nLab superconvex space




The notion of β€œsuperconvex spaces” generalizes the idea of convex spaces by replacing finite affine sums with countable affine sums.

Let 𝒒(β„•)\mathcal{G}(\mathbb{N}) denote the set of all probability measures on the set of natural numbers, hence every p\mathbf{p} can be represented as p=βˆ‘ iβˆˆβ„•p iΞ΄ i\mathbf{p} = \sum_{i \in \mathbb{N}} p_i \delta_i where βˆ‘ iβˆˆβ„•p i=1\sum_{i \in \mathbb{N}} p_i=1 with each p i∈[0,1]p_i \in [0,1]. Here 𝒒\mathcal{G} is the Giry monad, but because we want to forget the Οƒ\sigma-algebra associated with that measurable space we often write Ξ” β„•\Delta_{\mathbb{N}} to denote the underlying set of 𝒒(β„•)\mathcal{G}(\mathbb{N}). That set is the countably infinite-dimensional simplex. The set 𝒒(β„•)\mathcal{G}(\mathbb{N}) (= Ξ” β„•\Delta_{\mathbb{N}}) is the prototype space from which the axioms are abstracted.


Given any set AA, a sequence a:β„•β†’A\mathbf{a}: \mathbb{N} \rightarrow A, and any pβˆˆπ’’(β„•)\mathbf{p} \in \mathcal{G}(\mathbb{N}), we refer to the formal expression βˆ‘ iβˆˆβ„•p ia i\sum_{i \in \mathbb{N}} p_i a_i as a countably affine sum of elements of AA, and for brevity we use the notation βˆ‘ iβˆˆβ„•p ia i\sum_{i\in \mathbb{N}} p_i a_i to refer to a countably affine sum dropping the explicit reference to the condition that the limit of partial sums βˆ‘ i=0 Np i\sum_{i=0}^N p_i converges to one. An alternative notation to the countable affine sum notation is to use the integral notation

∫ β„•adp:=βˆ‘ iβˆˆβ„•p ia i. \int_{\mathbb{N}} \mathbf{a} \, d\mathbf{p} := \sum_{i \in \mathbb{N}} p_i a_i.

We say a set AA has the structure of a superconvex space if it comes equipped with a function

st A : 𝒒(β„•)Γ—Set(β„•,A) β†’ A : (p,a) ↦ ∫ β„•adp \begin{array}{ccccc} st_A&:& \mathcal{G}{(\mathbb{N})} \times \Set(\mathbb{N}, A) & \rightarrow& A \\ &:& (\mathbf{p}, \mathbf{a}) & \mapsto & \int_{\mathbb{N}} \mathbf{a} \, d\mathbf{p} \end{array}

such that the following two axioms are satisfied:

Axiom 1. For every sequence a:β„•β†’A\mathbf{a}: \mathbb{N} \rightarrow A and every jβˆˆβ„•j \in \mathbb{N} the property

∫ β„•adΞ΄ j=a j \int_{\mathbb{N}} \mathbf{a} \, \, d\delta_j = a_j


Axiom 2. If pβˆˆπ’’(β„•)\mathbf{p} \in \mathcal{G}(\mathbb{N}) and Q:ℕ→𝒒(β„•)\mathbf{Q}: \mathbb{N} \rightarrow \mathcal{G}(\mathbb{N}) is a sequence of probability measures on (β„•,𝒫(β„•))(\mathbb{N}, \mathcal{P}(\mathbb{N})) then

∫ jβˆˆβ„•(∫ β„•adQ j)dp=∫ β„•ad(∫ jβˆˆβ„•Q β€’dp). \int_{j \in \mathbb{N}} \big( \int_{\mathbb{N}} \mathbf{a} \, d\mathbf{Q}^j \big) \, d\mathbf{p} = \int_{\mathbb{N}} \mathbf{a} \, d( \int_{j \in \mathbb{N}} \mathbf{Q}^{\bullet} \, d\mathbf{p} \big).

