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 spacetimes 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.[1]
^Directed Algebraic Topology: Models of Non-Reversible Worlds, Marco Grandis, Cambridge University Press, ISBN 978-0-521-76036-2 Free download from authors website
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