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In mathematics, directed algebraic topology is a refinement of algebraic topology for directed spaces, topological spaces and their combinatorial counterparts equipped with some notion of direction. Some common examples of directed spaces are spactimes and simplicial sets. The basic goal is to find algebraic invariants that classify directed spaces up to directed analogues of homotopy equivalence. For example, homotopy groups and fundamental n-groupoids of spaces generalize to homotopy monoids and fundamental n-categories of directed spaces. Directed algebraic topology, like algebraic topology, is motivated by the need to describe qualitative properties of complex systems in terms of algebraic properties of state spaces, which are often directed by time. Thus directed algebraic topology finds applications in Concurrency (computer science), Network traffic control, General Relativity, Noncommutative Geometry, Rewriting Theory, and Biological systems.〔(Directed Algebraic Topology: Models of Non-Reversible Worlds ), Marco Grandis, Cambridge University Press, ISBN 978-0-521-76036-2 Free download from (authors website )〕 ==Directed Spaces== Many mathematical definitions have been proposed to formalise the notion of directed space. E. W. Dijkstra introduced a simple dialect to deal with semaphores, the so-called 'PV language', and to provide each PV program an abstract model: its 'geometric semantics'. Any such model admits a natural partially ordered space (or pospace) structure i.e. a topology and a partial order.〔Topology and Order. Leopoldo Nachbin, Van Nostrand Company, 1965〕 The points of the model should be thought of as the states of the program and the partial order as the 'causality' relation between states. Following this approach, the directed paths over the model i.e. the monotonic continuous paths, represent the execution traces of the program. From the computer science point of view, however, the resulting pospaces have a severe drawback. Because partial orders are by definition antisymmetric, their only directed loops i.e. directed paths which end where they start, are the constant loops. Inspired by smooth manifolds, L. Fajstrup, E. Goubault, and M. Raussen use the sheaf-theoretic approach to define local pospaces.〔(Algebraic Topology and Concurrency ) L. Fajstrup, E. Goubault, and M. Raussen, Theoretical Computer Science, 357, 2006, 241-278〕 Roughly speaking, a local pospace is a topological space together with an open covering whose elements are endowed with a partial order. Given two elements U and V of the covering, it is required that the partial orders on U and V match on the intersection. Though local pospaces allow directed loops, they form a category whose colimits—when they exist—may be rather ill-behaved. Noting that the directed paths of a (local) pospace appear as a by-product of the (local) partial order—even though they themselves contain most of the relevant information about direction—Marco Grandis defines d-spaces〔(Directed homotopy theory, I. The fundamental category ) Marco Grandis, Cahiers Top. Géom. Diff. Catég 44 (2003), 281-316〕 as topological spaces endowed with a collection of paths, whose members are said to be directed, such that any constant path is directed, the concatenation of two directed paths is still directed, and any subpath of a directed path is directed. D-spaces admit non-constant directed loops and form a category enjoying properties similar to the ones enjoyed by the category of topological spaces. As shown by Sanjeevi Krishnan, the drawbacks of local pospaces can be avoided if we extend the notion of pospaces by means of 'cosheaves'. The notion of stream〔(A Convenient Category of Locally Preordered Spaces ) Sanjeevi Krishnan, 2009, Applied Categorical Structures vol. 17, 5, 445-466〕 is defined thus. More precisely one considers preorders on open subsets and one requires that given any open subset U and any open covering Ω of U, the preorder associated with U is 'generated' by the preorders associated with each member of Ω. The resulting category behaves as nicely as the category of d-spaces. Indeed both of them one can define the directed geometric realization of cubical set (simplicial set) so that its underlying topological space is the (usual) geometric realisation. In fact there is a natural embedding G of the category of streams into the category of d-spaces. This embedding admits a left adjoint functor F. The images of F and G are isomorphic, an isomorphism being obtained by restricting F and G to those images. The category of d-spaces can thus be seen as one of the most general formalisations of the intuitive notion of directed space. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Directed algebraic topology」の詳細全文を読む スポンサード リンク
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