category theory

Addition is the most basic operation of arithmetic. In its simplest form, addition combines two numbers, the addends or terms, into a single number, the sum. Adding more than two numbers can be viewed as repeated addition; this procedure is known as summation and includes ways to add infinitely many numbers in an infinite series. Repeated addition of the number one is the most basic form of counting. Addition can also be defined for mathematical objects other than numbers — for example, matrices or ...

Including:

• Addition - Notation and terminology
• Addition - Interpretations
• Addition - Extending a measure
• Addition - Combining translations
• Addition - Properties
• Addition - Commutativity
• Addition - Associativity
• Addition - Zero and one
• Addition - Units
• Addition - Definitions and proofs for the real numbers
• Addition - Naturals
• Addition - Integers
• Addition - Rationals
• Addition - Reals
• Addition - Generalizations
• Addition - In algebra
• Addition - Addition of sets
• Addition - Related operations
• Addition - Notes

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There are a variety of recognized kinds of metalanguages including embedded, ordered, and nested or hierarchical. Embedded metalanguages, as their name suggests, are metalanguages embedded in an object language. They occur both formally and naturally. This idea is found in Douglas Hofstadter's book Gödel, Escher, Bach in his discussion of the relationship between formal languages and number theory: "...it is in the nature of any formalization of number theory that its metalanguage is embedded within it" (pg.270). They occur in ...

See also:

Metalanguage, Metalanguage - Kinds, Metalanguage - Role in metaphor, Metalanguage - Computing

Read more here: » Metalanguage: Encyclopedia II - Metalanguage - Kinds

There may be different notions of connectedness that are intuitively similar, but different as formally defined concepts. We might wish to call a topological space connected if each pair of points in it is joined by a path. However this concept turns out to be different from standard topological connectedness; in particular, there are connected topological spaces for which this property does not hold. Because of this, different terminology is used; ...

See also:

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

Read more here: » Connectedness: Encyclopedia II - Connectedness - Other notions of connectedness

All completeness properties are described along a similar scheme: one describes a certain class of subsets of a partial order that are required to have a supremum or infimum. Hence every completeness property has its dual, obtained by inverting the order-dependent definitions in the given statement. Some of the notions are usually not dualized while others may be self-dual (i.e. equivalent to their dual statements). ...

See also:

Completeness order theory, Completeness order theory - Types of completeness properties, Completeness order theory - Least and greatest elements, Completeness order theory - Finite completeness, Completeness order theory - Further completeness conditions, Completeness order theory - Relationships between completeness properties, Completeness order theory - Completions of domains, Completeness order theory - Completeness in terms of universal algebra, Completeness order theory - Completeness in terms of adjunctions, Completeness order theory - Notes, Completeness order theory - Reference

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In topology, a CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. The idea was to have a class of spaces that was broader than simplicial complexes (we could say now, had better categorical properties); but still retained a combinatorial nature, so that computational considerations were not ignored. The name itself is unrevealing: C ...

Including:

• CW complex - Attaching cells
• CW complex - CW complexes are defined inductively
• CW complex - 'The' homotopy category
• CW complex - Properties

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In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. Categories appear in virtually every branch of modern mathematics and are a central unifying notion. The study of categories in their own right is known as category theory. For more extensive motivational background and historical notes, see category theory and the list of category theory topics. Category mathematics - Definition. A category C consists of Including:

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

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Category may refer to: Categorization, a class of things, as in "the category of all living things" Categories, a text by the famous philosopher Aristotle. Pregnancy category Objective-C Categories permit to add methods to a class without having access to its source code Category (mathematics), from category theory, a collection of mathematical objects of the same kind, together with the structure-preserving processes between them category (topology), in mathmatics ...

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Nicolas Bourbaki is the collective allonym under which a group of mainly French 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for utmost rigour and generality, creating some new terminology and concepts along the way. While Nicolas Bourbaki is an invented personage, the Bourbaki group is officially known as the Association des collaborateurs de Nicolas Bourbaki< ...

Including:

• Nicolas Bourbaki - Books by Bourbaki
• Nicolas Bourbaki - Influence on mathematics in general
• Nicolas Bourbaki - The group
• Nicolas Bourbaki - The Bourbaki perspective and its limitations
• Nicolas Bourbaki - Dieudonné as speaker for Bourbaki
• Nicolas Bourbaki - The Bourbachique influence: education institutions trends

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In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. Such functors are ubiquitous in mathematics. Adjoint functors are studied in a branch of mathematics known as category theory. Like much of category theory, the general notion of adjoint functors arises at an abstract level beyond the everyday usage of mathematicians. Adjoint functors can be considered from several different points of view. This article starts with a number of introductory sections considering some ...

