Functors on posets, extra-fine sheaves, and interaction - - PowerPoint PPT Presentation

Functors on posets, extra-fine sheaves, and interaction decompositions

Juan Pablo Vigneaux Arizt´ ıa Join work D. Bennequin, O. Peltre, and G. Sergeant-Perthuis arXiv:2009.12646 Leipzig, October 30, 2020

October 30, 2020 1 / 32

Outline

1

Introduction and motivations

2

Extra-fine sheaves

3

Functors on posets

4

Equivalence of cohomologies

5

Global sections of measures

October 30, 2020 2 / 32

Basic definitions

A partially ordered set (poset) A as a small category such that:

1 there is at most one morphism between two objects; 2 if a → b and b → a, then a = b.

(So that the relation α → β is reflexive, anti-symmetric and transitive.) An hypergraph is a poset of subsets of a set I, such that S → S′ whenever S′ ⊆ S. An abstract simplicial complex K is an hypergraph with an additional property: if S belongs to K, then every subset of S belongs to K too. A presheaf on a category C is just a functor F : Cop → Sets. A copresheaf is F : C → Sets. A presheaf on a topological space X is a presheaf on its category of open sets Open(X), ordered by inclusion.

October 30, 2020 3 / 32

Review of the literature

Curry [4]: cellular (co)sheaves i.e. functors on the incidence poset of a regular cell complex. See also Ghrist and Hansen [6] for the spectral theory of cellular sheaves, Ghrist and Riess [5] for lattice-valued cellular sheaves. Applications: network coding, sensor networks, distributed consensus, flocking, synchronization, opinion dynamics. Robinson et al [9, 11, 10], Mansourbeigi [7]: sheaves on abstract simplicial complexes are a canonical model for the integration of information provided by interconnected sensors. Abramsky et al [2, 1]: functors on an abstract simplicial complex as models of local information that does not necessarily come from a “global state” (in quantum mechanics, data-base theory, etc.), a phenomenon called contextuality.

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The fundamental example

Index set I that models variables/observables Xi, i ∈ I. We consider a poset A of subsets of I, such that α → β iff β ⊂ α. A set α ⊂ I models the joint measurement of (Xa)a∈α. We consider a collection of finite sets {Ei} (micro-states/internal degrees

• f freedom), and associate to α ⊂ I the set Eα :=

i∈α Ei. This defines a

functor E : A → Sets; an arrow α → β is mapped to the canonical projection πβα : Eα → Eβ. Then we can introduce some “secondary” functors: The observables V : A

• p → Sets, such that α → REα and

V (α → β) : Vβ → Vα is precomposition by πβα. The probabilities P : A → Sets, such that Pα ≡ P(α) are the probabilities on Eα, and given P(α → β) maps p to p ◦ π−1

βα, which is

the marginal/image measure under πβα.

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Marginal problem

Given a section of P on a poset A of subsets of I i.e. a “coherent” collection of probabilities {pα}α∈Ob A, does there exist a probability q on E such that each pα is a marginal of q? This means ∀α ∈ Ob A, ∀xα ∈ Eα,

• xIα∈EI\α

q(xα, xI\α) = p(xα). How many solutions are there? The problem is NP-hard. But extensions q are known if q is allowed to be signed or complex measure (linearized marginal problem). Cf. Kellerer and Mat´ uˇ s. When I = {1, 2} and A is {1} → ∅ ← {2}, the law q is a transportation plan or a copula or a doubly stochastic matrix or a Markov operator...

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Interaction decomposition

How is the space of sections of P over XA (“pseudomarginals”)? Again, this seems difficult, but if the problem is relaxed to signed measures F = V ∗ or signed measures of total mass 1, ¯ F = F/K, we can compute the dimension of the space of global sections dim H0(XA, F), resp. dim H0(XA, ¯ F). Our path to do this is to decompose the sheaf V or its quotient V /K into a direct sum of sheaves. Under suitable hypotheses, Vα =

• β∈A

Sβ. Such V is an example of extra-fine sheaf.

