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Kan Extension
Kan extensions are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named after Daniel M. Kan, who constructed certain (Kan) extensions using limits in 1960. An early use of (what is now known as) a Kan extension from 1956 was in homological algebra to compute derived functors. In ''Categories for the Working Mathematician'' Saunders Mac Lane titled a section "All Concepts Are Kan Extensions", and went on to write that :The notion of Kan extensions subsumes all the other fundamental concepts of category theory. Kan extensions generalize the notion of extending a function defined on a subset to a function defined on the whole set. The definition, not surprisingly, is at a high level of abstraction. When specialised to posets, it becomes a relatively familiar type of question on constrained optimization. Definition A Kan extension proceeds from the data of three categor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of the integers from the natural numbers, of the rational numbers from the integers, of the real numbers from the rational numbers, and of polynomial rings from the field of their coefficients can all be done in terms of universal properties. In particular, the concept of universal property allows a simple proof that all constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property. Technically, a universal property is defined in terms of categories and functors by mean of a universal morphism (see , below). Universal morphisms can also be thought more abstractly as initial or terminal objects of a comma category ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Object (category Theory)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. '' Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
End (category Theory)
In category theory, an end of a functor S:\mathbf^\times\mathbf\to \mathbf is a universal extranatural transformation from an object ''e'' of X to ''S''. More explicitly, this is a pair (e,\omega), where ''e'' is an object of X and \omega:e\ddot\to S is an extranatural transformation such that for every extranatural transformation \beta : x\ddot\to S there exists a unique morphism h:x\to e of X with \beta_a=\omega_a\circ h for every object ''a'' of C. By abuse of language the object ''e'' is often called the ''end'' of the functor ''S'' (forgetting \omega) and is written :e=\int_c^ S(c,c)\text\int_\mathbf^ S. Characterization as limit: If X is complete and C is small, the end can be described as the equalizer in the diagram :\int_c S(c, c) \to \prod_ S(c, c) \rightrightarrows \prod_ S(c, c'), where the first morphism being equalized is induced by S(c, c) \to S(c, c') and the second is induced by S(c', c') \to S(c, c'). Coend The definition of the coend of a functor S:\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coend (category Theory)
In category theory, an end of a functor S:\mathbf^\times\mathbf\to \mathbf is a universal extranatural transformation from an object ''e'' of X to ''S''. More explicitly, this is a pair (e,\omega), where ''e'' is an object of X and \omega:e\ddot\to S is an extranatural transformation such that for every extranatural transformation \beta : x\ddot\to S there exists a unique morphism h:x\to e of X with \beta_a=\omega_a\circ h for every object ''a'' of C. By abuse of language the object ''e'' is often called the ''end'' of the functor ''S'' (forgetting \omega) and is written :e=\int_c^ S(c,c)\text\int_\mathbf^ S. Characterization as limit: If X is complete and C is small, the end can be described as the equalizer in the diagram :\int_c S(c, c) \to \prod_ S(c, c) \rightrightarrows \prod_ S(c, c'), where the first morphism being equalized is induced by S(c, c) \to S(c, c') and the second is induced by S(c', c') \to S(c, c'). Coend The definition of the coend of a functor S:\m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comma Category
In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objects in their own right. This notion was introduced in 1963 by F. W. Lawvere (Lawvere, 1963 p. 36), although the technique did not become generally known until many years later. Several mathematical concepts can be treated as comma categories. Comma categories also guarantee the existence of some limits and colimits. The name comes from the notation originally used by Lawvere, which involved the comma punctuation mark. The name persists even though standard notation has changed, since the use of a comma as an operator is potentially confusing, and even Lawvere dislikes the uninformative term "comma category" (Lawvere, 1963 p. 13). Definition The most general comma category construction involves two functors with the same codomain. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Category
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit (category Theory)
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 category. ... [...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] |
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