Morphism

A category is given by two pieces of data: a class of objects and, for any two objects X and Y, a set of morphisms from X to Y. Morphisms are often depicted as arrows between those objects. In the case of a concrete category, X and Y are sets of some kind and a morphism f is a function from X to Y satisfying some condition; this example supplies the notation f: X -> Y. But not every category is concrete, so these aren't the only types of morphisms.

Some examples of morphisms are homomorphisms from the categories studied in universal algebra (such as those of groups, rings, and so on), continuous functions between topological spaces, elements of a group when the group is thought of a special kind of category, paths in a single topological space (which form a groupoid), functors between categories, and many more.

Variants and subclasses of morphism:

• Every object X in every category has an identity morphism id_X which acts as an identity under the operation of composition.
• If f: X -> Y and g: Y -> X satisfy f o g  =  id_Y, then f is a retraction and g is a section.
• If f is both a retraction and a section, then it is an isomorphism. In this case, the objects X and Y should be thought of as completely equivalent for purposes of the category C.
• A morphism f: X -> X is an endomorphism of X.
• An endomorphism that is also an isomorphism is an automorphism.
• Suppose that whenever g: Y -> Z and h: Y -> Z and g o f  =  h o f, it always turns out that g  =  h. Then f is an epimorphism. Every retraction must be an epimorphism. It's also called an epi or an epic.

An epimorphism with a one-sided inverse is called a split epimorphism.

• Suppose that whenever g: W -> X and h: W -> X and f o g  =  f o h, it always turns out that g  =  h. Then f is a monomorphism. Every section must be a monomorphism. It's also called a mono or a monic.

An monomorphism with a one-sided inverse is called a split monomorphism.

• If f is both an epimorphism and a monomorphism, then f is a bimorphism. Note that not every bimorphism is an isomorphism! However, any morphism that is both an epimorphism and a section, or both a monomorphism and a retraction, must be an isomorphism.

• A homeomorphism is simply an isomorphism in the category of topological spaces.
• A diffeomorphism is simply an isomorphism in the category of differentiable manifolds.