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Coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products. Definition Let C be a category and let X_1 and X_2 be objects of C. An object is called the coproduct of X_1 and X_2, written X_1 \sqcup X_2, or X_1 \oplus X_2, or sometimes simply X_1 + X_2, if there exist morphisms i_1 : X_1 \to X_1 \sqcup X_2 and i_2 : X_2 \to X_1 \sqcup X_2 satisfying the following universal property: f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more elementary kind of structure, the abelian group. The direct sum of two abelian groups A and B is another abelian group A\oplus B consisting of the ordered pairs (a,b) where a \in A and b \in B. To add ordered pairs, we define the sum (a, b) + (c, d) to be (a + c, b + d); in other words addition is defined coordinate-wise. For example, the direct sum \Reals \oplus \Reals , where \Reals is real coordinate space, is the Cartesian plane, \R ^2 . A similar process can be used to form the direct sum of two vector spaces or two modules. We can also form direct sums with any finite number of summands, for example A \oplus B \oplus C, provided A, B, and C are the same kinds of algebraic structures (e.g., all abelian groups, or all vector spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product (category Theory)
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects. Definition Product of two objects Fix a category C. Let X_1 and X_2 be objects of C. A product of X_1 and X_2 is an object X, typically denoted X_1 \times X_2, equipped with a pair of morphisms \pi_1 : X \to X_1, \pi_2 : X \to X_2 satisfying the following universal property: * For every object Y and every pair of morphisms f_1 : Y \to X_1, f_2 : Y \to X_2, there exists a unique morphism f : Y \to X_1 \times X_2 such that the following diagram commutes: *: Whether a product exists may depend on C or on X_1 and X_2. If it does exist, it is unique up to canonical iso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjoint Union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A (that is, each element of A belongs to exactly one of these images). A disjoint union of a family of pairwise disjoint sets is their union. In category theory, the disjoint union is the coproduct of the category of sets, and thus defined up to a bijection. In this context, the notation \coprod_ A_i is often used. The disjoint union of two sets A and B is written with infix notation as A \sqcup B. Some authors use the alternative notation A \uplus B or A \operatorname B (along with the corresponding \biguplus_ A_i or \operatorname_ A_i). A standard way for building the disjoint union is to define A as the set of ordered pairs (x, i) such that x \in A_i, and the injection A_i \to A as x \mapsto (x, i). Example Consider the sets A_0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjoint Union (topology)
In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. Roughly speaking, in the disjoint union the given spaces are considered as part of a single new space where each looks as it would alone and they are isolated from each other. The name ''coproduct'' originates from the fact that the disjoint union is the categorical dual of the product space construction. Definition Let be a family of topological spaces indexed by ''I''. Let :X = \coprod_i X_i be the disjoint union of the underlying sets. For each ''i'' in ''I'', let :\varphi_i : X_i \to X\, be the canonical injection (defined by \varphi_i(x)=(x,i)). The disjoint union topology on ''X'' is defined as the finest topology on ''X'' for which all the canonical inje ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Product
In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, and is the “ universal” group having these properties, in the sense that any two homomorphisms from ''G'' and ''H'' into a group ''K'' factor uniquely through a homomorphism from to ''K''. Unless one of the groups ''G'' and ''H'' is trivial, the free product is always infinite. The construction of a free product is similar in spirit to the construction of a free group (the universal group with a given set of generators). The free product is the coproduct in the category of groups. That is, the free product plays the same role in group theory that disjoint union plays in set theory, or that the direct sum plays in module theory. Even if the groups are commutative, their free product is not, unless one of the two groups is the trivial gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions. Examples are the product of sets, groups (described below), rings, and other algebraic structures. The product of topological spaces is another instance. There is also the direct sum – in some areas this is used interchangeably, while in others it is a different concept. Examples * If we think of \R as the set of real numbers, then the direct product \R \times \R is just the Cartesian product \. * If we think of \R as the group of real numbers under addition, then the direct product \R\times \R still has \ as its underlying set. The difference between this and the preceding example is that \R \times \R is now a group, and so we have to also say how to add th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indexed Family
In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a ''family of real numbers, indexed by the set of integers'' is a collection of real numbers, where a given function selects one real number for each integer (possibly the same). More formally, an indexed family is a mathematical function together with its domain I and image X. (that is, indexed families and mathematical functions are technically identical, just point of views are different.) Often the elements of the set X are referred to as making up the family. In this view, indexed families are interpreted as collections of indexed elements instead of functions. The set I is called the ''index set'' of the family, and X is the ''indexed set''. Sequences are one type of families indexed by natural numbers. In general, the index set I is not restricted to be countable. For example, one could consider an uncountable family of subset ... [...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 compo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition of morphisms is the composition of functions. Many other categories (such as the category of groups, with group homomorphisms as arrows) add structure to the objects of the category of sets and/or restrict the arrows to functions of a particular kind. Properties of the category of sets The axioms of a category are satisfied by Set because composition of functions is associative, and because every set ''X'' has an identity function id''X'' : ''X'' → ''X'' which serves as identity element for function composition. The epimorphisms in Set are the surjective maps, the monomorphisms are the injective maps, and the isomorphisms are the bijective maps. The empty set serves as the initial object in Set with empty functions as morphisms. Ev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inclusion Map
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iota(x)=x. A "hooked arrow" () is sometimes used in place of the function arrow above to denote an inclusion map; thus: \iota: A\hookrightarrow B. (However, some authors use this hooked arrow for any embedding.) This and other analogous injective functions from substructures are sometimes called natural injections. Given any morphism f between objects X and Y, if there is an inclusion map into the domain \iota : A \to X, then one can form the restriction f \, \iota of f. In many instances, one can also construct a canonical inclusion into the codomain R \to Y known as the range of f. Applications of inclusion maps Inclusion maps tend to be homomorphisms of algebraic structures; thus, such inclusion maps are embeddings. More precisel ... [...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 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 functor; this functor assigns to every set ''S'' the free group on ''S.'' Categorical properties The monomorphisms in Grp are precisely the injective homomorphisms, the epimorphisms are precisely the surjective homomorphisms, and the isomorphisms are pre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Spaces
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
