The support is a morphism of monads Sharwin Rezagholi 1 Tobias Fritz - - PowerPoint PPT Presentation

The support is a morphism of monads

Sharwin Rezagholi1 Tobias Fritz2 Paolo Perrone1

1Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany 2Perimeter Institute for Theoretical Physics, Waterloo, Canada

March 28, 2019 SYCO 3, Oxford Preliminary paper: http://www.mis.mpg.de/publications/ preprints/2019/prepr2019-33.html

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A simple example

µ p q

• ♥

△

1 2 1 2 3 4 1 4

1

supp(µ) = supp 1 2 · p + 1 2 · q

• = supp(p) ∪ supp(q)

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Another simple example

Consider the sequence of probability vectors

• 1 −

1 i+1, 1 i+1

• i∈N.

Both entries are positive for every i ∈ N. i = 1 i = 3 i = 9 i → ∞ The support discontinuously shrinks: Lower semicontinuity in the order of set inclusion.

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The question

Probability → possibility: A morphism from a monad of probabilistic powerspaces to a monad of (possibilistic) powerspaces? Applications: Denotational semantics, Dynamical systems, ... This can also help us to better understand abstract notions of convexity... Main problem: How to encode the lower semicontinuity of the support? We work in the category Top of topological spaces and continuous maps.

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The hyperspace

Let X be a topological space.

Definition

Let A ⊆ X. We set Hit(A) := {C ⊆ X : C is closed and C ∩ A = ∅}.

Definition (Hyperspace)

The hyperspace of X is the set HX := {C ⊆ X : C is closed} equipped with the lower Vietoris topology with subbasis: {Hit(U) : U ⊆ X is open}.

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Duality theory for H

Theorem

There is an isomorphism of complete lattices between HX and Scott-continuous functionals φ : O(X) → S with the following two properties.

1 Strictness: φ(∅) = 0. 2 Modularity: φ(U ∩ V ) ∨ φ(U ∪ V ) = φ(U) ∨ φ(V ).

(S denotes the Sierpinski space.) We adopt functional-analytic coupling notation. C, U :=

• 1 if C hits U

0 otherwise

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The H-monad

H : Top → Top is a functor: X → HX, f → f♯ where f♯(C) = clf (C).

Definition (Unit)

The map σ : X → HX where σ(x) ∈ HX fulfills σ(x), U ≡        1 if x ∈ U 0 otherwise for every open U ⊆ X.

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Definition (Multiplication)

The map U : HHX → HX where for C ∈ HHX we have UC, U ≡ C, Hit(U) for every open U ⊆ X.

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HX HHX HX

σ U

HX HHX HX

σ♯ U

HHHX HHX HHX HX

U♯ U U U

The triple (H, σ, U) is a monad on Top. (In fact, it is even a 2-monad.)

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H-algebras

Theorem (Schalk 1993)

The category of H-algebras consists of complete lattices equipped with a sober topology whose specialization preorder equals the respective order. The structure maps are given by the join. The algebra-morphisms are continuous join-preserving maps. (Recall: The algebras of the powerset monad on the category of sets are complete semilattices.)

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Continuous subprobability valuations

Definition

A continuous map ν : O(X) → [0, 1] that satisfies the following four conditions.

1 Monotonicity: U ⊆ V implies ν(U) ≤ ν(V ). 2 Strictness: ν(∅) = 0. 3 Modularity: ν(U ∪ V ) + ν(U ∩ V ) = ν(U) + ν(V ). 4 Scott-continuity:

ν

α∈A

Uα

• =
• α∈A

ν(Uα) for any directed increasing net (Uα)α∈A.

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The space VX

Let X be a topological space.

Definition

We define the space VX to be the set of continuous subprobability valuations on X equipped with the topology for which the sets of the following form are a subbasis, θ(U, r) := {ν : ν(U) > r} for some open U ⊆ X and some r ∈ [0, 1). (This is very similar to the extended probabilistic powerdomain.)

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Duality theory for V

We denote the lower integral of the lower semicontinuous function f : X → [0, 1] with respect to the valuation ν by ν, f .

Theorem

There is a bijection between continuous valuations on the topological space X and Scott-continuous functionals L(X) → [0, 1] with the following two properties.

1 Strictness: v, 0 = 0. 2 Modularity: v, f ∧ g + v, f ∨ g = v, f + v, g.

(L(X) denotes the set of lower semicontinuous functions X → [0, 1].)

