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 Introduction and motivations 1 Extra-fine sheaves 2 Functors on posets 3 Equivalence of cohomologies 4 Global sections of measures 5 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 : C op → 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 . October 30, 2020 4 / 32
The fundamental example Index set I that models variables/observables X i , 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 ( X a ) a ∈ α . We consider a collection of finite sets { E i } (micro-states/internal degrees of freedom), and associate to α ⊂ I the set E α := � i ∈ α E i . This defines a functor E : A → Sets ; an arrow α → β is mapped to the canonical projection π βα : E α → E β . Then we can introduce some “secondary” functors: op → Sets , such that α �→ R E α and The observables V : A V ( α → β ) : V β → V α is precomposition by π βα . The probabilities P : A → Sets , such that P α ≡ P ( α ) are the probabilities on E α , and given P ( α → β ) maps p to p ◦ π − 1 βα , which is the marginal/image measure under π βα . October 30, 2020 5 / 32
Marginal problem Given a section of P on a poset A of subsets of I i.e. a “coherent” collection of probabilities { p α } α ∈ Ob A , does there exist a probability q on E such that each p α is a marginal of q ? This means � ∀ α ∈ Ob A , ∀ x α ∈ E α , q ( x α , x I \ α ) = p ( x α ) . x I α ∈ E I \ α 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... October 30, 2020 6 / 32
Interaction decomposition How is the space of sections of P over X A (“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 H 0 ( X A , F ), resp. dim H 0 ( X A , ¯ 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, � S β . V α = β ∈ A 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 X A (“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 H 0 ( X A , F ), resp. dim H 0 ( X A , ¯ 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, � S β . V α = β ∈ A Such V is an example of extra-fine sheaf. Remark that if α → β , then V β ֒ → V α . Introduce the boundary observables B α = � β : β � α V β . We define the interaction subspace S α as a supplement of B α , such that V α = S α ⊕ B α . A common choice is the orthogonal complement of B α for the standard canonical inner product on V α = R E α . October 30, 2020 7 / 32
Outline Introduction and motivations 1 Extra-fine sheaves 2 Functors on posets 3 Equivalence of cohomologies 4 Global sections of measures 5 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 { e U } U ∈ U of endomorphisms e U : F → F that is 1 Local: for every V ∈ U and W open, e V | 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 e V | W ( x ) � = 0, and x = � V ∈ U e V | W ( x ) . Here e V | W = e V ( W ). October 30, 2020 9 / 32
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 { e V } V ∈ V of endomorphisms e V : F → F that is 1 Super-local: for every V ∈ U and W open, e V | 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 e V | W ( x ) � = 0, and x = � V ∈ U e V | W ( x ) . 3 Orthogonal: for every V , W ∈ V such that V � = W , e V ◦ e W = e W ◦ e V = 0. Properties 2 and 3 imply that the e V are projectors: e 2 V = e V . We get an orthogonal decomposition of the sheaf F , as � V ∈ V S V , where S V = im e V . The maps e V | W are then projections on S V ( W ) parallel to � U � = V S U ( W ). October 30, 2020 10 / 32
Cech cohomology Let U be an open covering of a topological space X . Define � u = ( U 0 , ..., U n ) ∈ U n � � � U u := U 0 ∩ · · · ∩ U n � = ∅ K n ( U ) = . and then the space of n -cochains C n ( U , F ) = � u ∈ K n ( U ) F ( U u ). A coboundary operator δ = δ n +1 : C n ( U , F ) → C n +1 ( U , F ) is introduced as n follows: n +1 � ( − 1) i c ( U 0 , ..., ˆ ( δ c )( U 0 , ..., U n +1 ) = U i , ..., U n +1 ) | U u . i =0 By the usual arguments, δ ◦ δ = 0, so one can define the cohomology H n ( U , F ) = ker δ n +1 / im δ n n − 1 . n The Cech cohomology of F on X is H • ( X , F ) = colim U H • ( U , F ) . October 30, 2020 11 / 32
Acyclicity Theorem An extra-fine presheaf F is acyclic i.e for all n ≥ 1 , H n ( 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 { e V } V ∈ V . If ψ is an ( n + 1)-cocycle, choosing u = ( U 0 , ..., U n +1 ) we have n +1 � ( − 1) k +1 ψ ( U 0 , ..., ˆ ψ ( U 1 , ..., U n +1 ) | U u = U i , ..., U n ) | U u k =1 whenever U ∩ ... ∩ U n ⊂ U 0 , and otherwise the terms vanish. Therefore, �� � ψ = � U ∈ U e U ψ = � U ∈ U δφ U = δ U ∈ U φ U . October 30, 2020 12 / 32
Acyclicity Theorem An extra-fine presheaf F is acyclic i.e for all n ≥ 1 , H n ( 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 { e V } V ∈ V . If ψ is an ( n + 1)-cocycle, choosing u = ( U 0 , ..., U n +1 ) we have n +1 � ( − 1) k +1 e U 0 ψ ( U 0 , ..., ˆ e U 0 ψ ( U 1 , ..., U n +1 ) | U u = U i , ..., U n ) | U u k =1 whenever U ∩ ... ∩ U n ⊂ U 0 , and otherwise the terms vanish. Therefore, �� � ψ = � U ∈ U e U ψ = � U ∈ U δφ U = δ U ∈ U φ U . October 30, 2020 12 / 32
Acyclicity Theorem An extra-fine presheaf F is acyclic i.e for all n ≥ 1 , H n ( 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 { e V } V ∈ V . If ψ is an ( n + 1)-cocycle, choosing u = ( U 0 , ..., U n +1 ) we have n +1 � ( − 1) k +1 e U 0 ψ ( U 0 , ..., ˆ e U 0 ψ ( U 1 , ..., U n +1 ) | U u = U i , ..., U n ) | U u � �� � k =1 =: φ U 0 ( U 1 ,..., ˆ U i ,..., U n +1 ) whenever U ∩ ... ∩ U n ⊂ U 0 , and otherwise the terms vanish. Therefore, �� � ψ = � U ∈ U e U ψ = � U ∈ U δφ U = δ U ∈ U φ U . October 30, 2020 12 / 32
Outline Introduction and motivations 1 Extra-fine sheaves 2 Functors on posets 3 Equivalence of cohomologies 4 Global sections of measures 5 October 30, 2020 13 / 32
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