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Subterminal Object
In category theory, a branch of mathematics, a subterminal object is an object ''X'' of a category (mathematics), category ''C'' with the property that every object of ''C'' has at most one morphism into ''X''. If ''X'' is subterminal, then the pair of identity morphisms (1''X'', 1''X'') makes ''X'' into the product (category theory), product of ''X'' and ''X''. If ''C'' has a terminal object 1, then an object ''X'' is subterminal if and only if it is a subobject of 1, hence the name. The category of categories with subterminal objects and functors preserving them is not accessible category, accessible. References External links * Category theory {{cattheory-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] |
Category (mathematics)
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] |
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] |
Springer Science+Business Media
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product (category Theory)
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects. Definition Product of two objects Fix a category C. Let X_1 and X_2 be objects of C. A product of X_1 and X_2 is an object X, typically denoted X_1 \times X_2, equipped with a pair of morphisms \pi_1 : X \to X_1, \pi_2 : X \to X_2 satisfying the following universal property: * For every object Y and every pair of morphisms f_1 : Y \to X_1, f_2 : Y \to X_2, there exists a unique morphism f : Y \to X_1 \times X_2 such that the following diagram commutes: *: Whether a product exists may depend on C or on X_1 and X_2. If it does exist, it is unique up to canonical isomor ... [...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] |
Subobject
In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory,Mac Lane, p. 126 and subspaces from topology. Since the detailed structure of objects is immaterial in category theory, the definition of subobject relies on a morphism that describes how one object sits inside another, rather than relying on the use of elements. The dual concept to a subobject is a . This generalizes concepts such as quotient sets, quotient groups, quotient spaces, quotient graphs, etc. Definitions In detail, let ''A'' be an object of some category. Given two monomorphisms :u: S \to A \ \text \ v: T\to A with codomain ''A'', we define an equivalence relation by u \equiv v if there exists an isomorphism \phi: S \to T with u = v \circ \phi. Equivalently, we write u \leq v if u factors through ''v''—that is, if t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Accessible Category
The theory of accessible categories is a part of mathematics, specifically of category theory. It attempts to describe categories in terms of the "size" (a cardinal number) of the operations needed to generate their objects. The theory originates in the work of Grothendieck completed by 1969, and Gabriel and Ulmer (1971). It has been further developed in 1989 by Michael Makkai and Robert Paré, with motivation coming from model theory, a branch of mathematical logic. A standard text book by Adámek and Rosický appeared in 1994. Accessible categories also have applications in homotopy theory.J. RosickÃ"On combinatorial model categories" ''arXiv'', 16 August 2007. Retrieved on 19 January 2008.Rosický, J. "Injectivity and accessible categories." ''Cubo Matem. Educ'' 4 (2002): 201-211. Grothendieck continued the development of the theory for homotopy-theoretic purposes in his (still partly unpublished) 1991 manuscript ''Les dérivateurs''. Some properties of accessible categories dep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canadian Mathematical Society
The Canadian Mathematical Society (CMS) (french: Société mathématique du Canada) is an association of professional mathematicians dedicated to the interests of mathematical research, outreach, scholarship and education in Canada. It serves the national community through the publication of academic journals, community bulletins, and the administration of mathematical competitions. It was originally conceived in June 1945 as the Canadian Mathematical Congress. A name change was debated for many years; ultimately, a new name was adopted in 1979, upon its incorporation as a non-profit charitable organization. The society is also affiliated with various national and international mathematical societies, including the Canadian Applied and Industrial Mathematics Society and the Society for Industrial and Applied Mathematics. The society is also a member of the International Mathematical Union and the International Council for Industrial and Applied Mathematics. History The Canadian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
