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Higher-dimensional algebra
In mathematics, especially (higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra.
A first step towards defining higher dimensional algebras is the concept of 2-category of higher category theory, followed by the more 'geometric' concept of double category.
A higher level concept is thus defined as a category of categories, or super-category, which generalises to higher dimensions the notion of category – regarded as any structure which is an interpretation of Lawvere's axioms of the elementary theory of abstract categories (ETAC). Thus, a supercategory and also a super-category, can be regarded as natural extensions of the concepts of meta-category, multicategory, and multi-graph, k-partite graph, or colored graph (see a color figure, and also its definition in graph theory).
Supercategories were first introduced in 1970, and were subsequently developed for applications in theoretical physics (especially quantum field theory and topological quantum field theory) and mathematical biology or mathematical biophysics.
Other pathways in higher-dimensional algebra involve: bicategories, homomorphisms of bicategories, variable categories (also known as indexed or parametrized categories), topoi, effective descent, and enriched and internal categories.
In higher-dimensional algebra (HDA), a double groupoid is a generalisation of a one-dimensional groupoid to two dimensions, and the latter groupoid can be considered as a special case of a category with all invertible arrows, or morphisms.
Double groupoids are often used to capture information about geometrical objects such as higher-dimensional manifolds (or n-dimensional manifolds). In general, an n-dimensional manifold is a space that locally looks like an n-dimensional Euclidean space, but whose global structure may be non-Euclidean.
Double groupoids were first introduced by Ronald Brown in Double groupoids and crossed modules (1976), and were further developed towards applications in nonabelian algebraic topology. A related, 'dual' concept is that of a double algebroid, and the more general concept of R-algebroid.
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Higher-dimensional algebra
In mathematics, especially (higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra.
A first step towards defining higher dimensional algebras is the concept of 2-category of higher category theory, followed by the more 'geometric' concept of double category.
A higher level concept is thus defined as a category of categories, or super-category, which generalises to higher dimensions the notion of category – regarded as any structure which is an interpretation of Lawvere's axioms of the elementary theory of abstract categories (ETAC). Thus, a supercategory and also a super-category, can be regarded as natural extensions of the concepts of meta-category, multicategory, and multi-graph, k-partite graph, or colored graph (see a color figure, and also its definition in graph theory).
Supercategories were first introduced in 1970, and were subsequently developed for applications in theoretical physics (especially quantum field theory and topological quantum field theory) and mathematical biology or mathematical biophysics.
Other pathways in higher-dimensional algebra involve: bicategories, homomorphisms of bicategories, variable categories (also known as indexed or parametrized categories), topoi, effective descent, and enriched and internal categories.
In higher-dimensional algebra (HDA), a double groupoid is a generalisation of a one-dimensional groupoid to two dimensions, and the latter groupoid can be considered as a special case of a category with all invertible arrows, or morphisms.
Double groupoids are often used to capture information about geometrical objects such as higher-dimensional manifolds (or n-dimensional manifolds). In general, an n-dimensional manifold is a space that locally looks like an n-dimensional Euclidean space, but whose global structure may be non-Euclidean.
Double groupoids were first introduced by Ronald Brown in Double groupoids and crossed modules (1976), and were further developed towards applications in nonabelian algebraic topology. A related, 'dual' concept is that of a double algebroid, and the more general concept of R-algebroid.