Morphism

In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. 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 topology, continuous functions, and so on.

The study of morphisms and of the structures (called objects) over which they are defined, is central to category theory. Much of the terminology of morphisms, as well as the intuition underlying them, comes from concrete categories, where the objects are simply sets with some additional structure, and morphisms are structure-preserving functions.

Read more about MorphismDefinition, Some Specific Morphisms, Examples

Other articles related to "morphism, morphisms":

List Of Zero Terms - Zero Morphisms
... A zero morphism in a category is a generalised absorbing element under function composition any morphism composed with a zero morphism gives a zero morphism ... Specifically, if 0XY X → Y is the zero morphism among morphisms from X to Y, and f A → X and g Y → B are arbitrary morphisms, then g ∘ 0XY = 0XB and 0XY ∘ f = 0AY ... If a category has a zero object 0, then there are canonical morphisms X → 0 and 0 → Y, and composing them gives a zero morphism 0XY X → Y ...
Morphism - Examples
... algebra (groups, rings, modules, etc.), morphisms are usually homomorphisms ... In the category of topological spaces, morphisms are continuous functions and isomorphisms are called homeomorphisms ... In the category of smooth manifolds, morphisms are smooth functions and isomorphisms are called diffeomorphisms ...
Early Work On Cotangent Complexes
2, where Pierre Berthelot gave a definition when f is a smoothable morphism, meaning there is a scheme V and morphisms i X → V and h V → Y such that f = hi, i is ... For example, all projective morphisms are smoothable, since V can be taken to be a projective bundle over Y.) In this case, he defines the cotangent complex of f as an object in the derived category of coherent ... this definition is independent of the choice of V and that for a smoothable complete intersection morphism, this complex is perfect ...
Properties of The Cotangent Complex - Vanishing Properties
... If f is an étale morphism, then LB/A = 0 ... If f is a smooth morphism, then LB/A is quasi-isomorphic to ΩB/A ... If f is a local complete intersection morphism, then LB/A has projective dimension at most one ...
Finite Morphism
... In algebraic geometry, a branch of mathematics, a morphism of schemes is a finite morphism if has an open cover by affine schemes such that for each ...