# 4.3 Isomorphism as Sameness

## 4.3 Isomorphism as Sameness

Isomorphism is the mathematical notion of structural sameness. Two objects are isomorphic when there is a reversible map between them that preserves the relevant structure.

Equality is usually too strict. Two objects can have different representations and still behave identically for the purposes of a theory.

For example, the additive group of integers and the additive group of even integers are not equal as sets:

$$
\mathbb{Z} \neq 2\mathbb{Z}
$$

But they are isomorphic as groups. The map

$$
f : \mathbb{Z} \to 2\mathbb{Z}, \quad f(n) = 2n
$$

is bijective and preserves addition:

$$
f(n+m) = 2(n+m) = 2n + 2m = f(n) + f(m).
$$

So the two groups have the same group structure.

### Definition

An isomorphism is a morphism with an inverse morphism.

If $A$ and $B$ are structured objects, an isomorphism from $A$ to $B$ is a morphism

$$
f : A \to B
$$

such that there exists a morphism

$$
g : B \to A
$$

with

$$
g \circ f = \operatorname{id}_A
$$

and

$$
f \circ g = \operatorname{id}_B.
$$

When such a map exists, we write:

$$
A \cong B
$$

This means $A$ and $B$ are structurally the same.

### Isomorphism depends on structure

The meaning of isomorphism depends on the category of objects being studied.

| Structure         | Isomorphism                                            |
| ----------------- | ------------------------------------------------------ |
| Set               | Bijection                                              |
| Group             | Bijective group homomorphism                           |
| Ring              | Bijective ring homomorphism with compatible operations |
| Vector space      | Invertible linear map                                  |
| Topological space | Homeomorphism                                          |
| Metric space      | Isometry                                               |
| Graph             | Adjacency-preserving bijection with inverse            |

The same two underlying sets may be isomorphic in one structure and not in another.

For example, two spaces may be homeomorphic but not isometric. They have the same topological structure but different metric structure.

This shows that isomorphism is not absolute. It is relative to what structure we choose to preserve.

### Equality vs isomorphism

Equality says two objects are literally the same object in the underlying formal system.

Isomorphism says two objects have the same structure.

| Relation    | Meaning                 | Strength |
| ----------- | ----------------------- | -------- |
| $A = B$     | Literal equality        | Stronger |
| $A \cong B$ | Structural sameness     | Weaker   |
| $A \sim B$  | Some chosen equivalence | Weakest  |

In ordinary mathematical practice, isomorphic objects are often treated as the same once the isomorphism has been established. This is efficient, but it should be done carefully.

For example:

$$
V \cong k^n
$$

for every $n$-dimensional vector space $V$ over $k$. But $V$ is not literally equal to $k^n$ unless it has been defined that way.

Choosing a basis gives an isomorphism:

$$
V \to k^n.
$$

Different bases give different isomorphisms. So the identification is usually noncanonical.

### Canonical isomorphism

An isomorphism is canonical when it is determined naturally by the given structure, without arbitrary choices.

For example, a finite-dimensional vector space $V$ is naturally isomorphic to its double dual:

$$
V \cong V^{**}
$$

The map sends each vector $v \in V$ to the evaluation functional on $V^*$:

$$
\phi \mapsto \phi(v).
$$

This construction does not require choosing a basis.

By contrast, an isomorphism

$$
V \cong k^n
$$

requires choosing a basis. It is therefore noncanonical in general.

| Statement                                                      |        Requires choice? | Canonical? |
| -------------------------------------------------------------- | ----------------------: | ---------: |
| $V \cong k^n$                                                  |     Yes, choose a basis |         No |
| $V \cong V^{**}$ for finite-dimensional $V$                    |         No basis needed |        Yes |
| A finite set with $n$ elements is bijective with ${1,\dots,n}$ | Yes, choose an ordering |         No |

Canonical isomorphisms are safer to use as identifications because they do not depend on hidden choices.

### Invariants under isomorphism

An invariant is a property preserved by isomorphism.

If $A \cong B$, then any structural invariant of $A$ equals the corresponding invariant of $B$.

Examples:

| Structure               | Invariant                                                 |
| ----------------------- | --------------------------------------------------------- |
| Finite set              | Cardinality                                               |
| Vector space            | Dimension                                                 |
| Matrix under similarity | Trace, determinant, eigenvalues                           |
| Group                   | Order, abelian property, subgroup structure               |
| Topological space       | Compactness, connectedness                                |
| Graph                   | Number of vertices, degree sequence, connected components |

Invariants help prove that two objects are not isomorphic. If one invariant differs, no isomorphism can exist.

