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 of spaces generalize to homotopy monoids and fundamental 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 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. 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. 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 as topological spaces endowed with a collection of paths, whose members are said to