The second axiom uses the pushforward measure 𝒒(Q)pβˆˆπ’’ 2β„•\mathcal{G}(\mathbf{Q})\mathbf{p} \in \mathcal{G}^2{\mathbb{N}} and the natural transformation ΞΌ\mu of the Giry monad at component β„•\mathbb{N}, ΞΌ β„•:𝒒 2(β„•)→𝒒(β„•)\mu_{\mathbb{N}}: \mathcal{G}^2(\mathbb{N}) \rightarrow \mathcal{G}(\mathbb{N}), which yields the probability measure on the measurable space (β„•,𝒫ℕ)(\mathbb{N}, \mathcal{P}{\mathbb{N}}) whose value at the measurable set {j}\{j\} is given by the composite of measurable maps

Note the second axiom alone is sufficient because by choosing the constant sequence Q:ℕ→𝒒ℕ\mathbf{Q}: \mathbb{N} \rightarrow \mathcal{G}\mathbb{N} with value Ξ΄ j\delta_j it follows that the second axiom implies the first axiom.

A morphism of superconvex spaces, called a countably affine map, is a set function m:A→Bm: A \rightarrow B such that

m(∫ β„•adp)=∫ β„•(m∘a)dp, m\big( \int_{\mathbb{N}} \mathbf{a} \, d\mathbf{p}\big) = \int_{\mathbb{N}} (m \circ \mathbf{a}) \, d\mathbf{p},

where the composite m∘am \circ \mathbf{a} gives the sequence in BB with component m(a i)m(a_i). Composition of countably affine maps is the set-theoretical composition. Superconvex spaces with morphisms the countably affine maps form a category denoted SCvx\mathbf{SCvx}.

Probability Amplitudes

In physics, superconvex spaces have been referred to as strong convex spaces, and since probability amplitudes are employed there one makes use of the β„“ 2\ell_2-norm instead of the tradition β€˜β€™β„“ 1\ell_1-normβ€™β€˜ which is used above. By using

𝒒(β„•)={p:β„•β†’D 2|lim Nβ†’βˆž{βˆ‘ i=1 Np ip i ⋆}=1} \mathcal{G}(\mathbb{N}) = \{ \mathbf{p}: \mathbb{N} \rightarrow \mathbf{D}_2 | \lim_{N \rightarrow \infty} \{ \sum_{i=1}^N p_i p_i^{\star} \}= 1 \}

where D 2={re Δ±ΞΈβˆˆβ„‚|r∈[0,1],andθ∈[0,2Ο€)}\mathbf{D}_2 = \{r e^{\imath \theta} \in \mathbb{C} \, | r \in [0,1], and \theta \in [0,2 \pi) \} and p i ⋆p_i^{\star} is the complex conjugate of p ip_i, applied to the above axioms one obtains superconvex spaces useful for physics.


The most basic property of superconvex spaces, is


For AA any superconvex space every countably affine map m∈SCvx(Ξ” β„•,A)m \in \SCvx(\Delta_{\mathbb{N}}, A) is uniquely specified by a sequence in AA, hence we have SCvx(Ξ” β„•,A)β‰…Set(β„•,A)\SCvx(\Delta_{\mathbb{N}}, A) \cong \Set(\mathbb{N},A).


Every element pβˆˆΞ” β„•\mathbf{p} \in \Delta_{\mathbb{N}} has a unique representation as a countable affine sum p=βˆ‘ iβˆˆβ„•p iΞ΄ i\mathbf{p} = \sum_{i \in \mathbb{N}} p_i \delta_i, and hence a countably affine map m:Ξ” β„•β†’Am:\Delta_{\mathbb{N}} \rightarrow A is uniquely determined by where it maps each Dirac measure Ξ΄ i\delta_i. Thus i↦m(Ξ΄ i)i \mapsto m(\delta_i) specifies a sequence in AA.

