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Isomorphism Of Categories
In category theory, two categories ''C'' and ''D'' are isomorphic if there exist functors ''F'' : ''C'' → ''D'' and ''G'' : ''D'' → ''C'' that are mutually inverse to each other, i.e. ''FG'' = 1''D'' (the identity functor on ''D'') and ''GF'' = 1''C''. This means that both the objects and the morphisms of ''C'' and ''D'' stand in a one-to-one correspondence to each other. Two isomorphic categories share all properties that are defined solely in terms of category theory; for all practical purposes, they are identical and differ only in the notation of their objects and morphisms. Isomorphism of categories is a very strong condition and rarely satisfied in practice. Much more important is the notion of equivalence of categories; roughly speaking, for an equivalence of categories we don't require that FG be ''equal'' to 1_D, but only ''naturally isomorphic'' to 1_D, and likewise that GF be naturally isomorphic to 1_C. Properties As is true for any notion of isomorphism, we hav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in most areas of mathematics. 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 space (other), quotient spaces, direct products, completion, and duality (mathematics), duality. Many areas of computer science also rely on category theory, such as functional programming and Semantics (computer science), semantics. A category (mathematics), category is formed by two sorts of mathematical object, objects: the object (category theory), objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. Metapho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called scalar (mathematics), ''scalars''. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field (mathematics), field. Vector spaces generalize Euclidean vectors, which allow modeling of Physical quantity, physical quantities (such as forces and velocity) that have not only a Magnitude (mathematics), magnitude, but also a Orientation (geometry), direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix (mathematics), matrices, which ... [...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] |
Dual (category Theory)
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category ''C'' and the dual properties of the opposite category ''C''op. Given a statement regarding the category ''C'', by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite category ''C''op. (''C''op is composed by reversing every morphism of ''C''.) Duality, as such, is the assertion that truth is invariant under this operation on statements. In other words, if a statement ''S'' is true about ''C'', then its dual statement is true about ''C''op. Also, if a statement is false about ''C'', then its dual has to be false about ''C''op. (Compactly saying, ''S'' for ''C'' is true if and only if its dual for ''C''op is true.) Given a concrete category ''C'', it is often the case that the opposite category ''C''op per se is abstract. ''C''op need not b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slice Category
In mathematics, an overcategory (also called a slice category) is a construction from category theory used in multiple contexts, such as with covering spaces (espace étalé). They were introduced as a mechanism for keeping track of data surrounding a fixed object X in some category \mathcal. The dual notion is that of an undercategory (also called a coslice category). Definition Let \mathcal be a category and X a fixed object of \mathcalpg 59. The overcategory (also called a slice category) \mathcal/X is an associated category whose objects are pairs (A, \pi) where \pi:A \to X is a morphism in \mathcal. Then, a morphism between objects f:(A, \pi) \to (A', \pi') is given by a morphism f:A \to A' in the category \mathcal such that the following diagram commutes\begin A & \xrightarrow & A' \\ \pi\downarrow \text & \text &\text \downarrow \pi' \\ X & = & X \endThere is a dual notion called the undercategory (also called a coslice category) X/\mathcal whose objects are pairs (B, \p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Difference
In mathematics, the symmetric difference of two sets, also known as the disjunctive union and set sum, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \. The symmetric difference of the sets ''A'' and ''B'' is commonly denoted by A \operatorname\Delta B (alternatively, A \operatorname\vartriangle B), A \oplus B, or A \ominus B. It can be viewed as a form of addition modulo 2. The power set of any set becomes an abelian group under the operation of symmetric difference, with the empty set as the neutral element of the group and every element in this group being its own inverse. The power set of any set becomes a Boolean ring, with symmetric difference as the addition of the ring and intersection as the multiplication of the ring. Properties The symmetric difference is equivalent to the union of both relative complements, that is: :A\, \Delta\,B = \left(A \setminus B\ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Ring
In mathematics, a Boolean ring is a ring for which for all in , that is, a ring that consists of only idempotent elements. An example is the ring of integers modulo 2. Every Boolean ring gives rise to a Boolean algebra, with ring multiplication corresponding to conjunction or meet , and ring addition to exclusive disjunction or symmetric difference (not disjunction , which would constitute a semiring). Conversely, every Boolean algebra gives rise to a Boolean ring. Boolean rings are named after the founder of Boolean algebra, George Boole. Notation There are at least four different and incompatible systems of notation for Boolean rings and algebras: * In commutative algebra the standard notation is to use for the ring sum of and , and use for their product. * In logic, a common notation is to use for the meet (same as the ring product) and use for the join, given in terms of ring notation (given just above) by . * In set theory and logic it is also common to use f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Algebra (structure)
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution). Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle. __TOC__ History The term "Boolean algebra" honors George Boole (1815–1864), a self-educated E ... [...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] |
Additive Functor
In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-category C is a category such that every hom-set Hom(''A'',''B'') in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas: f\circ (g + h) = (f\circ g) + (f\circ h) and (f + g)\circ h = (f\circ h) + (g\circ h), where + is the group operation. Some authors have used the term ''additive category'' for preadditive categories, but this page reserves that term for certain special preadditive categories (see below). Examples The most obvious example of a preadditive category is the category Ab itself. More precisely, Ab is a closed monoidal category. Note that commutativity is crucial here; it ensures that the sum of two group homomorphisms is again a ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functor Category
In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object in the category). Functor categories are of interest for two main reasons: * many commonly occurring categories are (disguised) functor categories, so any statement proved for general functor categories is widely applicable; * every category embeds in a functor category (via the Yoneda embedding); the functor category often has nicer properties than the original category, allowing certain operations that were not available in the original setting. Definition Suppose C is a small category (i.e. the objects and morphisms form a set rather than a proper class) and D is an arbitrary category. The category of functors from C to D, written as Fun(C, D), Funct(C,D), ,D/math>, or D ^C, has as objects the covariant functors from C to D, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Preadditive Category
In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-category C is a category such that every hom-set Hom(''A'',''B'') in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas: f\circ (g + h) = (f\circ g) + (f\circ h) and (f + g)\circ h = (f\circ h) + (g\circ h), where + is the group operation. Some authors have used the term ''additive category'' for preadditive categories, but this page reserves that term for certain special preadditive categories (see below). Examples The most obvious example of a preadditive category is the category Ab itself. More precisely, Ab is a closed monoidal category. Note that commutativity is crucial here; it ensures that the sum of two group homomorphisms is again ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
