4.2 Morphisms and Mappings

A morphism is a map that preserves structure. It is the correct notion of transformation for a given mathematical setting.

A plain function sends elements of one set to elements of another set.

f : X \to Y

A morphism does more. It respects the operations, relations, or laws that define the structures on $X$ and $Y$ .

For groups, the morphisms are group homomorphisms. A map $f : G \to H$ is a group homomorphism if

f(a \ast b) = f(a) \cdot f(b)

for all $a,b \in G$ . The operation $\ast$ belongs to $G$ , and the operation $\cdot$ belongs to $H$ .

For vector spaces, the morphisms are linear maps. A map $T : V \to W$ is linear if

T(v+w) = T(v) + T(w)

and

T(\lambda v) = \lambda T(v)

for all $v,w \in V$ and all scalars $\lambda$ .

For topological spaces, the morphisms are continuous maps. A function $f : X \to Y$ is continuous if the inverse image of every open set in $Y$ is open in $X$ .

Structure	Morphism	Preserved data
Set	Function	Elements only
Group	Homomorphism	Operation
Ring	Ring homomorphism	Addition and multiplication
Vector space	Linear map	Addition and scalar multiplication
Topological space	Continuous map	Open-set structure
Metric space	Isometry, Lipschitz map, or continuous map	Distance-related structure
Graph	Graph homomorphism	Adjacency

The choice of morphism defines the kind of mathematics being done. If the morphisms change, the theory changes.

For example, a metric space can be studied with isometries, Lipschitz maps, or continuous maps. Each choice preserves a different amount of structure.

Map type	What it preserves
Isometry	Exact distance
Lipschitz map	Distance up to controlled distortion
Continuous map	Nearby points remain nearby in a topological sense

The same objects can therefore support different theories depending on which maps are allowed.

Morphisms as comparisons

Morphisms compare structures. A map from $A$ to $B$ tells us how information in $A$ can be represented inside $B$ .

An injective morphism often embeds one structure into another. A surjective morphism often collapses structure onto a quotient. An isomorphism gives a reversible comparison.

Type	Meaning
Injective morphism	Preserves distinction between elements
Surjective morphism	Covers the target
Isomorphism	Preserves structure reversibly
Endomorphism	Morphism from an object to itself
Automorphism	Isomorphism from an object to itself

For a group homomorphism $f : G \to H$ , the kernel measures what gets collapsed:

\ker(f) = {g \in G : f(g) = e_H}.

The image measures what part of $H$ is reached:

\operatorname{Im}(f) = {f(g) : g \in G}.

These constructions turn a morphism into structural data.

Composition

Morphisms compose. If $f : A \to B$ and $g : B \to C$ are morphisms of the same kind, then their composite is also a morphism:

g \circ f : A \to C.

Composition is defined by

(g \circ f)(a) = g(f(a)).

This simple rule is one of the main reasons morphisms are central. It allows long transformations to be built from short ones.

Composition is associative:

h \circ (g \circ f) = (h \circ g) \circ f.

Each object also has an identity morphism:

\operatorname{id}_A : A \to A.

The identity morphism does nothing:

\operatorname{id}_A(a) = a.

Together, objects, morphisms, identity maps, and composition form the basic pattern of category theory.

Preservation and reflection

A morphism may preserve a property, reflect a property, or both.

A map preserves a property if the property in the source implies the corresponding property in the target.

A map reflects a property if the property in the target implies the corresponding property in the source.

For example, a continuous function preserves convergence of sequences in many familiar spaces:

x_n \to x \quad \Rightarrow \quad f(x_n) \to f(x).

An isomorphism usually preserves and reflects all structure expressible in the language of the theory.

This is why isomorphic objects can be treated as structurally identical.

Transport through morphisms

Morphisms allow information to move.

If $f : A \to B$ is an isomorphism, any structure on $A$ can be transported to $B$ . The transported structure is defined so that $f$ becomes structure-preserving.

For a binary operation $\ast$ on $A$ , define an operation $\star$ on $B$ by

b_1 \star b_2 = f(f^{-1}(b_1) \ast f^{-1}(b_2)).

This makes $A$ and $B$ equivalent as structured objects.

Transport is common in linear algebra. Choosing a basis gives an isomorphism

V \cong k^n.

This transports abstract vectors into coordinate tuples. Computations can then be done in $k^n$ and interpreted back in $V$ .

Forgetful maps and added structure

Some maps forget structure. A group can be viewed as a set by ignoring its operation. A topological space can be viewed as a set by ignoring its open sets.

This operation is often called a forgetful functor in categorical language.

Structured object	Forgotten view
Group	Underlying set
Ring	Underlying additive group or set
Vector space	Underlying set
Topological space	Underlying set
Metric space	Underlying topological space

Forgetting structure makes more functions available but fewer theorems applicable.

A function between underlying sets need not be a morphism of the richer structures. A map between vector spaces may be a set function without being linear.

Automorphisms and symmetry

An automorphism is an isomorphism from an object to itself.

f : X \to X

Automorphisms describe symmetries. They are structure-preserving transformations that leave the object within the same structural type.

Examples:

Object	Automorphisms
Set	Permutations
Group	Group automorphisms
Vector space	Invertible linear maps
Graph	Vertex permutations preserving adjacency
Topological space	Homeomorphisms from the space to itself

The automorphisms of an object often form a group. This group measures the symmetries of the object.

For example, the symmetries of a square form a group under composition. Each symmetry preserves the square’s geometric structure.

Morphisms determine the theory

Choosing objects without choosing morphisms gives an incomplete theory.

For instance, suppose the objects are metric spaces. Several choices are possible:

Morphisms	Resulting emphasis
Isometries	Exact metric geometry
Lipschitz maps	Quantitative distortion
Continuous maps	Topology
Uniformly continuous maps	Uniform structure

Each choice produces a different notion of sameness.

With isometries, two spaces are equivalent only when distances are exactly preserved. With homeomorphisms, only open-set structure matters. A line segment and a curved arc may be different metrically but the same topologically.

Practical workflow

When studying a structure, always ask:

Question	Purpose
What are the objects?	Identifies the domain of study
What are the morphisms?	Identifies allowed transformations
What do morphisms preserve?	Identifies the structure
What are the isomorphisms?	Identifies sameness
What are the automorphisms?	Identifies symmetry

This workflow prevents a common mistake: treating any function as acceptable when the theory requires structure preservation.

A morphism is the disciplined form of a map. It carries objects from one setting to another while respecting the structure that makes those objects mathematical.