Morphisms must satisfy two axioms:

• IDENTITY: for every object X, there exists a morphism id_X : X → X called the identity morphism on X, such that for every morphism f : A → B we have $\{\backslash rm\; id\}\_B\backslash circ\; f=f=f\backslash circ\{\backslash rm\; id\}\_A$.
• ASSOCIATIVITY: $h\backslash circ(g\backslash circ\; f)=(h\backslash circ\; g)\backslash circ\; f$ whenever the operations are defined.

When C is a concrete category, composition is just ordinary composition of functions, the identity morphism is just the identity function, and associativity is automatic. (Functional composition is associative.)

Note that the domain and codomain are really part of the information determining the morphism. For example, in the category of sets, where morphisms are functions, two functions may be identical as sets of ordered pairs (having the same range), but have different codomains. These functions are considered distinct for the purposes of category theory. For this reason, many authors require that the hom-classes hom(X, Y) be disjoint. In practice, this is not a problem, because if they are not disjoint, the domain and codomain can be appended to the morphisms, (say, as the second and third components of an ordered triple), making them disjoint.

Some remarkable morphisms

---

• A morphism f : X → Y is called a monomorphism if $f\backslash circ\; g\_1=f\backslash circ\; g\_2$ implies g₁ = g₂ for all morphisms g₁, g₂ : Z → X. It is also called a mono or a monic. The morphism f has left-inverse if there is a morphism g:Y → X such that $g\backslash circ\; f=\{\backslash rm\; id\}\_X$. The left-inverse g is also called a retraction of f. Morphisms with left-inverses are always monomorphisms, but the converse is not always true in every category; a monomorphism may fail to have a left-inverse. A monomorphism which does have a left-inverse is called a split monomorphism. In concrete categories, a function which has left-inverse is injective. Thus in concrete categories, monomorphisms are often, but not always, injective. The condition of being an injection is stronger than that of being a monomorphism, but weaker than that of being a split monomorphism.
• Dually, a morphism f : X → Y is called an epimorphism if $g\_1\backslash circ\; f=g\_2\backslash circ\; f$ implies g₁ = g₂ for all morphisms g₁, g₂ : Y → Z. It is also called an epi or an epic. The morphism f has right-inverse if there is a morphism g:Y → X such that $f\backslash circ\; g=\{\backslash rm\; id\}\_Y$. The right-inverse g is also called a section of f. Morphisms with right-inverse are always epimorphisms, but the converse is not always true in every category; an epimorphism may fail to have a right-inverse. An epimorphism which does have a right-inverse is called a split epimorphism. In concrete categories, a function which has right-inverse is surjective. Thus in concrete categories, epimorphisms are often, but not always, surjective. The condition of being a surjection is stronger than that of being an epimorphism, but weaker than that of being a split epimorphism. In the category of sets, every surjection has a section. This result is equivalent to the axiom of choice.

  Note that if a split monomorphism f has a left-inverse g, then g is a split epimorphism and has right-inverse f.

• A morphism which is both an epimorphism and a monomorphism is called a bimorphism.
• A morphism f : X → Y is called an isomorphism if there exists a morphism g : Y → X such that $f\backslash circ\; g=\{\backslash rm\; id\}\_Y$ and $g\backslash circ\; f=\{\backslash rm\; id\}\_X$. If a morphism has both left-inverse and right-inverse, then the two inverses are equal, so f is an isomorphism, and g is called simply the inverse of f. Inverse morphisms, if they exist, are unique. The inverse g is also an isomorphism with inverse f. Two objects with an isomorphism between them are said to be isomorphic or equivalent.

  Note that every isomorphism is a bimorphism but, in general, not every bimorphism is an isomorphism. For example, in the category of commutative rings the inclusion Z → Q is a bimorphism which is not an isomorphism. However, any morphism that is both an epimorphism and a split monomorphism, or both a monomorphism and a split epimorphism, must be an isomorphism. A category in which every bimorphism is an isomorphism is a balanced category. For example, Set is a balanced category.

• Any morphism f : X → X is called an endomorphism of X.
• An endomorphism that is also an isomorphism is called an automorphism.
• If a split monomorphism h : X → Y has left-inverse g : Y → X, so that $g\backslash circ\; h=\{\backslash rm\; id\}\_X$, then $h\backslash circ\; g:\backslash ,\; Y\backslash to\; Y$ is idempotent, which means that $(h\backslash circ\; g)^2=h\backslash circ\; g$. More generally, any idempotent endomorphism f is said to be split if it admits a decomposition $f=h\backslash circ\; g$ with $g\backslash circ\; h=\backslash mathrm\{id\}$. In particular, the Karoubi envelope of a category splits every idempotent.

See also:

• zero morphisms
• normal morphisms
• holomorphism

Examples

---

• In the concrete categories studied in universal algebra (such as those of groups, rings, modules, etc.), morphisms are called homomorphisms. The terms isomorphism, epimorphism, monomorphism, endomorphism, and automorphism are all used in that specialized context as well.

• 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.

• Functors can be thought of as morphisms in the category of small categories.

• In a functor category the morphisms are natural transformations.

For more examples see the article on category theory.

Category theory

Morphismus | Morfismo | Morphisme | Morfismo | 射 (圏論) | Morfism | 态射