For example, vector spaces of different dimensions over the same field are not isomorphic:

$$
k^m \cong k^n \quad \text{if and only if} \quad m = n.
$$

The dimension is a complete invariant for finite-dimensional vector spaces over a fixed field.

### Complete and incomplete invariants

An invariant is complete if it fully classifies objects up to isomorphism.

Dimension is a complete invariant for finite-dimensional vector spaces over a fixed field.

Group order is not a complete invariant for finite groups. There can be non-isomorphic groups with the same number of elements.

| Field                                     | Invariant                    | Complete? |
| ----------------------------------------- | ---------------------------- | --------- |
| Finite-dimensional vector spaces over $k$ | Dimension                    | Yes       |
| Finite sets                               | Cardinality                  | Yes       |
| Finite groups                             | Order                        | No        |
| Graphs                                    | Number of vertices and edges | No        |
| Topological spaces                        | Connectedness                | No        |

A classification problem asks for complete invariants or canonical representatives.

### Isomorphism classes

Instead of studying individual objects, we often study isomorphism classes.

The isomorphism class of an object $A$ is the collection of all objects isomorphic to $A$:

$$
[A] = {B : B \cong A}.
$$

A classification theorem describes these classes.

For example:

Every finite-dimensional vector space over $k$ belongs to exactly one isomorphism class, determined by its dimension.

This means the classification is:

$$
0, 1, 2, 3, \dots
$$

where $n$ corresponds to $k^n$.

### Isomorphism and transport

An isomorphism allows structure and theorems to be transported.

If $f : A \to B$ is an isomorphism, then any structural fact about $A$ has a corresponding fact about $B$.

For example, if $G \cong H$ and $G$ is abelian, then $H$ is abelian.

Proof sketch:

Let $h_1,h_2 \in H$. Since $f$ is surjective, choose $g_1,g_2 \in G$ with

$$
f(g_1)=h_1,\quad f(g_2)=h_2.
$$

Then

$$
h_1h_2 = f(g_1)f(g_2)=f(g_1g_2)=f(g_2g_1)=f(g_2)f(g_1)=h_2h_1.
$$

So $H$ is abelian.

The property is preserved because the group operation is preserved.

### Automorphisms

An automorphism is an isomorphism from an object to itself.

$$
f : A \to A
$$

Automorphisms describe internal symmetries.

| Object            | Automorphism                           |
| ----------------- | -------------------------------------- |
| Set               | Permutation                            |
| Group             | Structure-preserving bijection         |
| Vector space      | Invertible linear map                  |
| Graph             | Vertex relabeling preserving edges     |
| Topological space | Homeomorphism from the space to itself |

The automorphisms of an object form a group under composition:

$$
\operatorname{Aut}(A).
$$

This group measures the symmetry of $A$.

### Structural practice

In structural mathematics, a statement should usually be invariant under isomorphism. If a property changes when an object is replaced by an isomorphic copy, then the property depends on representation rather than structure.

For example, the phrase “the first coordinate of $v$ is zero” depends on a chosen basis. It is not a structural property of a vector $v \in V$ unless a coordinate system is fixed.

By contrast, the statement “$v = 0$” is structural.

| Statement                                          |                 Structural? |
| -------------------------------------------------- | --------------------------: |
| $v = 0$                                            |                         Yes |
| The first coordinate of $v$ is zero                | No, unless a basis is fixed |
| $T$ is injective                                   |                         Yes |
| The matrix of $T$ has a zero in the top-left entry |  No, unless bases are fixed |
| $\dim V = n$                                       |                         Yes |

This distinction helps prevent representation-dependent arguments.

### Practical rule

When two objects appear similar, ask:

| Question                              | Purpose                                         |
| ------------------------------------- | ----------------------------------------------- |
| What structure is being preserved?    | Determines the meaning of isomorphism           |
| Is the map reversible?                | Distinguishes isomorphism from general morphism |
| Is the isomorphism canonical?         | Detects hidden choices                          |
| What invariants are preserved?        | Transfers properties                            |
| Are we using equality or isomorphism? | Prevents overstatement                          |

Isomorphism formalizes the idea that objects can be different in representation but identical in structure. It is one of the main tools that makes abstraction precise.