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Cosmos (category Theory)
In the area of mathematics known as category theory, a cosmos is a symmetric monoidal category, symmetric closed monoidal category that is complete category, complete and cocomplete category, cocomplete. Enriched category theory is often considered over a cosmos. References Category theory {{cattheory-stub ... [...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] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism compos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Monoidal Category
In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" \otimes is defined) such that the tensor product is symmetric (i.e. A\otimes B is, in a certain strict sense, naturally isomorphic to B\otimes A for all objects A and B of the category). One of the prototypical examples of a symmetric monoidal category is the category of vector spaces over some fixed field ''k,'' using the ordinary tensor product of vector spaces. Definition A symmetric monoidal category is a monoidal category (''C'', ⊗, ''I'') such that, for every pair ''A'', ''B'' of objects in ''C'', there is an isomorphism s_: A \otimes B \to B \otimes A that is natural in both ''A'' and ''B'' and such that the following diagrams commute: *The unit coherence: *: *The associativity coherence: *: *The inverse law: *: In the diagrams above, ''a'', ''l'' , ''r'' are the associativity isomorphism, the left unit isomorphism, and the right un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Monoidal Category
In mathematics, especially in category theory, a closed monoidal category (or a ''monoidal closed category'') is a category that is both a monoidal category and a closed category in such a way that the structures are compatible. A classic example is the category of sets, Set, where the monoidal product of sets A and B is the usual cartesian product A \times B, and the internal Hom B^A is the set of functions from A to B. A non- cartesian example is the category of vector spaces, ''K''-Vect, over a field K. Here the monoidal product is the usual tensor product of vector spaces, and the internal Hom is the vector space of linear maps from one vector space to another. The internal language of closed symmetric monoidal categories is linear logic and the type system is the linear type system. Many examples of closed monoidal categories are symmetric. However, this need not always be the case, as non-symmetric monoidal categories can be encountered in category-theoretic formulatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 (of all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cocomplete 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 (of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enriched Category Theory
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. It is motivated by the observation that, in many practical applications, the hom-set often has additional structure that should be respected, e.g., that of being a vector space of morphisms, or a topological space of morphisms. In an enriched category, the set of morphisms (the hom-set) associated with every pair of objects is replaced by an object in some fixed monoidal category of "hom-objects". In order to emulate the (associative) composition of morphisms in an ordinary category, the hom-category must have a means of composing hom-objects in an associative manner: that is, there must be a binary operation on objects giving us at least the structure of a monoidal category, though in some contexts the operation may also need to be commutative and perhaps also to have a right adjoint (i.e., making the category symm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