October 30, 2020 7 / 32

Interaction decomposition

How is the space of sections of P over XA (“pseudomarginals”)? Again, this seems difficult, but if the problem is relaxed to signed measures F = V ∗ or signed measures of total mass 1, ¯ F = F/K, we can compute the dimension of the space of global sections dim H0(XA, F), resp. dim H0(XA, ¯ F). Our path to do this is to decompose the sheaf V or its quotient V /K into a direct sum of sheaves. Under suitable hypotheses, Vα =

• β∈A

Sβ. Such V is an example of extra-fine sheaf. Remark that if α → β, then Vβ ֒ → Vα. Introduce the boundary

• bservables Bα =

β:βα Vβ. We define the interaction subspace Sα as a

supplement of Bα, such that Vα = Sα ⊕ Bα. A common choice is the

• rthogonal complement of Bα for the standard canonical inner product on

Vα = REα.

October 30, 2020 7 / 32

Outline

1

Introduction and motivations

2

Extra-fine sheaves

3

Functors on posets

4

Equivalence of cohomologies

5

Global sections of measures

October 30, 2020 8 / 32

Classical notion: Fine sheaves

Let X be a topological space.

Definition

A sheaf F : Open(X)op → Ab is fine if for every open covering U there exists a family {eU}U∈U of endomorphisms eU : F → F that is

1 Local: for every V ∈ U and W open, eV |W ↾W \ ¯

V = 0;

2 A partition of unity: for every open W and x ∈ F(W ), there exists

a finite number of V ∈ U such that eV |W (x) = 0, and x =

V ∈U eV |W (x).

Here eV |W = eV (W ).

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Extra-fine sheaves

Definition

A sheaf F : Open(X)op → Ab is extra-fine if for every open covering U there exists a finer covering V and a family {eV }V ∈V of endomorphisms eV : F → F that is

1 Super-local: for every V ∈ U and W open, eV |W = 0 then W ⊂ V ; 2 A partition of unity: for every open W and x ∈ F(W ), there exists

a finite number of V ∈ U such that eV |W (x) = 0, and x =

V ∈U eV |W (x).

3 Orthogonal: for every V , W ∈ V such that V = W ,

eV ◦ eW = eW ◦ eV = 0. Properties 2 and 3 imply that the eV are projectors: e2

V = eV . We get an

• rthogonal decomposition of the sheaf F, as

V ∈VSV , where

SV = im eV . The maps eV |W are then projections on SV (W ) parallel to

• U=V SU(W ).

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Cech cohomology

Let U be an open covering of a topological space X. Define Kn(U) =

• u = (U0, ..., Un) ∈ Un

Uu := U0 ∩ · · · ∩ Un = ∅

• .

and then the space of n-cochains C n(U, F) =

u∈Kn(U) F(Uu). A

coboundary operator δ = δn+1

n

: C n(U, F) → C n+1(U, F) is introduced as follows: (δc)(U0, ..., Un+1) =

n+1

• i=0

(−1)ic(U0, ..., ˆ Ui, ..., Un+1)|Uu. By the usual arguments, δ ◦ δ = 0, so one can define the cohomology Hn(U, F) = ker δn+1

n

/ im δn

n−1.

The Cech cohomology of F on X is H•(X, F) = colimU H•(U, F).

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Acyclicity

Theorem

An extra-fine presheaf F is acyclic i.e for all n ≥ 1, Hn(X, F) = 0. In fact, fine and super-local is enough.

Proof.

Given a covering U, choose V finer, with a super-local orthogonal decomposition {eV }V ∈V. If ψ is an (n + 1)-cocycle, choosing u = (U0, ..., Un+1) we have ψ(U1, ..., Un+1)|Uu =

n+1

• k=1

(−1)k+1ψ(U0, ..., ˆ Ui, ..., Un)|Uu whenever U∩... ∩ Un ⊂ U0, and otherwise the terms vanish. Therefore, ψ =

U∈U eUψ = U∈U δφU = δ

• U∈U φU
• .

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Acyclicity

Theorem

An extra-fine presheaf F is acyclic i.e for all n ≥ 1, Hn(X, F) = 0. In fact, fine and super-local is enough.

Proof.

Given a covering U, choose V finer, with a super-local orthogonal decomposition {eV }V ∈V. If ψ is an (n + 1)-cocycle, choosing u = (U0, ..., Un+1) we have eU0ψ(U1, ..., Un+1)|Uu =

n+1

• k=1

(−1)k+1eU0ψ(U0, ..., ˆ Ui, ..., Un)|Uu whenever U∩... ∩ Un ⊂ U0, and otherwise the terms vanish. Therefore, ψ =

U∈U eUψ = U∈U δφU = δ

• U∈U φU
• .

October 30, 2020 12 / 32

Acyclicity

Theorem

An extra-fine presheaf F is acyclic i.e for all n ≥ 1, Hn(X, F) = 0. In fact, fine and super-local is enough.

