# 4.2 Morphisms and Mappings

## 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.