We denote the bijective correspondence SCvx(Ξ” β„•,A)β‰…Set(β„•,A)\SCvx(\Delta_{\mathbb{N}}, A) \cong \Set(\mathbb{N},A) by ⟨aβŸ©β†”a\langle \mathbf{a} \rangle \leftrightarrow \mathbf{a}, where a:β„•β†’A\mathbf{a}: \mathbb{N} \rightarrow A specifies a sequence in AA.


A function f:β„•β†’β„•f: \mathbb{N} \rightarrow \mathbb{N} is a countably affine map if and only if ff is monotone, i<ji \lt j implies f(i)≀f(j)f(i) \le f(j).


Necessary condition. Suppose that f:β„•β†’β„•f: \mathbb{N} \rightarrow \mathbb{N} is a countably affine map. Let i<ji \lt j. By the superconvex space structure on β„•\mathbb{N} it follows, for all α∈(0,1)\alpha \in (0,1), that Ξ±i+(1βˆ’Ξ±)j=i\alpha i + (1-\alpha) j = i . If ff is not monotone then there exist a pair of elements i,jβˆˆβ„•i,j \in \mathbb{N} such that i<ji \lt j with f(j)<f(i)f(j) \lt f(i). This implies, for all α∈(0,1)\alpha \in (0,1), that f(j)=Ξ±f(i)+(1βˆ’Ξ±)f(j)<f(Ξ±i+(1βˆ’Ξ±)j)=f(i)f(j)= \alpha f(i) + (1-\alpha) f(j) \lt f(\alpha i + (1-\alpha) j ) =f(i), which contradicts our hypothesis that ff is a countably affine map.

Sufficient condition. Suppose ff is a monotone function, and that we are given an arbitrary countably affine sum βˆ‘ iβˆˆβ„•p ii=n\sum_{i \in \mathbb{N}} p_i i = n in β„•\mathbb{N}, so that for all i=0,1,…,nβˆ’1i=0,1,\ldots,n-1 we have p i=0p_i=0. Since the condition defining the superconvex structure is conditioned on the property p iβ‰ 0p_i \ne 0, the countably affine sum is not changed by removing any number of terms ii in the countable sum whose coefficient p i=0p_i=0. Hence for all jj such that n<jn\lt j it follows that f(n)≀f(j)f(n) \le f(j) so that

f(βˆ‘ i=0 ∞p ii)=f(n)=βˆ‘ i=n ∞p if(i) f( \sum_{i=0}^{\infty} p_i \, i) = f(n) = \sum_{i=n}^{\infty} p_i f(i)

where the last equality follows from the definition of the superconvex space structure on β„•\mathbb{N}.

The category SCvx\mathbf{SCvx} 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 Ξ” β„•\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(β‹…,A)∈Set Ξ” β„• op\widehat{A}=\mathbf{SCvx}(\cdot,A) \in \mathbf{Set}^{\Delta_{\mathbb{N}}^{op}}, where Ξ” β„•\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 0βˆˆβ„a_0 \in \mathcal{I} and βˆ‘ iβˆˆβ„•p 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 βˆ‘ iβˆˆβ„•p ia iβˆˆβ„\sum_{i \in \mathbb{N}}p_i a_i \in \mathcal{I}. Ideals are useful for defining functors to or from SCvx\mathbf{SCvx}.



A fundamental superconvex space is the set β„•\mathbb{N} with the superconvex space structure defined, for every sequence s:β„•β†’β„•\mathbf{s}: \mathbb{N} \rightarrow \mathbb{N} by

∫ β„•sdp=inf i{s i|p i>0} \int_{\mathbb{N}} \mathbf{s} d\mathbf{p} = \inf_i \{\mathbf{s}_i \, | \, p_i \gt 0 \}

That structure on β„•\mathbb{N} shows that the function Ο΅:𝒒(β„•)β†’β„•\epsilon: \mathcal{G}(\mathbb{N}) \rightarrow \mathbb{N} defined by βˆ‘ iβˆˆβ„•p iΞ΄ 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.