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The V monad

V : Top → Top is a functor: X → VX, f → f∗, the pushforward operation.

Definition (Unit)

The map δ : X → VX where x → δx where δx is the point-mass valuation characterized by δx, g ≡ g(x) for every lower semicontinuous g : X → [0, 1].

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Definition (Multiplication)

The map E : VVX → VX where for ξ ∈ VVX we have Eξ, g ≡ ξ, −, g for every lower semicontinuous g : X → [0, 1]. (Note that the map −, g : VX → [0, 1] is itself lower semicontinuous.)

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VX VVX VX

δ E

VX VVX VX

δ∗ E

VVVX VVX VVX VX

E∗ E E E

The triple (V , δ, E) is a monad on Top. (In fact, it is even a 2-monad.)

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V -algebras

“Probability-type” monads have “convex-type” algebras.

Definition (Category of convex spaces)

A set A with a map c : [0, 1] × A × A → A fulfilling

1 Unitality: c(0, x, y) = y, 2 Idempotency: c(λ, x, x) = x, 3 Parametric commutativity: c(λ, x, y) = c(1 − λ, y, x), 4 Parametric associativity: c(λ, c(µ, x, y), z) = c(λµ, x, c(ν, y, z)),

ν =       

λ(1−µ) 1−λµ if λ, µ = 1

• therwise arbitrary in [0, 1].

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Theorem

Every V -algebra is a convex space and every morphism of V -algebras is a map that preserves the convex structure (an affine map). (Compare: Goubault-Larrecq and Jia 2019, Arxiv-preprint.) The idea is simple: Let (A, a) be a V -algebra, set c(λ, x, y) := a

• λ · δx + (1 − λ) · δy
• .

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The support is a morphism in Top

Definition (Support of a valuation)

Let ν ∈ VX. The support is defined by supp(ν), U :=        1 if ν(U) > 0 0 otherwise. The support is a continuous map supp : VX → HX since supp−1(Hit(U)) = θ(U, 0).

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The support is a natural transformation

VX HX VY HY

supp f∗ f♯ supp

Proof: supp(f∗p), U = (f∗p)(U) > 0 = p(f −1(U)) > 0 = supp(p), f −1(U) = f♯(supp(p)), U.

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The support is a morphism of monads

VX X HX

supp δ σ

VVX VX HVX HHX HX

supp E supp supp♯ U

Theorem

The support induces a morphism of monads supp : (V , δ, E) → (H, σ, U).

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Theorem

Every H-algebra is also a V -algebra. It is a standard result that a morphism of monads induces a pullback functor between the respective categories of algebras. Here: Let (A, a) be an H-algebra, then (A, a) − → (A, a ◦ supp) yields a V -algebra with structure map ν − →

• supp(ν).

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The case of Borel probability measures

The functor P : Top → Top that assigns to a space X the set PX of τ-smooth Borel probability-measures with the A(lexandrov)-topology generates a submonad of V . For Tikhonov spaces, the P-construction is equivalent to assigning the weak topology. This includes all spaces usually studied in measure theory. We still have supp : P → H. This is the most general Borel-probability monad that we are aware of.

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Conclusions

A natural appearance of exotic convex spaces as V -algebras mediated by supp. Clear connection between probabilistic and possibilistic representations of systems, in denotational semantics, dynamical systems, entropy-theory, ... supp is induced by a morphism of effect monoids, general constructions are forthcoming. We work on a generalization to the category of locales. Preliminary paper: http://www.mis.mpg.de/publications/ preprints/2019/prepr2019-33.html

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Some literature

M.M. Clementino and W. Tholen. A characterization of the Vietoris

• topology. Topology Proceedings, 22: 71–95, 1997.
• J. Goubault-Larrecq and X. Jia. Algebras of the extended probabilistic

powerdomain monad. 2019. Available under http://arxiv.org/abs/1903.07472.

• R. Heckmann. Spaces of valuations, in: Annals of the New York Academy
• f Sciences 806, pp. 174–200, 1996.
• C. Jones and G. Plotkin. A probabilistic powerdomain of evaluations, in:

Proceedings of the 4th Annual Symposium on Logic in Computer Science,

• pp. 186–195, 1989.
• A. Schalk. Algebras for generalized power constructions. PhD thesis,

University of Darmstadt, 1993.

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