Including:

• Adjoint functors - Motivation
• Adjoint functors - Ubiquity of adjoint functors
• Adjoint functors - Deep problems formulated with adjoint functors
• Adjoint functors - Adjoint functors as solving optimization problems
• Adjoint functors - The case of partial orders
• Adjoint functors - Formal definitions
• Adjoint functors - Examples
• Adjoint functors - Properties
• Adjoint functors - Uniqueness of adjoints
• Adjoint functors - Relation to universal constructions
• Adjoint functors - Characterization via unit and co-unit
• Adjoint functors - Adjoints preserve certain limits
• Adjoint functors - Additivity
• Adjoint functors - Composition
• Adjoint functors - Adjoint pairs extend equivalences
• Adjoint functors - General existence theorem

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In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain a bigger one. A presheaf is similar to a sheaf, but it may not be possible to glue. Sheaves enable one to discuss in a refined way what is a local property, as appl ...

Including:

• 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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Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. Today, the natural sciences, engineering, economics, and medici ...

Including:

• Mathematics - History
• Mathematics - Inspiration pure and applied mathematics and aesthetics
• Mathematics - Notation language and rigor
• Mathematics - Is mathematics a science?
• Mathematics - Overview of fields of mathematics
• Mathematics - Major themes in mathematics
• Mathematics - Quantity
• Mathematics - Structure
• Mathematics - Space
• Mathematics - Change
• Mathematics - Foundations and methods
• Mathematics - Discrete mathematics
• Mathematics - Applied mathematics
• Mathematics - Important theorems
• Mathematics - Important conjectures
• Mathematics - History and the world of mathematicians
• Mathematics - Mathematics and other fields
• Mathematics - Common misconceptions

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Alexander Grothendieck (born March 28, 1928) was one of the most important mathematicians active in the 20th century. He was also one of its most extreme scientific personalities, with achievements over a short span of years that are still scarcely credible in their broad scope and sheer bulk, and an approach that antagonised even close followers. He made major contributions to algebraic geometry, homological algebra, and functional analysis. He was awarded the Fields Medal in 1966, and co-awarded the Crafoord Prize with Pierre Delig ...

Including:

• 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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Algebra is a branch of mathematics which studies structure and quantity. It may be roughly characterized as a generalization and abstraction of arithmetic, in which operations are performed on symbols rather than numbers. It includes elementary algebra, taught to high school students, as well as abstract algebra which covers such structures as groups, rings and fields. Along with geometry and analysis, it is one of the three main branches of mathematics. Algebra - History. The origins of algebra can be trac ...

Including:

• Algebra - History
• Algebra - Classification
• Algebra - Algebras

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This page gives some very general background to the mathematical idea of topos. This is an aspect of category theory, and has a reputation for being abstruse. The level of abstraction involved cannot be reduced beyond a certain point; but on the other hand context can be given. This is partly in terms of historical development, but also to some extent an explanation of differing attitudes to category theory. Other related archivescategory theory, topos

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In category theory, a functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of small categories. Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps. Nowadays, functors are used throughout modern mathematics to relate various categories. Functor - Definition. Let C and D be ca ...

Including:

• Functor - Definition
• Functor - Covariance and contravariance
• Functor - Examples
• Functor - Properties
• Functor - Relation to other categorical concepts

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In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be split naturally into connected pieces, each piece is usually called a component (or connected component). Many fields of mathematics include a formally defined property known as connectedness. In each field, the property may be defined differently. How ...

Including:

• Connectedness - Other notions of connectedness
• Connectedness - Connectivity

Read more here: » Connectedness: Encyclopedia - Connectedness

In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. Basically, this means the definition is the same as the product but with all arrows reversed. Despite this innocuous-looking change in the name and notation, coproducts can be dramatically different from products. The formal definition is as follows: Let C be a category and let {Xj | j ∈ J} be a indexed family of objects in C. The coproduct of the set {Xj} is an object ...

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In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. Universe mathematics - In a specific context. There are several precise versions of this general idea. Perhaps the simplest is that any set can be a universe, so long as you are studying that particular set. So if ...

Including:

• Universe mathematics - In a specific context
• Universe mathematics - In ordinary mathematics
• Universe mathematics - In set theory
• Universe mathematics - In category theory

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The word unit means any of several things: Unit of measurement, a fundamental quantity of measurement Units (computer program), a popular program that does unit conversion Functional unit, a component of a computer system such as the CPU Unit of action, a discrete piece of action (or beat) in a theatrical presentation Multiple unit, a passenger train whose carriages have their own motors United Nations Intelligence Taskforce, a fictional entity in the Doctor Who t

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Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. Universal algebra - Basic idea. From the point of view of universal algebra, an algebra (or abstract algebra) is a set A together with a collection of operations on A. An n-ary operation on A is a function that takes n elements of A and returns a single element of A. Thus, a 0-ary operation (or nullary operation) is simply an el ...

Including:

• Universal algebra - Basic idea
• Universal algebra - Examples
• Universal algebra - Groups
• Universal algebra - Modules
• Universal algebra - Further issues

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