Examples of cartesian closed categories include: The category Set of all sets, with functions as morphisms, is cartesian closed. The product X×Y is the cartesian product of X and Y, and ZY is the set of all functions from Y to Z. The adjointness is expressed by the following fact: the function f : X×Y → Z is naturally identified with the function g : X → ZY defined by gSee also:

Cartesian closed category, Cartesian closed category - Definition, Cartesian closed category - Examples, Cartesian closed category - Applications, Cartesian closed category - Equational theory

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A metric space is a 2-tuple (X,d) where X is a set and d is a metric on X, that is, a function d : X × X → R such that d(x, y) ≥ 0     (non-negativity) d(x, y) = 0   if and only if   x = y     (identity of indiscernibles) d(x, y) = d(y, x)     ...

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Metric space, Metric space - History, Metric space - Definition, Metric space - Examples, Metric space - Metric spaces as topological spaces, Metric space - Boundedness and compactness, Metric space - Separation properties and extension of continuous functions, Metric space - Distance between points and sets, Metric space - Equivalence of metric spaces, Metric space - Quotient metric space

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An input to a function is called argument of the function. For each argument x, the corresponding unique y in the codomain is called the function value at x, or the image of x under f. The image of x can be written as f(x) or as y. Written mathematics sometimes omits the parentheses around the argument, thus: sin x, but calculators and computers require parentheses around the argument. In some branches of mathematics, such as automata theory, th ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - The vocabulary of functions, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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Directly from the definition, one can show that the identity element e is unique. Then it is possible to define invertible elements: an element x is called invertible if there exists an element y such x*y = e and y*x = e. It turns out that the set of all invertible elements, together with the operation *, forms a group. In that ...

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Monoid, Monoid - Definition, Monoid - Examples, Monoid - Properties, Monoid - Monoid homomorphisms, Monoid - Relation to category theory

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Abelian variety - Definition. A complex torus of dimension g is a torus of real dimension 2g that carries the structure of a complex manifold. It can always be obtained as the quotient of a g-dimensional complex vector space by a lattice of rank 2g. A complex abelian variety of dimension g is a complex torus of dimension g that is also a projective algebraic variety over the field of complex numbers. Since they are complex tori, abelian varieties carry the structur ...

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Abelian variety, Abelian variety - History and motivation, Abelian variety - Analytic theory, Abelian variety - Definition, Abelian variety - Riemann conditions, Abelian variety - The Jacobian of an algebraic curve, Abelian variety - Abelian functions, Abelian variety - Algebraic definition, Abelian variety - Structure of the group of points, Abelian variety - Polarization and dual abelian variety, Abelian variety - Dual abelian variety, Abelian variety - Polarizations, Abelian variety - Polarizations over the complex numbers, Abelian variety - Abelian scheme

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Several equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function. Continuous function topology - Open and closed set definition. The most common one defines continuous functions as those functions where the preimages of open sets are open. Similar to the open set formulation is the closed set formulation, which says that preimages of closed sets are closed. Co ...

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Continuous function topology, Continuous function topology - Definitions, Continuous function topology - Open and closed set definition, Continuous function topology - Neighborhood definition, Continuous function topology - Closure and interior operator definition, Continuous function topology - Closeness relation definition, Continuous function topology - Useful properties of continuous maps, Continuous function topology - Other notes

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The study of categories is an attempt to capture what is commonly found in various classes of related mathematical structures. Consider the following example. The class Grp of groups consists of all objects having a "group structure". More precisely, Grp consists of all sets G endowed with a binary operation satisfying a certain set of axioms. One can proceed to prove theorems about groups by making logical deductions from the set of axioms. For example, it is immediately proved from the axioms t ...

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Category theory, Category theory - Background, Category theory - Historical notes, Category theory - Categories objects and morphisms, Category theory - Some properties of morphisms, Category theory - Functors, Category theory - Natural transformations and isomorphisms, Category theory - Universal constructions limits and colimits, Category theory - Equivalent categories, Category theory - Further concepts and results, Category theory - Higher-dimensional categories

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What precedes this point is intended as an informal introduction to the axioms of the primary arithmetic and primary algebra, and hopefully provides the reader with an intuitive sense of their motivation and rationale. The next order of business is to give the exact forms of the axioms that are used in the following more formal development, devolving from Peirce's various systems of logical graphs via Spencer-Brown's Laws of Form (LOF). In formal proofs, a variation of the annotation scheme from LOF will be used to mark each st ...

