Tensor and Fusion Categories — Examples

1 · Prerequisites

2 · Summary

These examples use finite-dimensional vector spaces and do not reconstruct their tensor product.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: Literature-sourcedVerification: Literature-sourcedjudge pass (gpt-5.6-terra)audited 2026-09-06‡ rests on unproved material (inherited)Open item page →

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Finite-dimensional vector spaces form a fusion category

Example

For a field $k$ , the category ${V e c t}_{k}^{f d}$ is a fusion category.

Facts & Assumptions

Given: A field $k$ .

[F1]

$k$ -modules are monoidal under $\otimes_{k}$ with unit $k$ (Modules over a commutative ring form a monoidal category).

[F2]

Every finite-dimensional $k$ -vector space is rigid (Finite-dimensional vector spaces are rigid).

[F3]

Fusion means finite semisimple tensor category (Fusion and multifusion categories).

Verification

technique · direct

1.1

By [F1] and [F2], the finite-dimensional subcategory is a rigid $k$ -linear monoidal category.

F1F2given

2.1

Every finite-dimensional vector space is a finite direct sum of copies of $k$ , so $k$ is the only simple isomorphism class and the category is finite semisimple. Its unit has endomorphism algebra $k$ .

step 1.1F3

3.1

These clauses are exactly those of [F3], so ${V e c t}_{k}^{f d}$ is fusion.

step 2.1F3∎

ExampleConstruction: Literature-sourcedVerification: AI-adaptedaudited 2026-09-06‡ rests on unproved material (inherited)Open item page →

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The Grothendieck ring of finite-dimensional vector spaces

Example

The dimension map identifies $K_{0} ({V e c t}_{k}^{f d})$ with $Z$ as a ring.

Facts & Assumptions

Given: A field $k$ and ${V e c t}_{k}^{f d}$ .

[F1]

This category is fusion (Finite-dimensional vector spaces form a fusion category). Every finite-dimensional vector space is isomorphic to a finite direct sum of copies of $k$ , and $k$ is simple, so $k$ is its only simple class.

[F2]

The Grothendieck group is generated by object classes modulo short-exact relations; its intended multiplication on generators is $[V] [W] = [V \otimes_{k} W]$ (The Grothendieck ring of a tensor category). We verify that this multiplication is well-defined in the present example.

Verification

technique · direct

1.1

Dimension is additive on short exact sequences: a basis of a subspace, together with lifts of a basis of the quotient, is a basis of the middle space. Thus $d ([V]) = \dim_{k} V$ respects every relation in [F2] and induces a group homomorphism $d : K_{0} \to Z$ . Split direct-sum sequences and [F1] give $[V] = (\dim_{k} V) [k]$ , including $[0] = 0$ . Hence $e : Z \to K_{0}$ , $m \mapsto m [k]$ , satisfies $d e = id$ and $e d = id$ , proving the asserted additive bijection on all classes, including negative virtual classes.

F1F2given

2.1

Transport integer multiplication along this bijection: set $a \cdot b = e (d (a) d (b))$ . This is a well-defined unital ring structure, with unit $e (1) = [k]$ . Tensor products of finite bases give $\dim_{k} (V \otimes_{k} W) = \dim_{k} V \dim_{k} W$ , so $[V] \cdot [W] = e (\dim_{k} (V \otimes_{k} W)) = [V \otimes_{k} W]$ . Thus the intended tensor multiplication in [F2] descends to the quotient; its bilinear extension is unique because object classes generate the group. The map $d$ is therefore an isomorphism of rings, with zero and unit preserved.

step 1.1F2algebra∎

ExampleConstruction: Literature-sourcedVerification: Literature-sourcedjudge pass (gpt-5.6-terra)audited 2026-09-06‡ rests on unproved material (inherited)Open item page →

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

A matrix multifusion category with nonsimple unit

Example

For $n > 1$ , finite $n \times n$ matrices of finite-dimensional vector spaces, with matrix multiplication using $\oplus$ and $\otimes_{k}$ , form a multifusion category whose unit is not simple.

Facts & Assumptions

Given: A field $k$ and an integer $n > 1$ .

[F1]

${V e c t}_{k}^{f d}$ is fusion (Finite-dimensional vector spaces form a fusion category).

[F2]

A multifusion category is finite semisimple multitensor category (Fusion and multifusion categories).

Verification

technique · direct

1.1

Let objects be $n \times n$ matrices $(V_{i j})$ and set $(V \otimes W)_{i j} = ⨁_{r} V_{i r} \otimes_{k} W_{r j}$ . The matrix $I$ with $k$ on the diagonal and $0$ off it is a unit.

F1givenconstruct

2.1

Entrywise semisimplicity and finite direct sums from [F1] give the finite semisimple rigid structure required in [F2].

step 1.1F1F2

3.1

But $I = E_{11} \oplus \dots \oplus E_{n n}$ is a nontrivial direct sum when $n > 1$ . Hence this is multifusion, not fusion.

step 1.1F2∎

ExampleConstruction: Literature-sourcedVerification: AI-adaptedaudited 2026-09-06‡ rests on unproved material (inherited)Open item page →

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Fusion rules for a supplied finite simple family

Example

For the supplied simple family $(k)$ of ${V e c t}_{k}^{f d}$ , the only fusion coefficient is $N_{11}^{1} = 1$ .

Facts & Assumptions

Given: The one-term family $(k)$ .

[F1]

${V e c t}_{k}^{f d}$ is a fusion category (Finite-dimensional vector spaces form a fusion category). Every finite-dimensional vector space is a finite direct sum of copies of the simple object $k$ , so $k$ is its sole simple class.

[F2]

Fusion coefficients are defined by products of simple classes (Fusion rules).

Verification

technique · direct

1.1

The unit isomorphism $k \otimes_{k} k ≅ k$ gives $[k] [k] = [k]$ .

F1given

2.1

Comparing this with the defining expansion in [F2] for the supplied one-element family yields $N_{11}^{1} = 1$ .

step 1.1F2∎

Tensor and Fusion Categories — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Finite-dimensional vector spaces form a fusion category

Example

Facts & Assumptions

Verification

The Grothendieck ring of finite-dimensional vector spaces

Example

Facts & Assumptions

Verification

A matrix multifusion category with nonsimple unit

Example

Facts & Assumptions

Verification

Fusion rules for a supplied finite simple family

Example

Facts & Assumptions

Verification

Sources