DIRECTED POINCARÉ DUALITY Contents 1. Introduction 1 2. Outline 2 3. Ditopology 5 3.1. Streams 5 3.2. Cubical Sets 7 3.3. Compa

DIRECTED POINCARE´ DUALITY

SANJEEVI KRISHNAN

Abstract. This paper generalizes ordinary Abelian sheaf (co)homology for sheaves of semimodules over locally preordered spaces. Motivating examples of locally preordered spaces are manifolds equipped with distinguished vector fields. Motivating examples of such homology and cohomology semimodules are the flows and Poincarésections, re- spectively, of a dynamical system. The main result is a PoincaréDuality for sheaves on spacetimes and generalizations admitting top-dimensional topological singularities, generalizing flow-cut dualities on directed graphs. Explicit calculations for timelike surfaces and formulas for directed graphs are given.

Contents 1. Introduction 1 2. Outline 2 3. Ditopology 5 3.1. Streams 5 3.2. Cubical sets 7 3.3. Comparisons 8 4. (Co)homology 8 4.1. Cubical semimodules 8 4.2. Homology 8 4.3. Cohomology 10 5. Duality 12 5.1. Smoothness 12 5.2. Main result 12 5.3. Examples 14 Appendix 14

1. Introduction Semigroups are to dynamics as groups are to topology. The flows on a directed graph form a semigroup under edgewise addition. The unbounded causal curves on a spacetime, up to homotopies through such flows, generate a semigroup under formal sums. These semigroups are defined like ordinary first integral homology, but with the added restrictions that the coefficients be natural numbers and the chains respect the given dynamics. In order to capture structure finer than mere topology, the restrictions on coefficients and chains are inseparable; without both, the homology-like construction is just the Borel-Moore homology of the underlying space. 1 2 SANJEEVI KRISHNAN

∗ Directed sheaf (co)homology semigroups H∗(X; F),H (X; F), introduced in this paper, describe dynamics on a directed space X [5] subject to constraints given by a sheaf F of semimodules on X. These semigroups are derived constructions like sheaf (co)homology [2], with semimodule-valued sheaves generalizing module-valued sheaves and cubical cosheaves of di- rections replacing the constant cosheaf. Directed (co)homology generalizes sheaf (co)homology [Propositions 4.5 and 4.11] and coincides with a non-Abelian cohomology theory for higher categories [27, 28] in low degrees [Example 4.9] only [Example ??]. Invariant under directed homotopy [5], directed (co)homology is easy to calculate for directed graphs with cellular sheaf coefficients [Proposition 4.8] and spacetime surfaces with constant coefficients [Figure 1 and Examples 5.8, 5.9, 5.10]. Sensitive to ordinary homotopy, directed (co)homology detects dynamical structure unseen by classical stalkwise weak equivalences and hence classically homotopical generalizations of sheaf cohomology [1]. A PoincaréDuality holds when the local dynamics only bifurcates in top dimensions. Just as weak homology manifolds [2] are spaces whose local homologies are free in top and zeroth dimensions and trivial otherwise, smooth streams are directed spaces of finite dimension whose local directed homology semimodules are projective in top and zeroth dimensions and have trivial local directed homotopy in middle dimensions. Examples of smooth streams over the natural numbers include spacetimes [Theorem 5.2] and directed graphs [Proposition 5.3]. Examples of smooth streams over the integers include weak homology manifolds equipped with their maximal directed structures [Proposition 5.4]. This paper formalizes and proves the following version of PoincaréDuality. Theorem 5.5. Fix a natural number n. There exist isomorphisms p ∼ H (X; O⊗S[Σp]F) = Hn−p(X; F), p = 0, 1, . . . , n natural in sheaves F of S-semimodules on n-smooth streams X, with O the nth local directed homology sheaf on X. The main idea of the proof is to approximate cocycles and cycles, locally and up to directed homotopy, as orthogonal subspaces in generalized tangent spaces (the stalks of orientation sheaves). Smoothness implies that (co)cycle sheaves in the case of interest satisfy a directed variant of descent, a homotopical local-to-global property identified along the way [Theorem ...]. The desired global duality hence follows from local dualities. The proof, like the proof of a PoincaréDuality for sheaves of spectra [1], provides a geometric alternative to algebraic derivations [2] of the following PoincaréDuality for sheaf (co)homology AbH, AbH on locally connected spaces. Corollary 5.6. Fix a natural number n. There exist isomorphisms p ∼ AbH X; O⊗S F) = AbHn−pX; F), p = 0, 1, . . . , n natural in sheaves F of S-modules on locally connected weak homology n-manifolds X, with O the nth local homology sheaf on X.

