[permalink] [id link]
In category theory, a branch of mathematics, a functor is a special type of mapping between categories.
from
Wikipedia
Some Related Sentences
category and theory
Falling somewhere in a category between Einstein's theory and sand fleas -- difficult to see but undeniably there, nevertheless -- is the tropical green `` city '' of Islandia, a string of offshore islands that has almost no residents, limited access and an unlimited future.
Because there is no canonical well-ordering of all sets, a construction that relies on a well-ordering may not produce a canonical result, even if a canonical result is desired ( as is often the case in category theory ).
There are several results in category theory which invoke the axiom of choice for their proof.
On the other hand, other foundational descriptions of category theory are considerably stronger, and an identical category-theoretic statement of choice may be stronger than the standard formulation, à la class theory, mentioned above.
The most general setting in which these words have meaning is an abstract branch of mathematics called category theory.
In category theory, an automorphism is an endomorphism ( i. e. a morphism from an object to itself ) which is also an isomorphism ( in the categorical sense of the word ).
This is a very abstract definition since, in category theory, morphisms aren't necessarily functions and objects aren't necessarily sets.
His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory into its foundations.
His emphasis on the role of universal properties brought category theory into the mainstream as an important organizing principle.
During this time he had officially as students Michel Demazure ( who worked on SGA3, on group schemes ), Luc Illusie ( cotangent complex ), Michel Raynaud, Jean-Louis Verdier ( cofounder of the derived category theory ) and Pierre Deligne.
Alexander Grothendieck's work during the ` Golden Age ' period at IHÉS established several unifying themes in algebraic geometry, number theory, topology, category theory and complex analysis.
He gave lectures on category theory in the forests surrounding Hanoi while the city was being bombed, to protest against the Vietnam War ( The Life and Work of Alexander Grothendieck, American Mathematical Monthly, vol.
The choice between the two definitions usually matters only in very formal contexts, like category theory.
In category theory, n-ary functions generalise to n-ary morphisms in a multicategory.
* Cone ( category theory ), a family of morphisms resembling a geometric cone
Notable theories falling into this category include the Holonomic brain theory of Karl Pribram and David Bohm, and the Orch-OR theory formulated by Stuart Hameroff and Roger Penrose.
In applying the elemental theory to beings that function on a cosmic scale ( e. g. Yog-Sothoth ) some authors created a separate category termed aethyr.
Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.
However it is important to note that the objects of a category need not be sets nor the arrows functions ; any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour of objects and arrows is a valid category, and all the results of category theory will apply to it.
category and branch
* A scientific field ( a branch of science ) – widely-recognized category of specialized expertise within science, and typically embodies its own terminology and nomenclature.
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space.
* A scientific field ( a branch of science ) – widely-recognized category of specialized expertise within science, and typically embodies its own terminology and nomenclature.
* Scientific field ( a branch of science ) – widely-recognized category of specialized expertise within science, and typically embodies its own terminology and nomenclature.
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits.
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure ( i. e. the composition of morphisms ) of the categories involved.
In category theory, a branch of mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets.
This is the central idea of category theory, a branch of mathematics which seeks to generalize all of mathematics in terms of objects and arrows, independent of what the objects and arrows represent.
It is studied in generality by the branch of mathematics known as category theory.
In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X.
Influenced by Nordic folk music, it is considered a category of folk metal, but it is a separate branch of that style, as there are notable differences between Viking metal and folk metal.
Lyre from various times and places are regarded by some organologists ( specialists in the history of musical instruments ) as a branch of the zither family, a general category that includes not only zithers, but many different stringed instruments, such as lutes, guitars, kantele, and psalteries.
In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category.
One issue is whether such a category could exist without violating the Geneva Conventions, and if such a category does exist, what steps the United States executive branch needs to take to comply with municipal laws as interpreted by the judicial branch of the United States government.
In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category C < sup > op </ sup >.
category and mathematics
General category theory, an extension of universal algebra having many new features allowing for semantic flexibility and higher-order logic, came later ; it is now applied throughout mathematics.
These foundational applications of category theory have been worked out in fair detail as a basis for, and justification of, constructive mathematics.
* Timeline of category theory and related mathematics
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds.
In mathematics, especially in category theory and homotopy theory, a groupoid ( less often Brandt groupoid or virtual group ) generalises the notion of group in several equivalent ways.
In certain areas of mathematics, notably category theory, it is valuable to distinguish between equality on the one hand and isomorphism on the other.
Other branches of mathematics, however, such as logic, number theory, category theory or set theory are accepted as part of pure mathematics, though they find application in other sciences ( predominantly computer science and physics ).
Category theory, another field within " foundational mathematics ", is rooted on the abstract axiomatization of the definition of a " class of mathematical structures ", referred to as a " category ".
Because of its applicability in diverse fields of mathematics, mathematicians including Saunders Mac Lane have proposed category theory as a foundational system for mathematics, independent of set theory.
In mathematics, specifically in category theory, the Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object.
* Core of a triangulated category in mathematics
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties.
As an alternative to set theory, others have argued for category theory as a foundation for certain aspects of mathematics.
In mathematics, a category is an algebraic structure that comprises " objects " that are linked by " arrows ".
0.397 seconds.