Abstract. As a practical foundation for a homotopy theory of abstract spacetime, we extend a category of certain compact partially ordered spaces to a convenient category of "locally preordered" spaces. In particular, we show that our new category is Cartesian closed and that the forgetful functor to the category of compactly generated spaces creates all limits and colimits.

Abstract. Considered is a class of pursuit-evasion games, in which an evader tries to avoid detection. Such games can be formulated as the search for sections to the complement of a coverage region in a Euclidean space over time. Prior results give homological criteria for evasion in the general case that are not necessary and sufficient. This paper provides a necessary and sufficient positive cohomological criterion for evasion in the general case. The principal tools are (1) a refinement of theČech cohomology of a coverage region with a positive cone encoding spatial orientation, (2) 1. Introduction. The motivation for this paper comes from a type of pursuit-evasion game. In such games, two classes of agents, pursuers and evaders, move in a fixed geometric domain over time. The goal of a pursuer is to capture an evader (e.g., by physical proximity or line-of-sight). The goal of an evader is to move in such a manner so as to avoid capture by any pursuer. This paper solves a feasibility problem of whether an evader can win in a particular setting under certain constraints.We focus on the setting of pursuers-as-sensors, in which, at each time, a certain region of space is "sensed" and any evader in this region is considered captured. Evasion, the successful evasion of all pursuers by an evader, corresponds to gaps in the sensed region over time. A coverage pair (E, C) over the time-axis R is, for each time t, the data of an ambient space E t ∼ = R n and a sensed or covered region C t ⊂ E t bundled to form an ambient space-time E ∼ = R n × R and a covered subspace C ⊂ E. The goal of the evader is to construct an evasion path, a section to the restriction E−C → R of the projection map E → R. This is not necessarily a topological problem. However, epistemic restrictions on C are natural in real-world settings. For example, a distributed sensor network may not directly perceive its geometric coverage region C but can often infer its (co)homology from local

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Topological spaces -such as classifying spaces, configuration spaces and spacetimes -often admit extra directionality. Qualitative invariants on such directed spaces often are more informative, yet more difficult, to calculate than classical homotopy invariants because directed spaces rarely decompose as homotopy colimits of simpler directed spaces. Directed spaces often arise as geometric realizations of simplicial sets and cubical sets equipped with ordertheoretic structure encoding the orientations of simplices and 1-cubes. We show that, under definitions of weak equivalences appropriate for the directed setting, geometric realization induces an equivalence between homotopy diagram categories of cubical sets and directed spaces and that its right adjoint satisfies a homotopical analogue of excision. In our directed setting, cubical sets with structure reminiscent of higher categories serve as analogues of Kan complexes. Along the way, we prove appropriate simplicial and cubical approximation theorems and give criteria for two different homotopy relations on directed maps in the literature to coincide.