2. Outline State spaces often come equipped with preorders describing causal structure. An example is the total order on 1-dimensional Minkowski spacetime R0,1 [Example 3.9]. The local causal preorders on a spacetime determine its underlying geometry up to conformal equivalence [26]. Streams [20], locally preordered spaces, generalize the salient structure of spacetime. While spacetimes are not closed under such topological constructions as compactifications and configuration spaces, streams are topologically fibered over spaces [20, Lemma 3.22] - DIRECTED POINCARE´ DUALITY 3 each topological construction lifts to a universal stream-theoretic construction. Section §3.1 recalls the relevant theory. Cubical sets combinatorially model streams. Edge orientations induce local preorders turning geometric realizations |C| of cubical sets C into stream realizations |C| [Definition 3.17]. Dually, locally monotone maps from hypercubes to streams X form singular cubical sets sing X [Definition 3.14]. While sing X captures the weak homotopy type underlying a stream X in nature, the failure for sing X to satisfy the Kan condition encodes extra order structure. Homotopy groups πn of based cubical sets satisfying the Kan condition generalize to directed homotopy monoids τn [[12]] of based cubical sets satisfying a milder condition. In particular, the monoid τn sing X? classifies maps from the terminal compactification n [Example 3.6] of R0,1 in the category of streams [Example 3.13] to a based stream X? up to directed homotopy [5]. Section §3.2 recalls cubical sets and §3.3 recalls constructions between streams and cubical sets. Intuitively, sheaf homology classifies stalks up to parallel transport in degree 0 and higher order parallel transport in higher degrees. Abelian sheaf homology are the global homotopy groups of generalized chains, a cubical sheaf constructed from a projective resolution of the constant cosheaf [2]. Directed homology are the global homotopy monoids of generalized directed chains, a cubical sheaf constructed from a precosheaf of cubical nerves. The directed homotopy monoids of the stalk at a point x, often represented by stream maps from the n terminal compactification of R0,1 [Example 3.13] sending ∞ outside of a neighborhood, naturally classifies infinitesimal n-dimensional flows at x [Figure ...]. Directed sheaf homology classifies global flows, coherent choices of infinitesimal flows, across a stream. Section §4.2 introduces the theory. Directed sheaf homology generalizes Abelian sheaf homology AbH for streams whose circulations generate their topologies in a certain sense [Proposition 4.5]. Corollary 4.6. There exist isomorphisms ∼ Hn(X; F) = AbHnX; F) natural in module-valued sheaves F on a locally connected space X equipped with its maximum circulation. Zeroth directed homology is the coproduct of coefficient stalks modulo parallel transport along locally monotone paths. Proposition 4.7. For each S, there exists an isomorphism Q F H0(X; F) = x∈X x /≡, natural in sheaves F on streams X, where ≡ is the smallest congruence with α ≡ β for all ... First directed homology is an equalizer, a generalized kernel, for cellular stalkwise flat sheaves on directed graphs. Proposition 4.8. There exist dotted maps making the diagram

σ7→σ0 Q ∗ / Q H1(X; F) / σ∈C([1]) Γ(σ F) / v∈C([0]) F|v| / H0(X; F) σ7→σ1 natural in cellular sheaves F of S-semimodules on 1-dimensional cubical sets C, commute such that the right three semimodules form a coequalizer diagram and for F ﬂat or stalkwise invertible, the left three semimodules form an equalizer diagram. 4 SANJEEVI KRISHNAN

Cohomology is formally dual to homology. While maps from algebraic models K(S, n) of spheres to chains define sheaves of cycles, maps from chains to algebraic models K(S, n) [Definition 4.1] of spheres define sheaves of cocycles. Just as directed homology classifies global cycles up to connected components [Proposition ...], directed cohomology classifies global cocycles up to connected components. Section §4.3 introduced the theory. Directed sheaf cohomology generalizes Abelian sheaf cohomology AbH for streams whose circulations generate their topologies in a certain sense [Proposition 4.11]. Corollary 4.12. There exist isomorphisms n ∼ n H (X; F) = AbH X; F) natural in sheaves F of modules over a ring on a locally connected space X equipped with its maximum circulation. Zeroth directed cohomology is the global sections functor for sheaves on streams whose global preorders reflect the connectivity of the underlying spaces. Proposition 4.13. There exists an isomorphism H0(X; F) = ΓF natural in sheaves F of semimodules on a stream X with 6X connected. Proposition 4.14. There exist dotted maps making the diagram

