4.17.4 Tensor Universal Property Commutative Relation

The Tensor Universal Property establishes a commutative relation by universal mapping, connecting tensor products to multilinear maps in algebra.

Tensor Universal Property Commutative Relation is the equation f = f̃ ∘ ⊗ that must hold between a multilinear map f, the canonical tensor map ⊗, and the induced linear map f̃, expressing that two different paths from V₁ × ... × Vₙ to W produce identical outputs on every input. It is called a commutative relation because it is the algebraic content behind the commuting triangle diagram associated with the tensor product's universal property, the same sense of "commuting" used throughout category theory for diagrams whose composite paths agree.

Stating the Relation

The Two Paths

Given vector spaces V₁, ..., Vₙ and W, a multilinear map f: V₁ × ... × Vₙ → W, the canonical map ⊗: V₁ × ... × Vₙ → V₁ ⊗ ... ⊗ Vₙ, and a linear map f̃: V₁ ⊗ ... ⊗ Vₙ → W, the commutative relation asserts

f = \tilde{f} \circ \otimes

as functions on V₁ × ... × Vₙ. Pointwise, for every tuple (v₁, ..., vₙ),

f (v_{1}, \dots, v_{n}) = \tilde{f} (\otimes (v_{1}, \dots, v_{n})) = \tilde{f} (v_{1} \otimes \dots \otimes v_{n})

As a Diagram

A diagram is called commutative when every directed path between two fixed objects, here V₁ × ... × Vₙ and W, yields the same composite map. The relation f = f̃ ∘ ⊗ is exactly the assertion that this diagram, with two paths from V₁ × ... × Vₙ to W, one direct and one through the tensor product, commutes.

Why the Relation Holds

Definition on Elementary Tensors

The relation is not an incidental coincidence but is built into the construction of f̃. When f̃ is defined on elementary tensors by setting f̃(v₁ ⊗ ... ⊗ vₙ) := f(v₁, ..., vₙ) and then extended linearly, the commutative relation holds by definition on elementary tensors, and since these span the tensor product, the well-definedness argument for f̃ guarantees the relation extends to hold everywhere f̃ is defined.

Dependence on Multilinearity

The commutative relation can only be satisfied by a linear f̃ because f is multilinear. If f failed to be multilinear in some argument, no single-valued linear map f̃ could exist making the triangle commute, because distinct representations of the same tensor as sums of elementary tensors would force f̃ to take contradictory values, since f would not respect the corresponding linear relations among tuples. The commutative relation is therefore inseparable from, and in a precise sense equivalent to, the multilinearity of f.

Verifying the Relation in Practice

Checking on a Spanning Set Suffices

Because elementary tensors span V₁ ⊗ ... ⊗ Vₙ, and both sides of f = f̃ ∘ ⊗ are determined once their values on elementary tensors are known, verifying the commutative relation reduces to checking it only on elementary tensors, that is, checking f̃(v₁ ⊗ ... ⊗ vₙ) = f(v₁, ..., vₙ) for all choices of v₁, ..., vₙ, rather than on arbitrary sums.

Checking on a Basis Suffices Further

If V₁, ..., Vₙ are finite-dimensional with chosen bases {e^{(1)}_{i₁}}, ..., {e^{(n)}_{iₙ}}, then by multilinearity of both sides of the relation, it is enough to verify

\tilde{f} (e_{i_{1}}^{(1)} \otimes \dots \otimes e_{i_{n}}^{(n)}) = f (e_{i_{1}}^{(1)}, \dots, e_{i_{n}}^{(n)})

on all tuples of basis vectors, since both f̃ ∘ ⊗ and f are determined by their action on basis tuples through multilinear extension. This gives a finite, computable check of the commutative relation for finite-dimensional spaces.

Failure of the Relation

What Failure Signals

If a proposed linear map g fails to satisfy g(v₁ ⊗ ... ⊗ vₙ) = f(v₁, ..., vₙ) for some tuple, then g is simply not the factorization associated to f; the commutative relation is what distinguishes the correct induced linear map from all other linear maps out of the tensor product. Failure of the relation for a candidate g does not contradict the universal property, since the universal property only asserts existence and uniqueness of some f̃ satisfying it, not that every linear map does.

Overdetermination Would Indicate Inconsistent Data

If two multilinear maps f₁ and f₂ both purported to induce the same linear map f̃ through the commutative relation, but disagreed on some tuple (v₁, ..., vₙ), this would contradict the relation itself, since f̃(v₁ ⊗ ... ⊗ vₙ) cannot simultaneously equal two different values. This shows the commutative relation encodes a strict compatibility condition linking f and f̃, not merely a loose association between them.

Relation to General Commutative Diagrams

Category-Theoretic Reading

In the language of category theory, the commutative relation is the statement that the triangle with vertices V₁ × ... × Vₙ, V₁ ⊗ ... ⊗ Vₙ, and W, and edges ⊗, f̃, and f, is a commutative diagram. The tensor product's universal property is typically phrased as: for every such f, there exists a unique f̃ making this specific triangle commute, situating the tensor commutative relation as one instance of the broader technique of defining objects by the diagrams they make commute.

Composability of Commutative Relations

If f̃₁: V₁ ⊗ ... ⊗ Vₙ → W satisfies the commutative relation for f, and h: W → W' is any linear map, then h ∘ f̃₁ satisfies the commutative relation for h ∘ f, since (h ∘ f̃₁) ∘ ⊗ = h ∘ (f̃₁ ∘ ⊗) = h ∘ f. This composability shows the commutative relation behaves well under post-composition, which underlies the functorial behavior of the tensor product construction with respect to maps out of W.

Consequences for Identifying Tensors

Distinguishing Elements via the Relation

The commutative relation gives a method for distinguishing elements of the tensor product: two tensors t₁, t₂ are shown to be different by exhibiting a multilinear map f whose factorization f̃ satisfies f̃(t₁) ≠ f̃(t₂), since the commutative relation guarantees f̃ is computable directly from f's values on elementary tensors of t₁ and t₂. This technique, testing tensors against multilinear maps via their factorizations, is a standard way to establish that elements built from different combinations of elementary tensors are genuinely distinct in the tensor product, rather than secretly equal due to some overlooked relation.

Canonical Isomorphisms Verified via the Relation

Standard identifications such as V ⊗ W ≅ W ⊗ V or (V ⊗ W) ⊗ U ≅ V ⊗ (W ⊗ U) are verified to be well-defined linear isomorphisms precisely by checking that the relevant commutative relation holds for the multilinear rearrangement map used to construct each one, confirming that the induced linear map behaves correctly on every elementary tensor before concluding it is an isomorphism.