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Coproducts
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 within a given category. 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 that satisfies the fol ... [...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. As an example, 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, the sum is defined (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. Direct sums can also be formed 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 spaces). That relies on the fact that the direct sum is associative up to isomorphism. That is, (A ... [...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] |
Product (category Theory)
In category theory, the product of two (or more) object (category theory), objects in a category (mathematics), category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of set (mathematics), sets, the direct product of group (mathematics), groups or ring (mathematics), rings, and the product topology, product of topological spaces. Essentially, the product of a indexed family, 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 follo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism. That is, an arrow such that for all objects and all morphisms , : f \circ g_1 = f \circ g_2 \implies g_1 = g_2. Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below. In the setting of posets intersections are idempotent: the intersection of anything with itself is itself. Monomorphisms generalize this property to arbitrary categories. A morphism is a monomorphism if it is idempotent with respect to pullbacks. The categorical dual of a monomorphism is an epimorphism, that is, a monomorphism in a categor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Product Of Algebras
In mathematics, the tensor product of two algebras over a commutative ring ''R'' is also an ''R''-algebra. This gives the tensor product of algebras. When the ring is a field, the most common application of such products is to describe the product of algebra representations. Definition Let ''R'' be a commutative ring and let ''A'' and ''B'' be ''R''-algebras. Since ''A'' and ''B'' may both be regarded as ''R''-modules, their tensor product :A \otimes_R B is also an ''R''-module. The tensor product can be given the structure of a ring by defining the product on elements of the form by :(a_1\otimes b_1)(a_2\otimes b_2) = a_1 a_2\otimes b_1b_2 and then extending by linearity to all of . This ring is an ''R''-algebra, associative and unital with the identity element given by . where 1''A'' and 1''B'' are the identity elements of ''A'' and ''B''. If ''A'' and ''B'' are commutative, then the tensor product is commutative as well. The tensor product turns the category of ''R''-alge ... [...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] |
Finite Set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the ''cardinality (or the cardinal number)'' of the set. A set that is not a finite set is called an '' infinite set''. For example, the set of all positive integers is infinite: Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set S is called finite if there exists a bijection for some natural number n (natural numbers are defined as sets in Zermelo-Fraenkel set theory). The number n ... [...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'', can be added together and multiplied ("scaled") by numbers called ''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. 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 matrices, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear equations. Vector spaces are characterized by their di ... [...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] |
Direct Product
In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abstraction of these notions in the setting of category theory. Examples are the product of sets, groups (described below), rings, and other algebraic structures. The product of topological spaces is another instance. The direct sum is a related operation that agrees with the direct product in some but not all cases. Examples * If \R is thought of as the set of real numbers without further structure, the direct product \R \times \R is just the Cartesian product \. * If \R is thought of as the group of real numbers under addition, the direct product \R\times \R still has \ as its underlying set. The difference between this and the preceding examples is that \R \times \R is now a group and so how to add their elements must also be s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