σ7→σ0 0 Q / Q ∗ 1 H (X; F) / v∈C([0]) F|v| / σ∈C([1]) Γ(σ F) / H (X; F) σ7→σ1 natural in cellular sheaves F of S-semimodules on 1-dimensional cubical sets C, commute such that the right three semimodules form a coequalizer diagram and the left three semimodules form an equalizer diagram. Just as local homological properties characterize weak homology manifolds, local directed homological properties of canonical cosheaves more generally [Proposition 5.4] characterize smooth streams. Section §5.1 formalizes smoothness and identiﬁes some examples of interest.

Proposition 5.3. Ditopological graphs are 1-smooth over N. Theorem 5.2. Each n-spacetime is n-smooth stream over N.

Proposition 2.1. An n-whmR with maximum circulation is n-smooth over R. A PoincaréDuality holds for directed (co)homology on smooth streams [Theorem 5.5]. Section §5.2 states and proves the duality [Theorem 5.5]. Section §5.3 calculates some examples of the duality on (1 + 1)-spacetimes [Examples 5.8, 5.9, 5.10]. A technical tool developed along the way is a directed homotopical local-to-global theorem [Theorem ...] for cubical presheaves. Scott-domains are posets that model the domains of computable partially recursive functions [..]. When the stalks of a cubical presheaf F admit the compatible structure of a Scott-domain, then there exists a Scott-continuous operator on the product of the stalks improving their coherence in a certain sense. The Scott Fixed Point Theorem then guarantees that the operator describes a terminating and natural algorithm for approximating a 0-chain of F by a global section of F. The existence of such an algorithm confers a constructive descent property on a cubical presheaf, used to infer a global Poincaré Duality from stalkwise PoincaréDualities. DIRECTED POINCARE´ DUALITY 5

3. Ditopology Directed spaces admit both topological and combinatorial descriptions. Streams, described in §3.1, are topological descriptions. Cubical sets, described in §3.2, are combinatorial descriptions. 3.1. Streams. We recall one [20] of several [5, 12] formalisms for causal structure on a general topological space. We write OX for the topology of a space X.A stream is a space whose open subsets are coherently preordered as follows.

Deﬁnition 3.1. A circulation on a space X is a function 6 assigning to each U ∈ OX a S preorder 6U on U such that for each collection V ⊂ OX , 6S V is the preorder on V with smallest graph containing [ (1) graph(6U ). U∈V A stream is a space X equipped with a circulation, which we write as 6, on X.A stream map is a continuous function f : X → Y of streams for which f(x) 6U f(y) whenever x 6f −1U y for each U ∈ OY . Example 3.2. Consider a stream X. For all open subsets U ⊂ V ⊂ X,

graph(6U ) ⊂ graph(6V ). In particular, 6 takes antisymmetric values if 6X is antisymmetric. Streams are closed under topological constructions. Proposition 3.3 ([20, Lemma 3.22]). The forgetful functor (2) D → T from the category D of streams and stream maps to the category T of spaces and maps is topological. Substreams and quotient streams denote subspaces and quotient spaces equipped with suitable universal circulations. 0 00 Deﬁnition 3.4. Fix a space X. On X, circulation 6 is less than circulation 6 if 0 00 graph(6U ) ⊂ graph(6U ),U ∈ OX . The maximum circulation on X is the largest circulation on X with respect to the afore- mentioned partial order. Lemma 3.5. For a connected and locally connected space X,

graph(6X ) = X × X for 6 the maximum circulation on X. Example 3.6 (Terminal compactifications). A compactification of a stream X is an inclusion of X into a compact Hausdorff stream as a dense substream. For each locally compact Hausdorff stream X, there exists a compactification X X,→ S of X, unique up to isomorphisms of streams under X, terminal among all compactifications of X. The underlying space of a terminal compactification is ordinary one-point compactification [Proposition 3.3]. 6 SANJEEVI KRISHNAN

Sometimes a global partial order on a space X uniquely extends to a circulation on X. The following proposition follows from [20, Proposition 5.11] and [25, Theorem 5]. Proposition 3.7. For a compact Hausdorﬀ space X, an assignment

X 7→6X of a partial order 6X on X uniquely extends to a circulation on X if graph(6X ) is closed in X × X and every maximal 6X -chain between two given points is connected as a subspace of X. This paper henceforth treats as streams all compact Hausdorff spaces X equipped with partial orders 6X having graphs graph(6X ) closed in X × X and every maximal 6X -chain between two given points connected. Example 3.8 (Topological lattices). A topological lattice is a lattice topologized so that meet and join operations are jointly continuous. On a compact Hausdorff connected topological lattice L, the partial order 6L has closed graph 2 graph(6L) = {(x, y) ∈ L | x ∨ y = y} n and every maximal 6L-chain is connected. In particular, the topological hypercube I , n equipped with coordinate-wise minimum and maximum operations, is a stream I0,1. Spacetimes are examples of streams. A time-orientation on a smooth manifold M equipped with Lorentzian metric g is a non-empty equivalence class of smooth vector fields V on M such that g(V,V ) < 0 everywhere, where two such vector fields V,W are equivalent if g(V,W ) < 0 everywhere. An n-spacetime is a Lorentzian manifold admitting a time- orientation. Each tangent vector space TpM of a spacetime M admits an additive partial order 6TpM whose cone of positive elements is defined by the values of all vector fields in the time-orientation of M at p. Spacetimes M are streams such that each inverse exponential map −1 expp : U,→ TpM from a convex open normal neighborhood U of p, preordered by the circulation 6U at U, defines an embedding of preordered spaces for each p ∈ M. Example 3.9 (Minkowski Spacetime). The Minkowski (n + 1)-spacetime n+1 Rn,1 = (R , g) is the manifold Rn+1 equipped with constant Lorentzian metric g defined by 2 2 2 2 2 dg ≡ dx1 + dx2 + ··· + dxn − dxn+1 and time-orientation containing the vector field ∂/∂x1 (and hence also ∂/∂x2,..., ∂/∂xn). The circulation on R0,1 sends each open interval to the standard total order on that interval. 2 ∼ n n As streams, R0,1 = R1,1 but R0,1 Rn,1 for all n > 1. The stream I0,1 from Example 3.8 is n a substream of R0,1. Example 3.10 (Circle). The spacetime circle 1 S1,0 = (S, g) is the unit circle S = {eiθ ∈ C | θ ∈ R} equipped with Lorentzian metric g satisfying dg2 = dθ2 and time-orientation containing the vector field ∂/∂θ. DIRECTED POINCARE´ DUALITY 7

Example 3.11 (Torus). The spacetime torus is the product stream 1 1 S1,0 × S1,0. Example 3.12 (Klein Bottle). The coequalizer diagram

(x,y)7→(x+1,y) 2 / R0,1 / R0,12 / K (x,y)7→(y,x+1) in the category of streams deﬁnes the spacetime Klein bottle K. Example 3.13 (Directed spheres). For each n > 0, the terminal compactiﬁcation n (3) SR0,1 n n of R0,1 [Example 3.6] based at the point not in R0,1 is based isomorphic to the quotient n n stream I0,1/∂I based at its quotiented point. For n = 1, (3) is isomorphic to the spacetime 1 circle S1,0 [Example 3.10]. For n > 1, ∞ has no open neighborhood U in (3) with 6U antisymmetric and hence (3) is not a spacetime. 3.2. Cubical sets. Just as connective chain complexes represent formal colimits in Abelian categories, cubical objects represent formal colimits of directed structures; the extrinsic orientations of abstract cubes represent intrinsic directionality in a hypothetical colimit. The subcategory of the Cartesian monoidal category of small categories and functors between them generated by the morphisms

δ−, δ+ : [0] → [1], δ−(0) = 0, δ+(0) = 1, σ : [1] → [0]. are the domains of cubical objects.

⊗n−1 ⊗n Deﬁnition 3.14. For each n and i = 1, 2, . . . , n, let δ±i ∈ ([1] , [1] ) denote ⊗n−i−1 ⊗n−i−1 δ−i = 1[1]⊗i−1 ⊗ δ− ⊗ [1] , δ+i = 1[1]⊗i−1 ⊗ δ+ ⊗ [1] . For each category C , let cC denote the functor category op cC = C .

The n-skeleton sk nC, if it exists, of a cC -object C is the smallest subfunctor of C such ⊗n ⊗n that (sk nC)([1] ) = C([1] ). The C -object π0C of components, if it exists, of a cC -object is deﬁned by the following coequalizer diagram in C :

C(δ−) / C([1]) / C([0]) / π0C C(δ+) A cubical set is an cS -object and a cubical function is a cS -morphism. Example 3.15. The representable cubical sets ⊗n ⊗n [1] = (−, [1] ), n = 0, 1, 2,... model n-dimensional hypercubes. The cubical nerve ner C of a small category C is the cubical set op ner = (−, ) op : → , C C C S where C is the category of small categories and functors between them. Inclusion ,→ C ⊗n ⊗n ⊗n induces a natural cubical inclusion [1] → ner[1] natural in -objects [1] . 8 SANJEEVI KRISHNAN

Definition 3.16. Let Cˆ and RC denotes the cubical sets ⊗n ! Z [1] ηCˆ ηex Cˆ Cˆ = C([1]⊗n) · ner[1]⊗n,RC = colim Cˆ / ex Cˆ / ··· η¯ / η¯ / Cˆ ex Cˆ natural in cubial sets C. 3.3. Comparisons. A stream realization functor and its right adjoint translate between the combinatorial and topological. Lattice operations on ordinals induce topological lattice operations, and hence circulations [Proposition 3.7], on topological hypercubes and more general geometric realizations. Definition 3.17. Let | − | denote the cocontinuous functor | − | : cS → D sending tensor products of finite cubical sets to binary Cartesian products, [0] to {0}, [1] to the unit interval equipped with its standard total order, and [δ]: [0] → [1] to the stream map having image δ(0) for δ = δ−, δ+.A stream realization is a stream in the image of | − | up to stream isomorphism. Let sing denote the right adjoint to | − |. Theorem 3.18. There exists an isomorphism | A | ∼ A π0|B| = π0(RB) . natural in cubical sets A and B with A finite. Corollary 3.19. There exists an isomorphism ∼ ( [1]⊗n,∂ [1]⊗n) τn(|C|, | ? |) = π0(RC,?) . natural in cubical sets C equipped with a distinguished vertex ?.

4. (Co)homology

Each stream X comes equipped with a natural cosheaf 6 of preorders and hence a natural copresheaf S[ner−] algebraicizing the information of 6 for each choice of ground semiring S. A (co)homology theory for sheaves of semimodules over streams reﬁnes classical sheaf (co)homology by replacing the role of the constant cosheaf over a locally connected space with S[ner].

op 4.1. Cubical semimodules. A cubical S-semimodule is a functor → MS and a cubical S-map is a natural transformation between cubical S-semimodules. Deﬁnition 4.1. Let K(S, n) denote the cubical S-semimodule

⊗n . S[ [1] ⊗n K(S, n) = ∂[[1] ] natural in semirings S.

4.2. Homology. For brevity, a (cubical pre)sheaf will mean a MS-valued (cubical pre)sheaf. Deﬁnition 4.2. Let (−)◦ denote the functor

op OX OX cMS → cMS . naturally sending a cubical copresheaf A to the cubical presheaf A(X) A(X \ −). DIRECTED POINCARE´ DUALITY 9

Deﬁnition 4.3. Let Hn(X; F) denote the semimodule ◦ Hn(X; F) = τnΓ(RS[ner] ⊗S F) natural in sheaves F of semimodules on streams X

A reformulation of directed homology amenable to dualization into a directed cohomology theory is as follows.

Proposition 4.4. There exist isomorphisms ∼ ◦ Hn(X; F) = π0Γ (homS(K(S, n),RS[ner] )⊗S F) natural in sheaves F of S-semimodules on streams X

The homology AbH (X; F) of a locally connected space X with coeﬃcients in a sheaf F of R ◦ modules over a ring R on X [2, V-3] is the homology of the chain complex (F⊗S (Lk ) ), where LkR denotes a projective resolution of the constant cosheaf on X taking the value of R.

Proposition 4.5. There exist isomorphisms ∼ Hn(X; F) = AbH (X; F) natural in continuous and sheaves F of R-modules on a space X equipped with a circulation ◦ ◦ such that πqR[ner] = 0 for q > 0 and ner is stalkwise connected. Proof. There exist natural isomorphisms ∼ −1 ◦ (4) Hn(X; F) = π0 sh homR(K(R, n), (S[ner][sk n−1S[ner]] ) )⊗S F ∼ ◦ = π0 homR(K(R, n), (R[ner] ⊗S F))(5) ∼ R ◦ = π0 homR(K(R, n), ((Lk ) ⊗S F))(6) ∼ R ◦ = Hn((Lk ) ⊗S F)(7) with (4) by deﬁnition, (5) by Ψ continuous, (6) by assumption on R[ner]◦, and (7) by the Dold-Kan theorem.

