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Dominant Functor
In category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ..., an abstract branch of mathematics, a dominant functor is a functor ''F'' : ''C'' → ''D'' in which every object of ''D'' is a retract of an object of the form ''F''(''x'') for some object ''X'' of ''C''.. References Functors {{categorytheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism compos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous function, continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a Linguistics, linguistic context; see function word. Definition Let ''C'' and ''D'' be category (mathematics), categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object X in ''C'' to an object F(X) in ''D' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deformation Retract
In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deformation retraction is a mapping that captures the idea of ''continuously shrinking'' a space into a subspace. An absolute neighborhood retract (ANR) is a particularly well-behaved type of topological space. For example, every topological manifold is an ANR. Every ANR has the homotopy type of a very simple topological space, a CW complex. Definitions Retract Let ''X'' be a topological space and ''A'' a subspace of ''X''. Then a continuous map :r\colon X \to A is a retraction if the restriction of ''r'' to ''A'' is the identity map on ''A''; that is, r(a) = a for all ''a'' in ''A''. Equivalently, denoting by :\iota\colon A \hookrightarrow X the inclusion, a retraction is a continuous map ''r'' such that :r \circ \iota = \operatorname_A, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