As an example of the utility of ideals in a superconvex space we note that for 𝒒\mathcal{G} the Giry monad the measurable space 𝒒(X)\mathcal{G}(X), with the smallest Οƒ\sigma-algebra such that the evaluation maps ev U:𝒒(X)β†’[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 𝒒X\mathcal{G}{X} are of the form ev U βˆ’1((0,1])ev_{U}^{-1}( (0,1]) or ev U βˆ’1([0,1))ev_U^{-1}([0,1)) for Uβ‰ 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 βˆ’1([0,1))ev_{U}^{-1}( [0,1)) and ev U βˆ’1((0,1])ev_U^{-1}((0,1]) are ideals in 𝒒X\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 βˆ’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 𝒒X\mathcal{G}{X} such that ev U βˆ’1([0,1))⫋︀ℐev_U^{-1}( [0,1)) \varsubsetneqq \mathcal{I}. Every element Pβˆˆβ„P \in \mathcal{I} which is not in ev U βˆ’1([0,1))ev_U^{-1}([0,1)) has the defining property that P(U c)=1P(U^c)=1. Now let Qβˆˆπ’’XQ \in \mathcal{G}{X}. If Qβˆ‰π’₯Q \notin \mathcal{J} and Qβˆ‰ev U βˆ’1([0,1))Q \notin ev_U^{-1}([0,1)) then Q(U c)β‰ 1Q(U^c) \ne 1 which implies Q∈ev U βˆ’1([0,1))β«‹οΈ€JQ \in ev_U^{-1}( [0,1)) \varsubsetneqq J which is self-contradictory. Thus π’₯\mathcal{J} must be all of 𝒒X\mathcal{G}{X} which shows ev U βˆ’1([0,1))ev_U^{-1}([0,1)) is a maximal (proper) ideal. The argument that the ideal ev U βˆ’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 𝒒(X)\mathcal{G}(X) is a countable intersection of maximal ideals, and hence measurable. This implies, for example, that every countably affine map k:𝒒(X)β†’β„•k: \mathcal{G}(X) \rightarrow \mathbb{N} is also a measurable function with β„•\mathbb{N} having the powerset Οƒ\sigma-algebra. (The only ideals of β„•\mathbb{N} are the principal ideals ↓0βŠ‚β†“1βŠ‚β€¦\downarrow \! 0 \subset \downarrow \! 1 \subset \ldots.)


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 βˆ‘ iβˆˆβ„•p 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 βˆžβ†’πŸšj: \mathbb{R}_{\infty} \rightarrow \mathbb{2} is given by j(u)=1j(u)=1 for all uβˆˆβ„u \in \mathbb{R} and j(∞)=0j(\infty)=0 (for the superconvex space structure on 2\mathbf{2} determined by 120Μ²+121Μ²=0Μ²\frac{1}{2} \underline{0} + \frac{1}{2} \underline{1} = \underline{0}).


A pathological space, useful for counterexamples, is given by the closed unit interval with the superconvex space structure defined by the infimum function, βˆ‘ iβˆˆβ„•p 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\}.


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 Ξ² K:𝒫Kβ†’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 pβˆˆπ’’β„•\mathbf{p} \in \mathcal{G}\mathbb{N}, countable affine sums by βˆ‘ iβˆˆβ„•p ik i:=Ξ² K(βˆ‘ iβˆˆβ„•p iΞ΄ 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 Ξ² K\beta_K a countably affine map.

To prove 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

βˆ‘ iβˆˆβ„•p iΞ² K(Ξ΄ k i)=βˆ‘ iβˆˆβ„•p ik i:=Ξ² K(βˆ‘ iβˆˆβ„•p iΞ΄ k i). \sum_{i \in \mathbb{N}} p_i \beta_K(\delta_{k_i}) = \sum_{i \in \mathbb{N}} p_i k_i := \beta_K(\sum_{i \in \mathbb{N}} p_i \delta_{k_i}).

To prove Ξ² K\beta_K is countably affine on 𝒫K\mathcal{P}K use the property that Ξ² K∘μ K=Ξ² Kβˆ˜π’«Ξ² 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.


