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Complete Category
In mathematics, a complete category is a category in which all small limits exist. That is, a category ''C'' is complete if every diagram ''F'' : ''J'' → ''C'' (where ''J'' is small) has a limit in ''C''. Dually, a cocomplete category is one in which all small colimits exist. A bicomplete category is a category which is both complete and cocomplete. The existence of ''all'' limits (even when ''J'' is a proper class) is too strong to be practically relevant. Any category with this property is necessarily a thin category: for any two objects there can be at most one morphism from one object to the other. A weaker form of completeness is that of finite completeness. A category is finitely complete if all finite limits exists (i.e. limits of diagrams indexed by a finite category ''J''). Dually, a category is finitely cocomplete if all finite colimits exist. Theorems It follows from the existence theorem for limits that a category is complete if and only if it has equalizers (o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Terminal Object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): is terminal if for every object in there exists exactly one morphism . Initial objects are also called coterminal or universal, and terminal objects are also called final. If an object is both initial and terminal, it is called a zero object or null object. A pointed category is one with a zero object. A strict initial object is one for which every morphism into is an isomorphism. Examples * The empty set is the unique initial object in Set, the category of sets. Every one-element set ( singleton) is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wheel Theory
A wheel is a type of algebra (in the sense of universal algebra) where division is always defined. In particular, division by zero is meaningful. The real numbers can be extended to a wheel, as can any commutative ring. The term ''wheel'' is inspired by the topological picture \odot of the real projective line together with an extra point ⊥ (bottom element) such that \bot = 0/0. A wheel can be regarded as the equivalent of a commutative ring (and semiring) where addition and multiplication are not a group but respectively a commutative monoid and a commutative monoid with Involution_(mathematics), involution. Definition A wheel is an algebraic structure (W, 0, 1, +, \cdot, /), in which * W is a set, * 0 and 1 are elements of that set, * + and \cdot are binary operations, * / is a unary operation, and satisfying the following properties: * + and \cdot are each commutative and associative, and have \,0 and 1 as their respective Identity element, identities. * / is an involut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of All Small Categories
In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories. Cat may actually be regarded as a 2-category with natural transformations serving as 2-morphisms. The initial object of Cat is the ''empty category'' 0, which is the category of no objects and no morphisms. The terminal object is the ''terminal category'' or ''trivial category'' 1 with a single object and morphism. at nLab The category Cat is itself a , and therefore not an object of itself. In order to avoid problems analogous to |
Compact Hausdorff Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces. One such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. Commutative rings appear in the following chain of subclass (set theory), class inclusions: Definition and first examples Definition A ''ring'' is a Set (mathematics), set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "+" and "\cdot"; e.g. a+b and a \cdot b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Modules
In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring of integers Z, it is the same thing as the category of abelian groups. The category of right modules is defined in a similar way. One can also define the category of bimodules over a ring ''R'' but that category is equivalent to the category of left (or right) modules over the enveloping algebra of ''R'' (or over the opposite of that). Note: Some authors use the term module category for the category of modules. This term can be ambiguous since it could also refer to a category with a monoidal-category action. Properties The categories of left and right modules are abelian categories. These categories have enough projectives and enough injectives. Mitchell's embedding theorem states every abelian category arises as a full sub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as field of rational functions, fields of rational functions, algebraic function fields, algebraic number fields, and p-adic number, ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many element (set), elements. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straighte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Vector Spaces
In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring of integers Z, it is the same thing as the category of abelian groups. The category of right modules is defined in a similar way. One can also define the category of bimodules over a ring ''R'' but that category is equivalent to the category of left (or right) modules over the enveloping algebra of ''R'' (or over the opposite of that). Note: Some authors use the term module category for the category of modules. This term can be ambiguous since it could also refer to a category with a monoidal-category action. Properties The categories of left and right modules are abelian categories. These categories have enough projectives and enough injectives. Mitchell's embedding theorem states every abelian category arises as a full subcate ... [...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] |
