nLab morphism

Idea

In category theory, the notion of morphism or map is an abstraction and generalisation of the notions of function, homomorphism (whence the name) and relation.

Morphisms $f$ are depicted as directed edges/arrows

$$x \overset{f}{\longrightarrow} y$$

between pairs of vertices — the morphism’s source (domain) and target (codomain) objects. Systems of such arrows are called diagrams.

Definition

For a pair of objects $x$, and $y$ in a (locally-small) category $A$, the set, $Hom _A (x,y)$, is called the hom-set, the elements being morphisms of domain $x$, and codomain $y$. For a morphism $f$, denote $f : x \rightarrow y$ to mean $f \in Hom _A (x,y)$. In enriched categories, the object, $Hom _A (x,y)$, is called the hom-object.

In higher category theory, a morphism is a 1-cell of 0-cells in n-categories.

Examples

A familiar example is the topos Set of sets and mappings; the objects are sets, and the morphisms are mappings, or functions.

In Grp, the locally-small category of groups, and group homomorphisms; the objects are set-theoretical groups, and the morphisms are group homomorphisms.

In Top, the locally-small category of topological spaces, and continuous functions; the objects are sets of topological structure, and the morphisms are continuous functions.

Variations

Morphisms are notable for being rich in stuff, structures, and properties. This is an incomplete list of just some of those variations. Henceforth denote by $A$, a category.

Monomorphism

A morphism $f \in hom _A (y,z)$, is a monomorphism, or is monic if it satisfies left-cancellation, as-in; for all morphisms $g, h \in hom _A (x,y)$, $f \circ g=f \circ h \Rightarrow g=h$. This is a generalisation of left-uniqueness in relations, injectivity in functions, and monomorphism in algebra. From strongest to weakest in properties:

• split monomorphism

• absolute monomorphism

• effective monomorphism

• normal monomorphism

• regular monomorphism

• strict monomorphism

• strong monomorphism

• extremal monomorphism

• monomorphism

Epimormphism

A morphism $f \in hom _A (x,y)$, is an epimorphism, or epic, if it satisfies right-cancellation, as-in; for all morphisms $g, h \in hom _A (y,z)$, $g \circ f=h \circ f \Rightarrow g=h$. This is a generalisation of right-totality in relations, surjectivity in functions, and epimorphism in abstract algebra. From strongest to weakest in properties:

• split epimorphism

• absolute epimorphism

• effective descent morphism

• descent morphism

• effective epimorphism

• normal epimorphism

• regular epimorphism

• strict epimorphism

• strong epimorphism

• extremal epimorphism

• epimorphism

Epic, and Monic

A morphism $f \in Hom _A (x,y)$, is a bimorphism, bimonomorphism, or is epic, and monic, if it satisfies left-cancellation, and right-cancellation. This is a generalisation of left-unique right-totality in relations, bijection in functions, and isomorphism in abstract algebra.

Isomorphism

A morphism, $f \in Hom _A (x,y)$, is an isomorphism, or is iso, if both it and its inverse morphism are epic, and monic. This is a generalisation of left-unique left-total right-unique right-totality in relations, bijection in functions, and isomorphism in abstract algebra.

Note, list is still incomplete.

Related concepts

• object

• morphism, multimorphism

  • inverse morphism

  • adjoint morphism

  • heteromorphism

  • endomorphism, automorphism, monomorphism, epimorphism, isomorphism

• 1-morphism

  horizontal morphism, vertical morphism

• 2-morphism

• 3-morphism

• k-morphism