The standard free space construction can be applied to superconvex spaces to obtain an adjoint pair β„±:Set⇆SCvx:𝒰\mathcal{F}:\mathbf{Set} \leftrightarrows \mathbf{SCvx}: \mathcal{U} where β„±(A)\mathcal{F}(A) consists of all formal countable affine sums, ∫ β„•adp:=βˆ‘ iβˆˆβ„•p ia i\int_{\mathbb{N}} \mathbf{a} \, d\mathbf{p} := \sum_{i \in \mathbb{N}}p_i a_i for pβˆˆπ’’β„•\mathbf{p} \in \mathcal{G}{\mathbb{N}} and a∈Set(β„•,A)\mathbf{a} \in \mathbf{Set}(\mathbb{N},A), modulo the relations

∫ jβˆˆβ„•(∫ iβˆˆβ„•adQ i)dpβ‰…βˆ« jβˆˆβ„•ad(∫ iβˆˆβ„•Q β€’dp)βˆ€Q∈Set(β„•,𝒒ℕ) \int_{j \in \mathbb{N}} \big( \int_{i \in \mathbb{N}} \mathbf{a} \, d\mathbf{Q}^i \big) \, d\mathbf{p} \cong \int_{j \in \mathbb{N}} \mathbf{a} \, d\big( \int_{i \in \mathbb{N}} \mathbf{Q}^{\bullet} \, d\mathbf{p} \big) \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.

One important aspect of the free space adjunction is that it implies

Lemma Let XX be an arbitrary set and ℱX\mathcal{F}X the free superconvex space on XX. Let the quotient map q^:ℱX→A\widehat{q}: \mathcal{F}{X} \rightarrow A be the coequalizer of the parallel pair 1 ℱX,k:ℱX→ℱX\mathbf{1}_{\mathcal{F}{X}}, k: \mathcal{F}{X} \rightarrow \mathcal{F}{X}. Then the quotient of the free space is the free space of the quotient, ℱX/q^≅ℱ(X/q)\mathcal{F}X / \widehat{q} \cong \mathcal{F}(X/q)

Proof The quotient map q^:β„±Xβ†’A\widehat{q}:\mathcal{F}X \rightarrow A is, up to isomorphism, just q^:β„±Xβ†’β„±X/q^\widehat{q}:\mathcal{F}X \rightarrow \mathcal{F}X/\widehat{q}. Because the quotient map is countably affine it is completely specified by where it sends the elements 1xβˆˆβ„±X1 x \in \mathcal{F}X. Because q(x)=q^(Ξ· X(x))=q^(1x)=[1x] q^=[k(1x)] q^q(x) = \widehat{q}(\eta_X(x)) = \widehat{q}(1x) = [1 x]_{\widehat{q}}=[k (1 x)]_{\widehat{q}} we have

q^(βˆ‘ iβˆˆβ„•p ix i)=βˆ‘ iβˆˆβ„•p i[1x i] q^=βˆ‘ iβˆˆβ„•p iq(x i). \hat{q}(\sum_{i \in \mathbb{N}} p_i x_i)=\sum_{i \in \mathbb{N}} p_i [1 x_i]_{\hat{q}} = \sum_{i \in \mathbb{N}} p_i q(x_i).

