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 Poincar´esections, re- spectively, of a dynamical system. The main result is a Poincar´eDuality for sheaves on spacetimes and generalizations admitting top-dimensional topological singularities, gen- eralizing 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 semi- modules 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 Poincar´eDuality 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 Poincar´eDuality. 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 Poincar´eDuality for sheaves of spectra [1], provides a geometric alternative to algebraic derivations [2] of the following Poincar´eDuality 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 x3.1 recalls the relevant theory. Cubical sets combinatorially model streams. Edge orientations induce local preorders turning geometric realizations jCj of cubical sets C into stream realizations jCj [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 x3.2 recalls cubical sets and x3.3 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 di- rected 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 1 outside of a neighborhood, nat- urally classifies infinitesimal n-dimensional flows at x [Figure ...]. Directed sheaf homology classifies global flows, coherent choices of infinitesimal flows, across a stream. Section x4.2 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 maxi- mum 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) = x2X 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) / σ2C([1]) Γ(σ F) / v2C([0]) Fjvj / 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 x4.3 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) / v2C([0]) Fjvj / σ2C([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 semi- modules 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 x5.1 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.