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Fundamental Theorem Of Topos Theory
In mathematics, The fundamental theorem of topos theory states that the slice \mathbf / X of a topos \mathbf over any one of its objects X is itself a topos. Moreover, if there is a morphism f : A \rightarrow B in \mathbf then there is a functor f^*: \mathbf / B \rightarrow \mathbf / A which preserves exponentials and the subobject classifier. The pullback functor For any morphism ''f'' in \mathbf there is an associated "pullback functor" f^* := - \mapsto f \times - \rightarrow f which is key in the proof of the theorem. For any other morphism ''g'' in \mathbf which shares the same codomain as ''f'', their product f \times g is the diagonal of their pullback square, and the morphism which goes from the domain of f \times g to the domain of ''f'' is opposite to ''g'' in the pullback square, so it is the pullback of ''g'' along ''f'', which can be denoted as f^*g. Note that a topos \mathbf is isomorphic to the slice over its own terminal object, i.e. \mathbf \cong \mathbf / 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...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] |
Topos
In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion of localization; they are a direct generalization of point-set topology. The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic. The mathematical field that studies topoi is called topos theory. Grothendieck topos (topos in geometry) Since the introduction of sheaves into mathematics in the 1940s, a major theme has been to study a space by studying sheaves on a space. This idea was expounded by Alexander Grothendieck by introducing the notion of a "topos". The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective, but an honest topological space is lacking; it is sometimes possible to find a topos formaliz ... [...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] |
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] |
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 object X in ''C'' to an object F(X) in ''D' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Object
In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. Categories (such as subcategories of Top) without adjoined products may still have an exponential law. Definition Let \mathbf be a category, let Z and Y be objects of \mathbf, and let \mathbf have all binary products with Y. An object Z^Y together with a morphism \mathrm\colon (Z^Y \times Y) \to Z is an ''exponential object'' if for any object X and morphism g \colon X\times Y \to Z there is a unique morphism \lambda g\colon X\to Z^Y (called the ''transpose'' of g) such that the following diagram commutes: This assignment of a unique \lambda g to each g establishes an isomorphism (bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subobject Classifier
In category theory, a subobject classifier is a special object Ω of a category such that, intuitively, the subobjects of any object ''X'' in the category correspond to the morphisms from ''X'' to Ω. In typical examples, that morphism assigns "true" to the elements of the subobject and "false" to the other elements of ''X.'' Therefore, a subobject classifier is also known as a "truth value object" and the concept is widely used in the categorical description of logic. Note however that subobject classifiers are often much more complicated than the simple binary logic truth values . Introductory example As an example, the set Ω = is a subobject classifier in the category of sets and functions: to every subset ''A'' of ''S'' defined by the inclusion function '' j '' : ''A'' → ''S'' we can assign the function ''χA'' from ''S'' to Ω that maps precisely the elements of ''A'' to 1 (see characteristic function). Every function from ''S'' to Ω arises in this fashion from prec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Timeline Of Category Theory And Related Mathematics
This is a timeline of category theory and related mathematics. Its scope ("related mathematics") is taken as: * Categories of abstract algebraic structures including representation theory and universal algebra; * Homological algebra; * Homotopical algebra; * Topology using categories, including algebraic topology, categorical topology, quantum topology, low-dimensional topology; * Categorical logic and set theory in the categorical context such as algebraic set theory; * Foundations of mathematics building on categories, for instance topos theory; * Abstract geometry, including algebraic geometry, categorical noncommutative geometry, etc. * Quantization related to category theory, in particular categorical quantization; * Categorical physics Categorical may refer to: * Categorical imperative, a concept in philosophy developed by Immanuel Kant * Categorical theory, in mathematical logic * Morley's categoricity theorem, a mathematical theorem in model theory * Categorical data ana ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deduction Theorem
In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs—to prove an implication ''A'' → ''B'', assume ''A'' as an hypothesis and then proceed to derive ''B''—in systems that do not have an explicit inference rule for this. Deduction theorems exist for both propositional logic and first-order logic. The deduction theorem is an important tool in Hilbert-style deduction systems because it permits one to write more comprehensible and usually much shorter proofs than would be possible without it. In certain other formal proof systems the same conveniency is provided by an explicit inference rule; for example natural deduction calls it implication introduction. In more detail, the propositional logic deduction theorem states that if a formula B is deducible from a set of assumptions \Delta \cup \ then the implication A \to B is deducible from \Delta ; in symbols, \Delta \cup \ \vdash B implies \Delta \vdash A \to B . In the sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
