MorphismIn mathematics, particularly in , a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in analysis and topology, continuous functions, and so on.
Category of modulesIn algebra, given a ring R, the category of left modules over R is the whose are all left modules over R and whose morphisms are all module homomorphisms between left R-modules. For example, when R is the ring of integers Z, it is the same thing as the . The category of right modules is defined in a similar way. One can also define the category of bimodules over a ring R but that category is equivalent to the category of left (or right) modules over the enveloping algebra of R (or over the opposite of that).
Homotopy categoryIn mathematics, the homotopy category is a built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed below. More generally, instead of starting with the category of topological spaces, one may start with any and define its associated homotopy category, with a construction introduced by Quillen in 1967. In this way, homotopy theory can be applied to many other categories in geometry and algebra.
Category of ringsIn mathematics, the category of rings, denoted by Ring, is the 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 , meaning that the class of all rings is proper. The category Ring is a meaning that the objects are sets with additional structure (addition and multiplication) and the morphisms are functions that preserve this structure.
TensorIn mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product.
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).
Section (category theory)In , a branch of mathematics, a section is a right inverse of some morphism. , a retraction is a left inverse of some morphism. In other words, if and are morphisms whose composition is the identity morphism on , then is a section of , and is a retraction of . Every section is a monomorphism (every morphism with a left inverse is left-cancellative), and every retraction is an epimorphism (every morphism with a right inverse is right-cancellative). In algebra, sections are also called split monomorphisms and retractions are also called split epimorphisms.
Category of small categoriesIn mathematics, specifically in , the category of small categories, denoted by Cat, is the whose objects are all and whose morphisms are functors between categories. Cat may actually be regarded as a with natural transformations serving as 2-morphisms. The initial object of Cat is the empty category 0, which is the category of no objects and no morphisms. The terminal object is the terminal category or trivial category 1 with a single object and morphism. The category Cat is itself a , and therefore not an object of itself.
Rigid categoryIn , a branch of mathematics, a rigid category is a where every object is rigid, that is, has a dual X* (the internal Hom [X, 1]) and a morphism 1 → X ⊗ X* satisfying natural conditions. The category is called right rigid or left rigid according to whether it has right duals or left duals. They were first defined (following Alexander Grothendieck) by Neantro Saavedra Rivano in his thesis on . There are at least two equivalent definitions of a rigidity.
Variety (universal algebra)In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras, and (direct) products.