The function qq specifies an equivalence relation on XX, yielding the set X/qX/q. Because qq is surjective we can thus endow the set X/qX/q with the superconvex space structure of AA using the bijective map of sets q˜:X/q→𝒰(A)\widetilde{q}: X/q \rightarrow \mathcal{U}(A). Hence the above equation extends (trivially) on the right by βˆ‘ iβˆˆβ„•p iq(x i)=βˆ‘ iβˆˆβ„•p iq˜([x i] q)=q˜(βˆ‘ iβˆˆβ„•p i[x i] q)\sum_{i \in \mathbb{N}} p_i q(x_i) = \sum_{i \in \mathbb{N}} p_i \, \widetilde{q}([x_i]_q) = \widetilde{q}(\sum_{i \in \mathbb{N}} p_i [x_i]_q ).
The map β„±X/q^β†’β„±(𝒰(A))\mathcal{F}X / \hat{q} \rightarrow \mathcal{F}( \mathcal{U}(A)) specified by βˆ‘ iβˆˆβ„•p i[1x i] q^β†¦βˆ‘ iβˆˆβ„•p iq˜([x i] q)\sum_{i \in \mathbb{N}}p_i [1 x_i]_{\widehat{q}} \mapsto \sum_{i \in \mathbb{N}}p_i \widetilde{q}( [x_i]_q ) specifies a SCvx\mathbf{SCvx}-isomorphism. By the isomorphism q˜\widetilde{q} we can write this as β„±X/q^β‰…β„±(X/q)\mathcal{F}{X}/\widehat{q} \cong \mathcal{F}(X/q). That completes the proof.

Now let XX denote a standard measurable space, so 𝒒X\mathcal{G}{X} is standard also. Forgetting the measurable structure on 𝒒X\mathcal{G}{X} we can take the free space of that set, β„±(𝒰(𝒒X))\mathcal{F}( \mathcal{U}( \mathcal{G}{X})) which consist of all countable affine sums ∫ β„•Pdp\int_{\mathbb{N}} \mathbf{P} \, d\mathbf{p} for all P∈Set(β„•,𝒰(𝒒X))\mathbf{P} \in \mathbf{Set}(\mathbb{N}, \mathcal{U}(\mathcal{G}{X})) and all pβˆˆπ’’β„•\mathbf{p} \in \mathcal{G}{\mathbb{N}}. These countable affine sums are defined pointwise (∫ β„•Pdp)U:=βˆ‘ iβˆˆβ„•p iP i(U)(\int_{\mathbb{N}} \mathbf{P} \, d\mathbf{p})U := \sum_{i \in \mathbb{N}} p_i P_i(U) for all U∈Σ XU \in \Sigma_X.

We observe that the free space β„±(𝒰(𝒒X))\mathcal{F}(\mathcal{U}(\mathcal{G}{X})) can also be viewed as the image of the functor 𝒫:Measβ†’SCvx\mathcal{P}: \mathbf{Meas} \rightarrow \mathbf{SCvx} on XX, where 𝒫\mathcal{P} is the Giry monad (functor) viewed as a functor into SCvx\mathbf{SCvx}.

We would like to extend the free space adjunction β„±βŠ£π’°\mathcal{F} \dashv \mathcal{U} with an adjunction 𝒫:Std⇆SCvx:Ξ£\mathcal{P}: \mathbf{Std} \leftrightarrows \mathbf{SCvx}: \mathbf{\Sigma} such that the composite is the Giry monad on Std\mathbf{Std}. The category SCvx ⋆\mathbf{SCvx}_{\star} denotes some subcategory of SCvx\mathbf{SCvx} so that the adjunction can be constructed. If such an adjunction exists the counit of the adjunction says every space AA is the quotient of a free space, i.e., AA 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 13:1β†’[0,1]\frac{1}{3}: \mathbf{1} \rightarrow [0,1] and 23:1β†’[0,1]\frac{2}{3}: \mathbf{1} \rightarrow [0,1], where [0,1]β‰…P2[0,1] \cong \mathbf{P}{\mathbf{2}}. The coequalizer of those pair of points is the discrete space A={0,u,1}A=\{0,u,1\} with the structure defined by pu+(1βˆ’p)0=up u + (1-p) 0 = u for all p∈(0,1]p \in (0,1], pu+(1βˆ’p)1=up u + (1-p) 1 = u for all p∈(0,1]p \in (0,1], and p0+(1βˆ’p)1=up 0 + (1-p) 1 = u for all p∈(0,1)p \in (0,1).

