|
Outline Of Category Theory
The following outline is provided as an overview of and guide to category theory, the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of ''objects'' and ''arrows'' (also called morphisms, although this term also has a specific, non category-theoretical sense), where these collections satisfy certain basic conditions. 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. Essence of category theory * Category * Functor * Natural transformation Branches of category theory * Homological algebra * Diagram chasing * Topos theory * Enriched category theory * Higher category theory * Categorical logic * Applied category theory Specific categories *Category of sets **Con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Applied Category Theory
Applied category theory is an academic discipline in which methods from category theory are used to study other fields including but not limited to computer science, physics (in particular quantum mechanics), natural language processing, control theory, probability theory and causality. The application of category theory in these domains can take different forms. In some cases the formalization of the domain into the language of category theory is the goal, the idea here being that this would elucidate the important structure and properties of the domain. In other cases the formalization is used to leverage the power of abstraction in order to prove new results about the field. List of applied category theorists * Samson Abramsky * John C. Baez * Bob Coecke * Joachim Lambek * Valeria de Paiva * Gordon Plotkin * Dana Scott * David Spivak See also * Categorical quantum mechanics * ZX-calculus * DisCoCat * Petri net * Univalent foundations * String diagrams External ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Rings
In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings is large, meaning that the class of all rings is proper. As a concrete category The category Ring is a concrete category meaning that the objects are sets with additional structure (addition and multiplication) and the morphisms are functions that preserve this structure. There is a natural forgetful functor :''U'' : Ring → Set for the category of rings to the category of sets which sends each ring to its underlying set (thus "forgetting" the operations of addition and multiplication). This functor has a left adjoint :''F'' : Set → Ring which assigns to each set ''X'' the free ring generated by ''X''. One can also view the category of rings as a concrete category over Ab (the category of abelian groups) or over Mon (the category of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Abelian Groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object of Ab is the trivial group which consists only of its neutral element. The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms. Ab is a full subcategory of Grp, the category of ''all'' groups. The main difference between Ab and Grp is that the sum of two homomorphisms ''f'' and ''g'' between abelian groups is again a group homomorphism: :(''f''+''g'')(''x''+''y'') = ''f''(''x''+''y'') + ''g''(''x''+''y'') = ''f''(''x'') + ''f''(''y'') + ''g''(''x'') + ''g''(''y'') : = ''f''(''x'') + ''g''(''x'') + ''f''(''y'') + ''g''(''y'') = (''f''+''g'')(''x'') + (''f''+''g'')(''y' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Groups
In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. Relation to other categories There are two forgetful functors from Grp, M: Grp → Mon from groups to monoids and U: Grp → Set from groups to sets. M has two adjoints: one right, I: Mon→Grp, and one left, K: Mon→Grp. I: Mon→Grp is the functor sending every monoid to the submonoid of invertible elements and K: Mon→Grp the functor sending every monoid to the Grothendieck group In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a group homomorp ... of that monoid. The forgetful functor U: Grp → Set has a left adjoint given by the composite KF: Set→Mon→Grp, where F is the Free_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Preordered Sets
In mathematics, the category PreOrd has preordered sets as objects and order-preserving functions as morphisms. This is a category because the composition of two order-preserving functions is order preserving and the identity map is order preserving. The monomorphisms in PreOrd are the injective order-preserving functions. The empty set (considered as a preordered set) is the initial object of PreOrd, and the terminal objects are precisely the singleton preordered sets. There are thus no zero objects in PreOrd. The categorical product in PreOrd is given by the product order on the cartesian product. We have a forgetful functor PreOrd → Set that assigns to each preordered set the underlying set, and to each order-preserving function the underlying function. This functor is faithful, and therefore PreOrd is a concrete category. This functor has a left adjoint (sending every set to that set equipped with the equality relation) and a right adjoint (sending every set to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Metric Spaces
