EpimorphismIn , an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism f : X → Y that is right-cancellative in the sense that, for all objects Z and all morphisms , Epimorphisms are categorical analogues of onto or surjective functions (and in the the concept corresponds exactly to the surjective functions), but they may not exactly coincide in all contexts; for example, the inclusion is a ring epimorphism. The of an epimorphism is a monomorphism (i.e. an epimorphism in a C is a monomorphism in the Cop).
Pushout (category theory)In , a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit of a consisting of two morphisms f : Z → X and g : Z → Y with a common domain. The pushout consists of an P along with two morphisms X → P and Y → P that complete a commutative square with the two given morphisms f and g. In fact, the defining universal property of the pushout (given below) essentially says that the pushout is the "most general" way to complete this commutative square.
Adjoint functorsIn mathematics, specifically , adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e.
Closed monoidal categoryIn mathematics, especially in , a closed monoidal category (or a monoidal closed category) is a that is both a and a in such a way that the structures are compatible. A classic example is the , Set, where the monoidal product of sets and is the usual cartesian product , and the internal Hom is the set of functions from to . A non- example is the , K-Vect, over a field . 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.
Empty productIn mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question), just as the empty sum—the result of adding no numbers—is by convention zero, or the additive identity. When numbers are implied, the empty product becomes one. The term empty product is most often used in the above sense when discussing arithmetic operations.
Category of abelian groupsIn mathematics, the Ab has the abelian groups as and group homomorphisms as morphisms. This is the prototype of an : indeed, every can be embedded in Ab. The zero object of Ab is the trivial group {0} 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 of Grp, the .
Tensor algebraIn mathematics, the tensor algebra of a vector space V, denoted T(V) or T^•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property (see below). The tensor algebra is important because many other algebras arise as quotient algebras of T(V).
Characteristic (algebra)In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero. That is, char(R) is the smallest positive number n such that: if such a number n exists, and 0 otherwise.
Automorphism groupIn mathematics, the automorphism group of an object X is the group consisting of automorphisms of X under composition of morphisms. For example, if X is a finite-dimensional vector space, then the automorphism group of X is the group of invertible linear transformations from X to itself (the general linear group of X). If instead X is a group, then its automorphism group is the group consisting of all group automorphisms of X. Especially in geometric contexts, an automorphism group is also called a symmetry group.
Zero morphismIn , a branch of mathematics, a zero morphism is a special kind of morphism exhibiting properties like the morphisms to and from a zero object. Suppose C is a , and f : X → Y is a morphism in C. The morphism f is called a constant morphism (or sometimes left zero morphism) if for any W in C and any g, h : W → X, fg = fh. Dually, f is called a coconstant morphism (or sometimes right zero morphism) if for any object Z in C and any g, h : Y → Z, gf = hf. A zero morphism is one that is both a constant morphism and a coconstant morphism.
Product of ringsIn mathematics, a product of rings or direct product of rings is a ring that is formed by the Cartesian product of the underlying sets of several rings (possibly an infinity), equipped with componentwise operations. It is a direct product in the . Since direct products are defined up to an isomorphism, one says colloquially that a ring is the product of some rings if it is isomorphic to the direct product of these rings.
CoequalizerIn , a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary . It is the categorical construction to the equalizer. A coequalizer is a colimit of the diagram consisting of two objects X and Y and two parallel morphisms f, g : X → Y. More explicitly, a coequalizer of the parallel morphisms f and g can be defined as an object Q together with a morphism q : Y → Q such that q ∘ f = q ∘ g.