Proof.

Given a covering U, choose V finer, with a super-local orthogonal decomposition {eV }V ∈V. If ψ is an (n + 1)-cocycle, choosing u = (U0, ..., Un+1) we have eU0ψ(U1, ..., Un+1)|Uu =

n+1

• k=1

(−1)k+1 eU0ψ(U0, ..., ˆ Ui, ..., Un)|Uu

• =:φU0(U1,..., ˆ

Ui,...,Un+1)

whenever U∩... ∩ Un ⊂ U0, and otherwise the terms vanish. Therefore, ψ =

U∈U eUψ = U∈U δφU = δ

• U∈U φU
• .

October 30, 2020 12 / 32

Outline

1

Introduction and motivations

2

Extra-fine sheaves

3

Functors on posets

4

Equivalence of cohomologies

5

Global sections of measures

October 30, 2020 13 / 32

Alexandrov topology

Let A be a poset. The lower Alexandrov topology on the set Ob A is generated by the basis {Uα}α∈A, where Uα := { β | α → β } is the “open star” at α. We denote the resulting space XA. The upper Alexandrov topology is the lower topology on A

• p i.e. with

basis Uα := { β | β → α }. The resulting space is X A. Remark that if α → β in A, then Uα ⊂ Uβ and Uα ⊃ Uβ.

Proposition

f : Ob A → Ob B is order preserving iff f is continuous for the lower (or upper) topology.

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Induced sheaves

Remark that if α → β in A, then Uα ⊂ Uβ and Uα ⊃ Uβ.

Proposition

Every functor F : A → Sets can be extended uniquely to a sheaf ˜ F on XA.

Proof.

Given U = ∪α∈UUα, then F(U) := limα∈U F(α). In particular, F(Uα) = F(α), and if s : α → β, then F(s) : F(α) → F(β) is the restriction map from Uα to Uβ.

October 30, 2020 15 / 32

Canonical extra-fineness

When there exists a super-local orthogonal decomposition of ˜ F subordinated to the covering U

A = {Uα}α∈A we say that ˜

F is canonically extra-fine. In fact, since U

A is finer than any other covering, such ˜

F is extra-fine. When A not stable by coproducts, a sheaf can be extra fine without being fine. If {eα}α∈A is the super-local orthogonal decomposition associated to U

A,

then F =

α∈ASα, with Sα = im eα.

Proposition

When ˜ F is canonically extra-fine, H0(XA, ˜ F) =

• H0(U

A, Sα).

October 30, 2020 16 / 32

Interaction decomposition

Now we restrict our attention to injective presheaves: functors V : A

• p → VectK such that for every α → β in A, jαβ := V (α → β) is
• injective. Set Vαβ = jαβVβ.

Definition

An interaction decomposition of an injective presheaf V is a family {Sγ}γ∈A such that Sγ ⊂ Vγ and for all α ∈ A, Vα =

• α→β

jαβSβ. If we define a presheaf Sγ such that Sγ(α) = jαβSβ =: Sαβ if α → β and zero otherwise, we have V =

γ∈ASγ.

October 30, 2020 17 / 32

Proposition

If {Sγ}γ defines an interaction decomposition of V , the family of endomorphism eγ(α) : V (α) → V (α) defined as projection on Sαβ parallel to

β′=β Sαβ′ forms a super-local orthogonal decomposition subordinated

to U

A.

October 30, 2020 18 / 32

Condition G: ∀α, β ∈ A such that α → β, Vαβ ∩      

• γ:α

=

− →γ

γ→β

Vαγ       ⊂

• γ:α

=

− →γ

β

=

− →γ Vαγ, (G) where β

=

− → γ means that β → γ and β = γ. Locally finite (descending) dimension: for e very α, the lengths of all nondegenerate chains α → β1 → β2 → · · · βn → · · · are uniformly bounded.

Theorem

Let V be an injective presheaf on a poset A.

1 If the the condition G is satisfied and A is of locally finite dimension,

the sheaf defined by V on X A is canonically extra-fine;

2 If the sheaf induced by V on X A is canonically extra-fine, the

condition G is satisfied.

October 30, 2020 19 / 32

Duality

An injective presheaf V : A

• p → VectK defines a sheaf ˜

V on X A that we suppose extra-fine. The transposed maps teα|β : V ∗(β) → V ∗(β) do not define however a super-local orthogonal decomposition; super-locality fails, but still

teα ◦ teβ = 0 when α = β.

Introduce the following hypothesis: there exists F : A → VectK with surjective maps πβα = F(α → β), such that for all α and α → β, Vα = F ∗

α and tπβα = jαβ. Then Fα ֒

→ V ∗

α, since f (v) = f , v = v(f ).

Define e∗

α = teα|F.

Proposition

idF =

α∈Ae∗ α (for each f ∈ F, only a finite number of terms are

nontrivial)

October 30, 2020 20 / 32

Duality (continued)

One deduces that F =

α∈AT α, where T α = im e∗ α.

These presheaves T are also acyclic, and H0(XA, F) =

α H0(XA, T α) = Tα, the last equality given by the fact

that induced maps are isomorphisms.

October 30, 2020 21 / 32

Example: local observables on hypergraphs

We specialize our discussion to the framework in the introduction, where A is a poset of subsets of an index set I, Vα are the functions REα. In this case, F = R(Eα), finite linear combinations of elements of Eα. A satisfies the strong intersection property if ∀α, β ∈ A, α ∩ β ∈ A; the weak intersection property asks for α ∩ β ∈ A whenever α ∩ β = ∅.

Proposition

If A has the strong (resp. weak) intersection property, the condition G is satisfied by the free presheaf V (resp. the reduced free presheaf ¯ V = V /K).

October 30, 2020 22 / 32

Example: local observables on hypergraphs (continued)

Hence there is also an orthogonal decomposition for the pre-dual F of V (resp. its reduced version ¯ F, given by annihilation of the sum of the coordinates).

Theorem

Suppose A satisfies the weak intersection property, then H•(XA, F) = H0(XA, ¯ F) ⊕ H•(XA, K).

Proof.

Write F ≃ ¯ F ⊕ K, and remark that H•(XA, F) = H•(XA, ¯ F) ⊕ H•(XA, K).

October 30, 2020 23 / 32

Outline

1

Introduction and motivations

2

Extra-fine sheaves

3

Functors on posets

4

Equivalence of cohomologies

5

Global sections of measures

October 30, 2020 24 / 32

Sheaf cohomology

G : A

• p → Sets induces a presheaf on X A, that has an associated Cech

cohomology H(XA, G). But G is also an element of the topos PSh(A). In topos theory, it is customary to study the derived functors H•

d (A, −) of ΓA(−) = HomAb(A)(Z, −) (sheaf

cohomology).

October 30, 2020 25 / 32

Sheaf cohomology

G : A

• p → Sets induces a presheaf on X A, that has an associated Cech

cohomology H(XA, G). But G is also an element of the topos PSh(A). In topos theory, it is customary to study the derived functors H•

d (A, −) of ΓA(−) = HomAb(A)(Z, −) (sheaf

cohomology). The standard resolution [3, Ex. V.2.3.6] gives an explicit presentation, cf. [8, Prop. 6.1]. The n-cochains are C n(A, G) =

a0→···→an in AG(a0) and the coboundary

δ : C n−1(A, G) → C n(A, G) is given by (δg)a0→···→an =

n−1

• i=0

(−1)igdi(a0→···an) + (−1)nG(ϕn)gdn(a0→···an), (1) where ϕn is the A-morphism from an to an−1 in the sequence an → · · · → a0, and di(c0

f1

→ · · ·

fn

→ cn) =      c1 → · · · cn if i = 0 c0 → · · · ci−1

fi+1◦fi

→ ci+1 → · · · → cn if 0 < i < n c0 → · · · → cn−1 if i = n . (2)

October 30, 2020 25 / 32

Comparison

So we’re in presence of two different types simplicial sets: nerves of the coverings {Kn(U)}n and nerves of categories {Nn(A)}n.

Theorem

If products exist conditionally in A, there is an isomorphism H•(XA, G) and H•

d(A, G)

October 30, 2020 26 / 32

Outline

1

Introduction and motivations

2

Extra-fine sheaves

3

Functors on posets

4

Equivalence of cohomologies

5

Global sections of measures

October 30, 2020 27 / 32

M¨

• bius inversion

Suppose that for each i ∈ I, |Ei| is an integer Ni. Then Nα := |Eα| =

a∈α Na.

Moreover, if A satisfies the strong intersection property, Vα =

β⊂α Sαβ.

Define Dβ = dim Sβ. Since Nα =

β⊂α Dβ, M¨

• bius inversion yields

Dα =

β⊂α µα,βNβ.