In category theory, Met is a category that has metric spaces as its objects and metric maps ( continuous functions between metric spaces that do not increase any pairwise distance) as its morphisms. This is a category because the composition of two metric maps is again a metric map. It was first considered by . Arrows The monomorphisms in Met are the injective metric maps. The epimorphisms are the metric maps for which the domain of the map has a dense image in the range. The isomorphisms are the isometries, i.e. metric maps which are injective, surjective, and distance-preserving. As an example, the inclusion of the rational numbers into the real numbers is a monomorphism and an epimorphism, but it is clearly not an isomorphism; this example shows that Met is not a balanced category. Objects The empty metric space is the initial object of Met; any singleton metric space is a terminal object. Because the initial object and the terminal objects differ, there are no zero objec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Topological Spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology. N.B. Some authors use the name Top for the categories with topological manifolds, with compactly generated spaces as objects and continuous maps as morphisms or with the category of compactly generated weak Hausdorff spaces. As a concrete category Like many categories, the category Top is a concrete category, meaning its objects are sets with additional structure (i.e. topologies) and its morphisms are functions preserving this structure. There is a natural forgetful functor to the category of sets which assigns to each topological space the underlyin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Sets And Relations
In mathematics, the category Rel has the class of sets as objects and binary relations as morphisms. A morphism (or arrow) ''R'' : ''A'' → ''B'' in this category is a relation between the sets ''A'' and ''B'', so . The composition of two relations ''R'': ''A'' → ''B'' and ''S'': ''B'' → ''C'' is given by :(''a'', ''c'') ∈ ''S'' o ''R'' ⇔ for some ''b'' ∈ ''B'', (''a'', ''b'') ∈ ''R'' and (''b'', ''c'') ∈ ''S''. Rel has also been called the "category of correspondences of sets". Properties The category Rel has the category of sets Set as a (wide) subcategory, where the arrow in Set corresponds to the relation defined by .This category is called SetRel by Rydeheard and Burstall. A morphism in Rel is a relation, and the corresponding morphism in the opposite category to Rel has arrows reversed, so it is the converse relation. Thus Rel contains its opposite and is self-dual. The involution represented by taking the converse relation provides the dagger to mak ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Finite Dimensional Hilbert Spaces
In mathematics, the category FdHilb has all finite-dimensional Hilbert spaces for objects and the linear transformations between them as morphisms. Whereas the theory described by the normal category of Hilbert spaces, Hilb, is ordinary quantum mechanics, the corresponding theory on finite dimensional Hilbert spaces is called fdQM. Properties This category * is monoidal, * possesses finite biproducts, and * is dagger compact. According to a theorem of Selinger, the category of finite-dimensional Hilbert spaces is complete in the dagger compact category In category theory, a branch of mathematics, dagger compact categories (or dagger compact closed categories) first appeared in 1989 in the work of Sergio Doplicher and John E. Roberts on the reconstruction of compact topological groups from thei .... Many ideas from Hilbert spaces, such as the no-cloning theorem, hold in general for dagger compact categories. See that article for additional details. References Categorie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Chain Complexes
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathematics), image of each homomorphism is contained in the kernel (algebra)#Group homomorphisms, kernel of the next. Associated to a chain complex is its Homology (mathematics), homology, which is (loosely speaking) a measure of the failure of a chain complex to be Exact sequence, exact. A cochain complex is similar to a chain complex, except that its homomorphisms are in the opposite direction. The homology of a cochain complex is called its cohomology. In algebraic topology, the singular chain complex of a topological space X is constructed using continuous function#Continuous functions between topological spaces, continuous maps from a simplex to X, and the homomorphisms of the chain complex capture how these maps restrict to the boundary of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Graded Vector Spaces
In mathematics, a graded vector space is a vector space that has the extra structure of a ''grading'' or ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces, generally indexed by the integers. For "pure" vector spaces, the concept has been introduced in homological algebra, and it is widely used for graded algebras, which are graded vector spaces with additional structures. Integer gradation Let \mathbb be the set of non-negative integers. An \mathbb-graded vector space, often called simply a graded vector space without the prefix \mathbb, is a vector space together with a decomposition into a direct sum of the form : V = \bigoplus_ V_n where each V_n is a vector space. For a given ''n'' the elements of V_n are then called homogeneous elements of degree ''n''. Graded vector spaces are common. For example the set of all polynomials in one or several variables forms a graded vector space, where the homogeneous elements of degree ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |