathbb{N}}} \mathbf{a} \, d\mathbf{Q}^i \big) \, d\mathbf{p} \;\cong\; \textstyle{\int_{j \in \mathbb{N}}} \mathbf{a} \, d \big( \textstyle{\int_{i \in \mathbb{N}}} \mathbf{Q}^{\bullet} \, d\mathbf{p} \big) \quad\quad\quad \forall \, \mathbf{Q} \in \mathbf{Set}( \mathbb{N}, \mathcal{G}{\mathbb{N}}) \,,

so that the sole necessary axiom of a superconvex space is satisfied.

Since the notion of “superconvex spaces” generalizes the idea of convex spaces by replacing finite affine sums with countable affine sums, we note that the free space construction generalizes the distribution monad on the category of sets, for which the category of algebras is the category of convex spaces. In short, the value of the distribution monad DD on a set XX consists of all affine sums of elements of XX, and if f:X→Yf \colon X \rightarrow Y is a function then D(f)D(f) is the ‘’pushforward probability measure’‘ (which coincides with the affine map on an affine sum).

By replacing (finite) affine sums with countable affine sums we obtain the monad D^\hat{D} on Set\mathbf{Set} defined by D^(X)\hat{D}(X) consisting of all countable affine sums of elements of XX, and for the function f:X→Yf \colon X \rightarrow Y we have once again the pushforward map, which is in fact a countably affine map. Now a quick glance at the proof that the category of algebras for the distribution monad is the category of convex spaces, quickly shows that upon replacing the term finite sums with the term countable affine sums yields the result that the category of algebras for the monad obtained from the adjunction given by the free space construction defined above is the category of superconvex spaces. \end{example}

• totally convex space

• convex space

References

The notion of superconvex spaces originates with:

• Dieter Pumplün, Banach spaces and superconvex modules, in Symposia Gaussiana – Conference A: Mathematics and Theoretical Physics, De Gruyter (1995) [[doi:10.1515/9783110886726.323](https://doi.org/10.1515/9783110886726.323)]

The proof that SCvx\mathbf{SCvx} has no cogenerator is due to:

• Reinhard Börger, Ralf Kemper, There is no cogenerator for total convex spaces, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 35 4 (1994) 335-338 [[numdam:CTGDC_1994__35_4_335_0](http://www.numdam.org/item/CTGDC_1994__35_4_335_0)]

The fact that SCvx\mathbf{SCvx} 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

• Xiao-qing Meng, Categories of convex sets and of metric spaces with applications to stochastic programming and related areas, PhD thesis (djvu)

Proposition 1.2 in

• Reinhard Börger, Ralf Kemper, Cogenerators for convex spaces, Applied Categorical Structures 2 (1994) 1-11 [[doi:10.1007/BF00878499](https://doi.org/10.1007/BF00878499)]

is particularly useful for viewing superconvex spaces as positively convex spaces which are somewhat easier to work with because the condition ∑ i∈ℕp i=1\sum_{i \in \mathbb{N}}p_i=1 is replaced by the inequality ≤1\le 1.

The fact that the functor Σ:Ω→Std 2\mathbf{\Sigma}: \mathbf{\Omega} \rightarrow \mathbf{Std}_2 is a codense functor can be found in

• Kirk Sturtz, Giry algebras for standard measurable spaces [[arXiv:2202:10819]]

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.

• Tobias Fritz, Convex spaces I: definition and examples [[arXiv/0903.5522](http://arxiv.org/abs/0903.5522)]

The term strong convex space was employed in:

• George Mackey, p. 68 of: The Mathematical Foundations of Quantum Mechanics: a Lecture-note Volume, Mathematical physics monograph series, Benjamin (1963), Dover (2004) [[google books](https://books.google.de/books?id=qlpb2mWYmfYC&printsec=frontcover&hl=de&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false)]