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Essentially Surjective Functor
In mathematics, specifically in category theory, a functor :F:C\to D is essentially surjective if each object d of D is isomorphic to an object of the form Fc for some object c of C. Any functor that is part of an equivalence of categories is essentially surjective. As a partial converse, any full and faithful functor In category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties is called a fully faithful functor. Formal definitions Explicitly, let ''C'' and ' ... that is essentially surjective is part of an equivalence of categories.Mac Lane (1998), Theorem IV.4.1 Notes References * * External links * Functors {{Categorytheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in most areas of mathematics. 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 space (other), quotient spaces, direct products, completion, and duality (mathematics), duality. Many areas of computer science also rely on category theory, such as functional programming and Semantics (computer science), semantics. A category (mathematics), category is formed by two sorts of mathematical object, objects: the object (category theory), objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. Metapho ... [...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 Mathematical object, object X in ''C'' to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Of Categories
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two Category (mathematics), categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation. If a category is equivalent to the dual (category theory), opposite (or dual) of another category then one speaks of a duality of categories, and says that the two categories are dually equivalent. An equivalence of categories consists of a functor betwe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Full And Faithful Functor
In category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties is called a fully faithful functor. Formal definitions Explicitly, let ''C'' and ''D'' be (locally small) categories and let ''F'' : ''C'' → ''D'' be a functor from ''C'' to ''D''. The functor ''F'' induces a function :F_\colon\mathrm_(X,Y)\rightarrow\mathrm_(F(X),F(Y)) for every pair of objects ''X'' and ''Y'' in ''C''. The functor ''F'' is said to be *faithful if ''F''''X'',''Y'' is injectiveJacobson (2009), p. 22 *full if ''F''''X'',''Y'' is surjectiveMac Lane (1971), p. 14 *fully faithful (= full and faithful) if ''F''''X'',''Y'' is bijective for each ''X'' and ''Y'' in ''C''. Properties A faithful functor need not be injective on objects or morphisms. That is, two objects ''X'' and ''X''′ may map to the same object in ''D'' (which is why the range of a full and faithful functor is not necessarily isomorphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categories For The Working Mathematician
''Categories for the Working Mathematician'' (''CWM'') is a textbook in category theory written by American mathematician Saunders Mac Lane, who cofounded the subject together with Samuel Eilenberg. It was first published in 1971, and is based on his lectures on the subject given at the University of Chicago, the Australian National University, Bowdoin College, and Tulane University. It is widely regarded as the premier introduction to the subject. Contents The book has twelve chapters, which are: :Chapter I. Categories, Functors, and Natural Transformations. :Chapter II. Constructions on Categories. :Chapter III. Universals and Limits. :Chapter IV. Adjoints. :Chapter V. Limits. :Chapter VI. Monads and Algebras. :Chapter VII. Monoids. :Chapter VIII. Abelian Categories. :Chapter IX. Special Limits. :Chapter X. Kan Extensions. :Chapter XI. Symmetry and Braiding in Monoidal Categories :Chapter XII. Structures in Categories. Chapters XI and XII were added in the 1998 s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