Here µα,β are the M¨

• bius coefficients of the poset, defined recursively by

µα,α = 1 for every α ∈ A, µα,β = −

• γ:α→γ→β,γ=β

µα,γ whenever α → β and α = β, and µα,β vanishes otherwise. For instance, when A = 2I, I finite, then µα,β = (−1)|α|−|β| if β ⊂ α, with the convention |∅| = −1.

October 30, 2020 28 / 32

Sections of V

Proposition

If A is finite and satisfies the strong intersection property, and each Ei is finite, then dimK H0(U

A, V ) =

• α,β∈A

µα,βNβ.

Proof.

Since A satisfies the strong intersection property, V is canonically extra-fine, V =

α∈ASα.

Then H0(U

A, V ) = α∈AH0(U A, Sα) = α∈ASα.

Use dim Sα = Dα =

β⊂α µα,βNβ to conclude.

October 30, 2020 29 / 32

Index formula

Theorem

Suppose that A is finite and satisfies the weak intersection property, and that all Ei are finite. Then, χ(A, F) :=

∞

• k=0

(−1)k dim Hk(XA, F) =

• α,β∈A

µα,βNβ = dim H0(U

A, ¯

F) + χ(A).

October 30, 2020 30 / 32

Index formula

Theorem

Suppose that A is finite and satisfies the weak intersection property, and that all Ei are finite. Then, χ(A, F) :=

∞

• k=0

(−1)k dim Hk(XA, F) =

• α,β∈A

µα,βNβ = dim H0(U

A, ¯

F) + χ(A).

Proof.

First remark that F = ¯ F ⊕ K, and then that H∗(XA, F) = H∗(XA, ¯ F) ⊕ H∗(XA, K). Since A satisfies the weak intersection property, ¯ F is acyclic. Additionally, H0(U

A, ¯

F) =

α∈A ¯

S∗

α and its dimension if

• α dim ¯

S∗

α = α dim ¯

Sα =

α,β µα,β(Nβ − 1).

Finally,

k≥0(−1)k dim Hk(U A, K) equals χ(A) = α,β µα,β. In fact,

H•(U

A, K) is also the simplicial cohomology of the nerve N(A).

October 30, 2020 30 / 32

• S. Abramsky, R. S. Barbosa, K. Kishida, R. Lal, and S. Mansfield,

Contextuality, Cohomology and Paradox, in 24th EACSL Annual Conference on Computer Science Logic (CSL 2015), S. Kreutzer, ed., vol. 41 of Leibniz International Proceedings in Informatics (LIPIcs), Dagstuhl, Germany, 2015, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, pp. 211–228.

• S. Abramsky and A. Brandenburger, The sheaf-theoretic structure of

non-locality and contextuality, New Journal of Physics, 13 (2011), p. 113036.

• M. Artin, A. Grothendieck, and J.-L. Verdier, Th´

eorie des Topos et Cohomologie ´ Etale des Sch´ emas: S´ eminaire de G´ eom´ etrie Alg´ ebrique du Bois-Marie 1963/64 - SGA 4. Tome 2, Lecture notes in mathematics, Springer-Verlag, 1972.

• J. Curry, Sheaves, cosheaves and applications, PhD thesis, The University of

Pennsylvania, 2013. arXiv:1303.3255.

• R. Ghrist and H. Riess, Cellular sheaves of lattices and the tarski laplacian,

arXiv preprint arXiv:2007.04099, (2020).

• J. Hansen and R. Ghrist, Toward a spectral theory of cellular sheaves, arXiv

preprint arXiv:1808.01513, (2018).

October 30, 2020 31 / 32

• S. M.-H. Mansourbeigi, Sheaf Theory as a Foundation for Heterogeneous Data

Fusion, PhD thesis, Utah State University, 2010.

• I. Moerdijk, Classifying Spaces and Classifying Topoi, Lecture Notes in

Mathematics, Springer Berlin Heidelberg, 2006.

• M. Robinson, Understanding networks and their behaviors using sheaf theory, in

2013 IEEE Global Conference on Signal and Information Processing, IEEE, 2013,

• pp. 911–914.

, Sheaves are the canonical data structure for sensor integration, Information Fusion, 36 (2017), pp. 208–224.

• M. Robinson, C. Joslyn, E. Hogan, and C. Capraro, Conglomeration of

heterogeneous content using local topology, tech. rep., American University, Mar, 2015.

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