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Existence Theorem For Limits
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits. Limits and colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it is helpful to first study the specific examples these concepts are meant to generalize. Definition Limits and colimits in a category C are defined by means of diagrams in C. Formally, a diagram of shape J in C is a functor from J to C: :F:J\to C. The category J is thought of as an index category, and the diagram F is thought of as indexing a collection of objects and morphisms in C patterned on J. One is most often interested in the case where the category J is a small or even finite cat ... [...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] |
Morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, ''morphism'' is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to function composition. A morphism in category theory is an abstraction of a homomorphism. The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. Much of the terminology of morphisms, as well as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Co-cone
In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well. Definition Let ''F'' : ''J'' → ''C'' be a diagram in ''C''. Formally, a diagram is nothing more than a functor from ''J'' to ''C''. The change in terminology reflects the fact that we think of ''F'' as indexing a family of objects and morphisms in ''C''. The category ''J'' is thought of as an "index category". One should consider this in analogy with the concept of an indexed family of objects in set theory. The primary difference is that here we have morphisms as well. Thus, for example, when ''J'' is a discrete category, it corresponds most closely to the idea of an indexed family in set theory. Another common and more interesting example takes ''J'' to be a span. ''J'' can also be taken to be the empty category, leading to the simplest cones. Let ''N'' be an object of ''C''. A cone from ''N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functor Co-cone (extended)
In mathematics, specifically category theory, a functor is a mapping between 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 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 linguistic context; see function word. Definition Let ''C'' and ''D'' be 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'', * associates each morphism f \colon X \to Y in ''C'' to a morphism F(f) \colon F(X) \to F(Y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Co-cone
In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well. Definition Let ''F'' : ''J'' → ''C'' be a diagram in ''C''. Formally, a diagram is nothing more than a functor from ''J'' to ''C''. The change in terminology reflects the fact that we think of ''F'' as indexing a family of objects and morphisms in ''C''. The category ''J'' is thought of as an "index category". One should consider this in analogy with the concept of an indexed family of objects in set theory. The primary difference is that here we have morphisms as well. Thus, for example, when ''J'' is a discrete category, it corresponds most closely to the idea of an indexed family in set theory. Another common and more interesting example takes ''J'' to be a span. ''J'' can also be taken to be the empty category, leading to the simplest cones. Let ''N'' be an object of ''C''. A cone from ''N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual (category Theory)
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category ''C'' and the dual properties of the opposite category ''C''op. Given a statement regarding the category ''C'', by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite category ''C''op. Duality, as such, is the assertion that truth is invariant under this operation on statements. In other words, if a statement is true about ''C'', then its dual statement is true about ''C''op. Also, if a statement is false about ''C'', then its dual has to be false about ''C''op. Given a concrete category ''C'', it is often the case that the opposite category ''C''op per se is abstract. ''C''op need not be a category that arises from mathematical practice. In this case, another category ''D'' is also termed to be in duality with ''C'' if ''D'' and ''C''op are e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Up To
Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' are equal. This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count. For example, ''x'' is unique up to ''R'' means that all objects ''x'' under consideration are in the same equivalence class with respect to the relation ''R''. Moreover, the equivalence relation ''R'' is often designated rather implicitly by a generating condition or transformation. For example, the statement "an integer's prime factorization is unique up to ordering" is a concise way to say that any two lists of prime factors of a given integer are equivalent with respect to the relation ''R'' that relates two lists if one can be obtained by reordering (permutation) from the other. As anot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Cones
In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well. Definition Let ''F'' : ''J'' → ''C'' be a diagram in ''C''. Formally, a diagram is nothing more than a functor from ''J'' to ''C''. The change in terminology reflects the fact that we think of ''F'' as indexing a family of objects and morphisms in ''C''. The category ''J'' is thought of as an "index category". One should consider this in analogy with the concept of an indexed family of objects in set theory. The primary difference is that here we have morphisms as well. Thus, for example, when ''J'' is a discrete category, it corresponds most closely to the idea of an indexed family in set theory. Another common and more interesting example takes ''J'' to be a span. ''J'' can also be taken to be the empty category, leading to the simplest cones. Let ''N'' be an object of ''C''. A cone from ''N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Terminal Object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): is terminal if for every object in there exists exactly one morphism . Initial objects are also called coterminal or universal, and terminal objects are also called final. If an object is both initial and terminal, it is called a zero object or null object. A pointed category is one with a zero object. A strict initial object is one for which every morphism into is an isomorphism. Examples * The empty set is the unique initial object in Set, the category of sets. Every one-element set (singleton) is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category. * In t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Cone
In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well. Definition Let ''F'' : ''J'' → ''C'' be a diagram in ''C''. Formally, a diagram is nothing more than a functor from ''J'' to ''C''. The change in terminology reflects the fact that we think of ''F'' as indexing a family of objects and morphisms in ''C''. The category ''J'' is thought of as an "index category". One should consider this in analogy with the concept of an indexed family of objects in set theory. The primary difference is that here we have morphisms as well. Thus, for example, when ''J'' is a discrete category, it corresponds most closely to the idea of an indexed family in set theory. Another common and more interesting example takes ''J'' to be a span. ''J'' can also be taken to be the empty category, leading to the simplest cones. Let ''N'' be an object of ''C''. A cone from ''N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functor Cone (extended)
In mathematics, specifically category theory, a functor is a mapping between 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 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 linguistic context; see function word. Definition Let ''C'' and ''D'' be 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'', * associates each morphism f \colon X \to Y in ''C'' to a morphism F(f) \colon F(X) \to F(Y) in ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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