Directed algebraic topology
Encyclopedia
In mathematics
, directed algebraic topology is a form of algebraic topology
that studies topological spaces equipped with a family of directed paths, closed under some operations.
The term d-space is applied to these spaces. Directed algebraic topology is a subject that emerged in the 1990s. Its domain is distinguished from classical algebraic topology by the principle that directed spaces have privileged directions and that the directed paths therein need not be reversible. Its homotopical tools, corresponding to ordinary homotopies, fundamental group and fundamental n-groupoids, are similarly 'non-reversible': directed homotopies, fundamental monoids and fundamental ncategories. Its applications deal with domains where privileged directions appear, like concurrent
processes
, traffic networks
, spacetime
models, noncommutative geometry
, rewriting systems
and the modelling of biological system
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, directed algebraic topology is a form of algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
that studies topological spaces equipped with a family of directed paths, closed under some operations.
The term d-space is applied to these spaces. Directed algebraic topology is a subject that emerged in the 1990s. Its domain is distinguished from classical algebraic topology by the principle that directed spaces have privileged directions and that the directed paths therein need not be reversible. Its homotopical tools, corresponding to ordinary homotopies, fundamental group and fundamental n-groupoids, are similarly 'non-reversible': directed homotopies, fundamental monoids and fundamental ncategories. Its applications deal with domains where privileged directions appear, like concurrent
processes
Concurrency (computer science)
In computer science, concurrency is a property of systems in which several computations are executing simultaneously, and potentially interacting with each other...
, traffic networks
Network traffic control
In computer networking, network traffic control is the process of managing, prioritising, controlling or reducing the network traffic, particularly Internet bandwidth, used by network administrators, to reduce congestion, latency and packet loss. This is part of bandwidth management...
, spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
models, noncommutative geometry
Noncommutative geometry
Noncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...
, rewriting systems
Rewriting
In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. What is considered are rewriting systems...
and the modelling of biological system
Biological system
In biology, a biological system is a group of organs that work together to perform a certain task. Common systems, such as those present in mammals and other animals, seen in human anatomy, are those such as the circulatory system, the respiratory system, the nervous system, etc.A group of systems...
s.
Further reading
- Directed homotopy theory, I. The fundamental category, Marco Grandis
- Directed homotopy theory, II. Homotopy Constructs, Marco Grandis, Theory and Applications of Categories, Vol. 10, No. 14, 2002, pp. 369–391
- A Few Points On Directed Algebraic Topology, Marco Grandis
- Directed combinatorial homology and noncommutative tori, Marco Grandis, Math. Proc. Cambridge Philos. Soc. 138 (2005), 233-262