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 Morphism: Definition, Some Specific Morphisms, Examples

### Other articles related to "morphism, morphisms":

**Morphism**s

... 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 ...

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 ...

... 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 ...

**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 ...