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Logical graph, Logical graph - Abstract point of view, Logical graph - In lieu of a beginning, Logical graph - Duality logical and topological, Logical graph - Computational representation, Logical graph - Quick tour of the neighborhood, Logical graph - Primary arithmetic as semiotic system, Logical graph - Primary algebra as pattern calculus, Logical graph - Formal development, Logical graph - Axioms, Logical graph - Frequently used theorems, Logical graph - Exemplary proofs

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A category C must have zero morphisms for the concept of normality to make complete sense. In that case, we say that a monomorphism is normal if it is the kernel of some morphism, and an epimorphism is normal (or conormal) if it is the cokernel of some morphism. C itself is normal if every monomorphism is normal. C is conormal if every epimorphism is normal. Finally, C is binormal if it's both normal and conormal. But note that some authors will use only the word "normal" t ...

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Normal morphism, Normal morphism - Definition, Normal morphism - Examples

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Directly from the definition, one can show that the identity element e is unique. Then it is possible to define invertible elements: an element x is called invertible if there exists an element y such x*y = e and y*x = e. The element y is called the inverse of x and is usually written x−1. Associativity guarantees that inverses, if they exist, are unique. The set of all invertible elements in a monoid M, together with the operation *, forms a group. In that ...

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Monoid, Monoid - Definition, Monoid - Examples, Monoid - Properties, Monoid - Monoid homomorphisms, Monoid - Relation to category theory

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A category C consists of a class ob(C) of objects: a class hom(C) of morphisms. Each morphism f has a unique source object a and target object b where a and b are in ob(C). We write f: a → b, and we say "f is a morphism from a to b". We write hom(a, b) (or homC(a, b)) to denote the hom-class of all morphisms from a to b. (Some a ...

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Category mathematics, Category mathematics - Definition, Category mathematics - Examples, Category mathematics - Types of morphisms, Category mathematics - Types of categories

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Analogy - Identity of relation. In ancient Greek the word αναλογια (analogia) originally meant proportionality, in the mathematical sense, and it was indeed sometimes translated to Latin as proportio. From there analogy was understood as identity of relation between any two ordered pairs, whether of mathematical nature or not. Kant's Critique of Judgment held to this notion. Kant argued that there can be exactly the same relation between two completely different objects. ...

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Analogy, Analogy - Models and theories of analogy, Analogy - Identity of relation, Analogy - Shared abstraction, Analogy - Special case of induction, Analogy - Hidden deduction, Analogy - Shared structure, Analogy - High-level perception, Analogy - Applications and types of analogy, Analogy - Linguistics, Analogy - Mathematics, Analogy - Artificial intelligence, Analogy - Anatomy, Analogy - Law, Analogy - Engineering

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The concept of "average profit" suggested that a process of competition and market-balancing had already established a uniform (or ruling average, or normal) profit rate previously; yet, paradoxically, what profit volumes would be (and consequently profit rates) could be established only after sales, by deducting costs from gross revenues. An output was produced before it was definitively valued in markets, yet the quantity of value produced affected the total price for which it was sold. This was a dynami ...

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Prices of production, Prices of production - Basic definition, Prices of production - Two interpretations of production prices, Prices of production - Three types of production prices, Prices of production - Production prices and the transformation problem, Prices of production - Value and price, Prices of production - Facts and logic

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Homological methods and sheaf theory had already been introduced in algebraic geometry by Jean-Pierre Serre, after sheaves had been invented by Kiyoshi Oka and Jean Leray. Grothendieck took them to a higher level, changing the tools and the level of abstraction. Amongst his insights, he shifted attention from the study of individual varieties to the relative point of view (pairs of varieties related by a morphism), allowing a broad generalization of many classical theorems. This he applied first to the Riemann-Roch theorem, aro ...

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Alexander Grothendieck, Alexander Grothendieck - Mathematical achievements, Alexander Grothendieck - Major mathematical topics from Récoltes et Semailles, Alexander Grothendieck - Life, Alexander Grothendieck - Childhood and studies, Alexander Grothendieck - Politics and retreat from scientific community, Alexander Grothendieck - Manuscripts written in the 1980s, Alexander Grothendieck - Disappearance

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Sheaves are used in topology, algebraic geometry and differential geometry whenever one wants to keep track of algebraic data that vary with every open set of the given geometrical space. They are a global tool to study objects which vary locally (that is, depend on the open sets). As such, they are a natural instrument to study the global behaviour of entities which are of local nature, such as op ...