Corollary 4.6. There exist isomorphisms ∼ Hn(X; F) = AbH (X; F) natural in continuous and sheaves F of R-modules on a locally connected space X equipped with its maximal circulation.

Proof. Proposition 4.5 applies by Lemma ??.

Proposition 4.7. There exists an isomorphism Q F (8) H0(X; F) = x∈X x /≡, natural in sheaves F of semimodules on streams X, where ≡ denotes the smallest congruence satisfying α ≡ β if there exists a directed path γ in X such that α = σ0 and β = σ1 for ∗ σ ∈ Γ(γ F)0. 10 SANJEEVI KRISHNAN

Proof. There exist solid vertical morphisms deﬁning natural isomorphisms

σ7→σ L y L F(U) / Fx x U y / x∈X 6 σ7→σx ∼= ∼= d−1⊗S F ◦ / ◦ Γc((S[ner] )([1])⊗S F) / Γc((S[ner] )([0])⊗S F) d+1⊗S F of coequalizer diagrams above. Hence the coequalizer of the bottom row, the left hand side of (8), is naturally isomorphic to the coequalizer of the top row, the right hand side of (8).

An S-semimodule M is ﬂat if −⊗S M preserves ﬁnite limits. Proposition 4.8. There exist dotted maps making the diagram

A proof appears in [22].

Example 4.9 (Parity Complexes). The homology of the parity complex [28]

γ7→γ0 / L ∗ / L / 0 / σ∈C([1]) Γ(σ F) / v∈C([0]) F|v| / 0 γ7→γ1 of semimodules over a semiring associated to a cellular sheaf F on a digraph C coincides c with the directed homology semimodule H∗(X; F) [Propositions 4.7 and 4.8]. 4.3. Cohomology. Cohomology classiﬁes twisted cocycles up to homotopy.

Deﬁnition 4.10. Let Hn(X; F) denote the semimodule

n H (X; F) = π0Γ (homS(S[ner],RK(S, n))⊗S F) , natural in sheaves F of semimodules on a stream X.

The cohomology AbH (X; F) of a locally connected space X with coeﬃcients in a sheaf F of R-modules over X [2, V-3] is the cohomology of the chain complex ((LkR)∗ ⊗ F), with (LkR) a projective resolution of the constant cosheaf on X taking values in R [2, Proposition . . . ].

Proposition 4.11. There exist isomorphisms n ∼ H (X; F) = AbH (X; F) natural in continuous and sheaves F of R-modules on a space X equipped with a circulation ◦ ◦ such that πqR[ner] = 0 for q > 0 and ner is stalkwise connected. DIRECTED POINCARE´ DUALITY 11

Proof. There exist natural isomorphisms

n ∼ H (X; F) = π0 (sh homR(S[ner],K(R, n))⊗S F)(9) ∼ = π0 (homR(R[ner],K(R, n))⊗S F)(10) ∼ R ∗ = π0 homR(((Lk ) ⊗S F),K(R, n))(11) ∼ n R ∗ = H ((Lk ) ⊗S F)(12) with (9) by deﬁnition, (10) by Ψ continuous, (11) by assumption on R[ner]◦, and (12) by the Dold-Kan theorem.

Corollary 4.12. There exist isomorphisms

Hn(X; F) =∼ Hn(X; F) natural in continuous and sheaves F of R-modules on a locally connected space X equipped with its maximum circulation.

Proof. Proposition 4.11 applies by Lemma ??.

Classical zeroth sheaf cohomology is the global sections functor. Zeroth directed sheaf cohomology also is the global sections functor, at least when the global preorder on the stream is connected. A preorder 6X on a set X is connected if the equivalence relation generated by 6X has graph X × X. Proposition 4.13. There exists an isomorphism

H0(X; F) = ΓF natural in sheaves F of semimodules on a stream X with 6X connected. Proof. There exist natural isomorphisms

0 ∼ −1 (13) H (X; F) = π0(sh homS S[ner],K(S, 0)[sk n−1K(S, 0)] ⊗S F) ∼ = π0(sh homR(S[ner],K(R, 0))⊗S F)(14) ∼ = π0(kS⊗S F)(15) ∼ = π0(F)(16) (17) =∼ F with (14) by deﬁnition, (14) by R a ring, (15) by ner connected by assumption on X, (16) by kS a unit for ⊗S , and (17) by F discrete.