If the functor Ξ£:SCvx ⋆→Std\mathbf{\Sigma}: \mathbf{SCvx}_{\star} \rightarrow \mathbf{Std} exists it should recognize that that superconvex space arises as a quotient space of 𝒫2\mathcal{P}{\mathbf{2}}, and hence should have the barycenter map Ο΅ A:𝒫2β†’A\epsilon_A: \mathcal{P}{\mathbf{2}} \rightarrow A given by the mapping Ξ΄ 0↦0\delta_0 \mapsto 0, Ξ΄ 1↦1\delta_1 \mapsto 1, and rΞ΄ 0+(1βˆ’r)Ξ΄ 1↦ur \delta_0 + (1-r) \delta_1 \mapsto u for all r∈(0,1)r \in (0,1).

Let Ξ©\mathbf{\Omega} denote the full subcategory of SCvx\mathbf{SCvx} consisting of the two objects β„•\mathbb{N} and Ξ” β„•\Delta_{\mathbb{N}}, and let ΞΉ:Ξ©β†’SCvx\iota:\mathbf{\Omega} \rightarrow \mathbf{SCvx} be the inclusion functor.

If we look at the category A↓ιA \downarrow \iota there are two countably affine maps,

Ο• 0,u : A β†’ β„• : a ↦ {0 fora∈{0,u} 1 fora=1 \begin{array}{lcccc} \phi_{0,u} &:& A & \rightarrow & \mathbb{N} \\ &:& a & \mapsto & \left\{ \begin{array}{ll} 0 & for \, a \in \{0,u\} \\ 1 & for \, a=1 \end{array} \right. \end{array}


Ο• 1,u : A β†’ β„• : a ↦ {0 fora∈{1,u} 1 fora=0, \begin{array}{lcccc} \phi_{1,u} &:& A & \rightarrow & \mathbb{N} \\ &:& a & \mapsto & \left\{ \begin{array}{ll} 0 & for \, a \in \{1,u\} \\ 1 & for \, a=0 \end{array} \right. \end{array},

from which every other countably affine map A→ℕA \rightarrow \mathbb{N} can be obtained by composing either ϕ 0,u\phi_{0,u} or ϕ 1,u\phi_{1,u} with a monotonic function ϕ:ℕ→ℕ\phi: \mathbb{N} \rightarrow \mathbb{N}. Because AA is discrete there are no non-constant countably affine maps A→Δ ℕA \rightarrow \Delta_{\mathbb{N}}.

The coequalizer of those two maps yields the discrete space 2\mathbf{2} for which 𝒫2\mathcal{P}{\mathbf{2}} is the space we are trying to find - it is the smallest free space such that there is a countably affine map 𝒫2β†’A\mathcal{P}{\mathbf{2}} \rightarrow A.

Whether probing a space AA with arrows to objects and products of objects in Ξ©\mathbf{\Omega} is sufficient is unknown but a clue is given in the fact that every standard space is Std\mathbf{Std}-isomorphic to either [0,1][0,1] or a countable discrete space. Hence SCvx ⋆\mathbf{SCvx}_{\star} should intuitively consists of quotients of ∏ iβˆˆβ„•Ξ” β„•\prod_{i \in \mathbb{N}} \Delta_{\mathbb{N}} (and smaller) - the countability condition being the critical condition. That condition rules out spaces such as ∏ i∈[0,1]Ξ” β„•\prod_{i \in [0,1]}\Delta_{\mathbb{N}} and the pathological space of Example 5.4.


The notion of superconvex spaces originates with:

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

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

is particularly useful for viewing superconvex spaces as positively convex spaces which are somewhat easier to work with because the condition βˆ‘ iβˆˆβ„•p i=1\sum_{i \in \mathbb{N}}p_i=1 is replaced by the inequality ≀1\le 1.

The fact that the functor Ξ£:Ξ©β†’Std 2\mathbf{\Sigma}: \mathbf{\Omega} \rightarrow \mathbf{Std}_2 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:

  • 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]

Last revised on August 19, 2022 at 09:12:06. See the history of this page for a list of all contributions to it.