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Sheaf mathematics, Sheaf mathematics - Introduction, Sheaf mathematics - The formal definition, Sheaf mathematics - Definition of a presheaf, Sheaf mathematics - The gluing axiom, Sheaf mathematics - Examples, Sheaf mathematics - Morphisms of sheaves, Sheaf mathematics - Stalks of a sheaf at a point and germs of functions, Sheaf mathematics - The étale space of a sheaf, Sheaf mathematics - Generalizations, Sheaf mathematics - History

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Universal property - Existence and uniqueness. Defining a quantity does not guarantee its existence. Given a functor U and an object X as above, there may or may not exist a universal morphism from X to U (or from U to X). If, however, a universal morphism (A, φ) does exists then it is unique up to a unique isomorphism. That is, if (A′, φ′) is another such pair then there exists a unique isomorphism g : A → A′ such ...

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Universal property, Universal property - Formal definition, Universal property - Properties, Universal property - Existence and uniqueness, Universal property - Equivalent formulations, Universal property - Relation to adjoint functors, Universal property - Examples, Universal property - Tensor algebras, Universal property - Kernels, Universal property - Limits and colimits, Universal property - What is it good for?, Universal property - History

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Proper map - Definition. A morphism f : X → Y of algebraic varieties or schemes is called universally closed if all its fiber products f × Id: X × Z → Y × Z are closed maps of the underlying topological spaces. A morphism f : X → Y of algebraic varieties or is called proper if it is separated and universally closed. A morphism of schemes is called proper if it is separated, of finite type and unive ...

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Proper map, Proper map - Topological spaces, Proper map - Definition, Proper map - Properties, Proper map - Examples, Proper map - Algebraic varieties and schemes, Proper map - Definition, Proper map - Examples, Proper map - Valuative criterion of properness, Proper map - Stein factorization

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Main articles: universal property, limit (category theory) Using the language of category theory, many areas of mathematical study can be cast into appropriate categories, such as the categories of all sets, groups, topologies, and so on. These categories surely have some objects that are "special" in a certain way, such as the empty set or the product of two topologies. Yet, in the definition of a category, objects are considered to be atomic; i.e. we do not know, whether an object A is a set, a topology, or any other a ...

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Category theory, Category theory - Background, Category theory - Historical notes, Category theory - Categories, objects, and morphisms, Category theory - Some properties of morphisms, Category theory - Functors, Category theory - Natural transformations and isomorphisms, Category theory - Universal constructions, limits, and colimits, Category theory - Equivalent categories, Category theory - Further concepts and results, Category theory - Higher-dimensional categories

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Main articles: category, morphism A category C consists of a class ob(C) of objects: a class hom(C) of morphisms. Each morphism f has a unique source object a and target object b. We write f: a → b, and we say "f is a morphism from a to b". We write hom(a, b) [or Hom(a, b), or homC(a, b)] to denote the hom-class of all morphisms from < ...

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Category theory, Category theory - Background, Category theory - Historical notes, Category theory - Categories, objects, and morphisms, Category theory - Some properties of morphisms, Category theory - Functors, Category theory - Natural transformations and isomorphisms, Category theory - Universal constructions, limits, and colimits, Category theory - Equivalent categories, Category theory - Further concepts and results, Category theory - Higher-dimensional categories

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Properties and parameters based on the idea of connectedness often involve the word connectivity. For example, in graph theory, a connected graph is one from which we must remove at least one vertex to create a disconnected graph. In recognition of this, such graphs are also said to be 1-connected. Similarly, a graph is 2-connected if we must remove at least two vertices from it, to create a disconnected graph. A 3-connected graph requires the removal of at least three vertices, and so on. The connectivity ...

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Connectedness, Connectedness - Other notions of connectedness, Connectedness - Connectivity

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Most mathematicians define a binary relation (and hence a function) as an ordered triple (X, Y, G), where X and Y are the domain and codomain sets, and G is the graph of f. However, some mathematicians define a relation as being simply the set of pairs G, without explicitly giving the domain and co-domain. There are advantages and disadvantages to each definition, but either of them is satisfactory for most uses of functions in mathematics. The explicit domain and ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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The condition for a binary relation f from X to Y to be a function can be split into two conditions: f is total, or entire: for each x in X, there exists some y in Y such that x is related to y. f is single-valued: for each x in X, there is at most one y in Y such that x is related to y. In some contexts, a relation that satisfies condition (1), but not necessarily (2) ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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