Proposition 4.14. There exist dotted maps making the diagram

5. Duality 5.1. Smoothness. Smoothness generalizes certain properties of spacetimes. Deﬁnition 5.1. A stream X is n-smooth over S if: −1 ◦ (1) For each q > 0, πq sh homS(K(S, n), (S[ner][sk n−1S[ner]] ) ) = 0. (2) For each p 6 n, x ∈ X, and commutative solid diagram of the form ⊗p−1 S[[1] ] / 0 ,

S[∂ [1]⊗p] S[ner]◦ 8/ x

⊗p S[[1] ] there exists a dotted arrow making the entire diagram commute.

The condition πq = 0 for all q ensures that the stream has dimension bounded by n. The right lifting property ensures that the stream does not bifurcate in lower dimensions.

Theorem 5.2. Each n-spacetime is n-smooth over N. Proposition 5.3. Stream realizations of digraphs are 1-smooth over N.

An weak homology n-manifold over a ring R (n-whmR) is a space X whose sheaf of local p-homology groups with coeﬃcients in R is trivial for p 6= n and stalkwise free for p = n.

Proposition 5.4. An n-whmR with maximum circulation is n-smooth over R. Proof. Fix a point x in a weak homology n-manifold. There exist isomorphisms −1 ◦ ∼ ◦ πq sh homR(K(R, n), (S[ner][sk n−1S[ner]] ) ) = πq+nR[ner] ∼ = Hq+n =∼ 0

For each 0 < p < n, the epi cubical homomorphism R[ner] → 0 of cubical groups with connections is a Kan ﬁbration and hence has the right lifting property ⊗p ⊗n with respect to ∂[1] → [1] . ⊗n 5.2. Main result. The natural action of Σn on ner([1] ) induces a natural action of Σn on Hn. This paper hence regards Hn as equipped with the action from Σn and hence also Σp for all p 6 n by the inclusion Σp ,→ Σn regarding a p-permutation as an n-permutation ﬁxing the last (n − p)-letters. Theorem 5.5. Fix a natural number n. There exist isomorphisms p ∼ H (X; O⊗S[Σp]F) = Hn−p(X; F), p = 0, 1, . . . , n with O = H~ n, natural in sheaves F of S-semimodules on n-smooth streams. DIRECTED POINCARE´ DUALITY 13

q Proof. Let Z C,ZqC denote the cubical S-semimodules q −1 Z C = homS(C,K(S, q)[sk q−1K(S, q)] ) −1 ZqC = homS(K(S, q),C[sk q−1C] ) natural in cubical S-semimodules C. Let Ap be the cubical S-subpresheaf

p ⊗n p A ⊂ (S[ner])([1] )⊗S Z S[ner] p U p such that A (U) is generated by all tensors zn ⊗ zU satisfying p U p U p U zU (d±1zn ) = zU (d±2zn ) = ··· = zU (d±nzn ) = 0. q U U Let zU , zq , aq denote elements q q U U q zU ∈ Z S[ner U], zq ∈ ZqS[ner U], aq ∈ A (U).

It suﬃces construct level-wise π0-isomorphisms

p ι ◦ p A / ZnS[ner] ⊗S Z S[ner]

∆

◦ Zn−pS[ner] natural in sheaves F on X. For then there would exist isomorphisms p ∼ p H (X; O ⊗S[Σp] F) = Hn(X; Z S[ner] ⊗S[Σp] F)(18) ∼ ◦ p = π0(ZnS[ner] ⊗S[Σp]Z S[ner]⊗S F)(19) ∼ p = π0(A ⊗S F)(20) ∼ ◦ = π0(Zn−pS[ner] ⊗S F)(21) ∼ (22) = Hn−p(X; F), natural in sheaves F on X, with (18) by definition, (20) by (sh ι⊗S F) an isomorphism, (21) by (sh ∆⊗S F) an isomorphism, and (22) by definition. Fix an open subset U ⊂ X. Define the U-components of ι, ∆ by the rules U p U U p U p ∆U an ⊗ zU = (an )∗ ◦ δ, ιU (an ⊗ zU ) = zn ⊗ zU , ⊗p ⊗n where δ is the unique retraction [1] ,→ [1] in to projection onto the first n−p factors. ⊗n The map ι is a level-wise π0-isomorphism by |[1] | a topological lattice. The map ∆ is a level-wise π0-isomorphism by a straightforward application of n-smoothness. Corollary 5.6. Fix a natural number n. There exist isomorphisms ∼ (23) AbH (X; O⊗S F) = AbH (X; F), p = 0, 1, . . . , n natural in sheaves F of S-modules on weak homology n-manifolds X, with O the nth local homology sheaf on X.

p Proof. In (23), the left side is naturally isomorphic to H (X; O⊗S F) [Proposition 4.5], the right side is naturally isomorphic to Hp(X; F) [Proposition 4.11], and hence the desired isomorphism follows [Theorem 5.5] because X is n-smooth over Z [Proposition 5.4] and hence S. 14 SANJEEVI KRISHNAN

5.3. Examples. The following calculations demonstrate the duality and its limits.

1 Example 5.7. For the 1-spacetime circle S1,0, ( c 1 n 1 N, n = 0, 1 Hn(S1,0; N) = Hc (S1,0; N) = 0, n > 2. Example 5.8. For the Klein 2-spacetime K,  N, n = 0 ( c  n N, n = 0, 1, 2 Hn(K; N) = C22, n = 1 ,Hc (K; N) = 0, n > 2, 0, n > 1. where C22 denotes the free commutative monoid presented by two generators g1, g2 and single relation 2g1 = 2g2. 1 1 Example 5.9. For the 2-spacetime torus S1,0 × S1,0,  N, n = 0, 2 c 1 1 n 1 1  Hn(S1,0 × S1,0; N) = Hc (S1,0 × S1,0; N) = N ⊕ N, n = 1 0, n > 2.

∼ −1 The following calculation demonstrates that the dual isomorphism AbH (X; F) = AbH (X; O ⊗S F) in (co)homology with locally constant and ﬁnitely generated module-valued coeﬃcients does not readily generalize.

Example 5.10. For the punctured Minkowski 2-spacetime R2,1 \ 0, c 1 H1(R2,1 \ 0; N) = 0,Hc (R2,1 \ 0; N) = Z.

Appendix A. Semimodules A semiring is a monoid object in the closed monoidal category M of commutative monoids and monoid homomorphisms, with closed structure M (A, B) deﬁned point-wise by multiplication of B. Concretely, a semiring is a set R with associative multiplications +R, ×R on R having respective units 0R, 1R and satisfying

a +R b = b +R a

r ×R (a +R b) = (r ×R a) +R (r ×R b)

0R ×R r = 0R

A semiring R is commutative as a monoid object if ×R is commutative. Example A.1 (Free monogenic semiring). The natural numbers

N = {0, 1, 2,...} under ordinary addition and multiplication forms a semiring. Example A.2 (Tropical semiring). The tropical semiring is the set

T = {0, 1, 2,...} equipped with binary operations +T, ×T deﬁned by x +T y = min(x, y) and x ×T y = x + y. DIRECTED POINCARE´ DUALITY 15

A S-semimodule is a module object over a semiring S. Concretely, an S-semimodule is a set M equipped with an associative binary operation +M having unit 0M and function · : R × M → M satisfying

0R · x = 0M

1R · x = x

r · (a +M b) = (r · a) +M (r · b)

(r +R s) · x = (r · x) +M (s · b),

Proposition A.3. The category MS admits a unique tensor product

⊗S : MS × MS → MS turning MS into a closed monoidal category whose closed structure homS(A, B) sends each pair A, B of MS-objects to the set of semimodule homomorphisms A → B equipped with structure pointwise inherited from B. For each semiring S, the left adjoint

S[−]: S → MS to the forgetful functor MS → S sends a set X to the free S-semimodule S[X] generated by X.

Appendix B. Preorders

Preorders naturally encode causal structure. Formally, a preorder 6X on a set X is a binary relation on X satisfying: (1) For all x ∈ X, x 6X x. (2) For all x, y, z ∈ X, x 6X z whenever x 6X y and y 6X z. A preorder 6X on a set X is a partial order if 6X further satisﬁes: (1) For all x, y ∈ X, x = y whenever x 6X y and y 6X x. A preordered set is a set X equipped with a preorder, written 6X in this paper. A poset is a preordered set X such that 6X is a partial order. Theorem [Scott]. For a Scott-domain X, a Scott-continuous map X → X has a least ﬁxed point. Theorem [1]. Globally hyperbolic spacetimes are bicontinuous as posets.

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