4.17.2 Tensor Universal Property Factorization

Tensor Universal Property Factorization is the specific claim, within the universal mapping property of the tensor product, that every multilinear map factors through the canonical tensor map: for any multilinear map f: V₁ × ... × Vₙ → W, there is a linear map f̃ making the equation f = f̃ ∘ ⊗ hold, where ⊗ is the canonical map into the tensor product V₁ ⊗ ... ⊗ Vₙ. Factorization is the "existence" half of the universal property, describing the act of routing an arbitrary multilinear map through the tensor product without altering the values it produces.

What Factorization Means

Passing Through an Intermediate Object

To say that f factors through V₁ ⊗ ... ⊗ Vₙ is to say that computing f can be broken into two stages: first apply the canonical map ⊗ to obtain a tensor, and then apply the linear map f̃ to that tensor to obtain the same output f would have given directly. Symbolically,

for all tuples (v₁, ..., vₙ). The intermediate object, V₁ ⊗ ... ⊗ Vₙ together with ⊗, is said to be "universal" for multilinear maps precisely because every such map factors through it in this way, and does so through a map that is linear rather than merely some other kind of function.

The Commutative Triangle

Factorization is exactly the statement that this triangle commutes: the diagonal arrow f equals the composite of the two other arrows, ⊗ followed by f̃. The diagonal arrow is given data; the vertical and slanted arrows are what the universal property supplies.

Constructing the Factorization

Reduction to a Linear Problem

The tensor product V₁ ⊗ ... ⊗ Vₙ is built, in one standard construction, as the quotient of the free vector space Free(V₁ × ... × Vₙ) on the underlying set V₁ × ... × Vₙ by the subspace R generated by all multilinearity relations, such as

Any function f out of the set V₁ × ... × Vₙ extends uniquely to a linear map F out of the free vector space, by linearity on basis elements. The factorization through V₁ ⊗ ... ⊗ Vₙ = Free(V₁ × ... × Vₙ)/R is then possible exactly when F vanishes on R, since a linear map on a quotient exists precisely when the original map annihilates the subspace being quotiented by. Multilinearity of f is exactly the condition that makes F vanish on each generator of R, so F vanishes on all of R, and the factorization f̃ is the induced map on the quotient.

The First Isomorphism Theorem Perspective

This last step is an application of the standard fact that a linear map F: U → W with R ⊆ ker(F) factors uniquely as F = f̃ ∘ π, where π: U → U/R is the quotient projection. Setting U = Free(V₁ × ... × Vₙ) and identifying π with the canonical map ⊗ recovers the factorization of the tensor universal property as a special case of factoring linear maps through quotients.

Uniqueness of the Factorization

Determined by Values on Elementary Tensors

Because the elementary tensors v₁ ⊗ ... ⊗ vₙ span V₁ ⊗ ... ⊗ Vₙ, any two linear maps that agree on all elementary tensors agree everywhere. Since the factorization condition f̃(v₁ ⊗ ... ⊗ vₙ) = f(v₁, ..., vₙ) pins down the value of f̃ on every elementary tensor, and hence on every spanning element, there is only one linear map that can serve as the factorization of a given f.

Failure Without Linearity

If the requirement that the factorization be linear is dropped, uniqueness fails: many different functions defined on V₁ ⊗ ... ⊗ Vₙ could agree with f on elementary tensors while disagreeing on non-elementary tensors, since these admit multiple representations as sums of elementary tensors. Linearity is the condition that forces all such representations to be treated consistently, collapsing the many possible factorizations to a single one.

Factorization as a Universal Property

Initial Object Formulation

In category-theoretic language, factorization through the tensor product exhibits (V₁ ⊗ ... ⊗ Vₙ, ⊗) as an initial object in the category whose objects are pairs (U, g) consisting of a vector space U and a multilinear map g: V₁ × ... × Vₙ → U, and whose morphisms (U, g) → (U', g') are linear maps h: U → U' with g' = h ∘ g. Initiality means that from (V₁ ⊗ ... ⊗ Vₙ, ⊗) there is exactly one morphism to every other object (W, f) in this category, and that morphism is precisely the factorization f̃.

Uniqueness Up to Canonical Isomorphism

Initial objects in any category are unique up to a unique isomorphism. Consequently, if two different constructions both satisfy the tensor universal property with respect to the same multilinear maps, the factorizations each provides for the other's canonical map compose to the identity, and the two constructions are canonically identified. This is why factorization is treated as characterizing the tensor product rather than merely as a derived fact about one particular construction of it.

Illustrative Instances

Factoring a Bilinear Form

A bilinear form f: V × W → F factors through V ⊗ W as a linear functional f̃ ∈ (V ⊗ W)*. Concretely, if f(v, w) = v^T A w for a matrix A in coordinates, the factorization f̃ acts on a tensor ∑ vᵢ ⊗ wᵢ by summing vᵢ^T A wᵢ over the terms of any representation of that tensor, and the factorization property guarantees this sum is independent of which representation is used.

Factoring the Multiplication Map of an Algebra

For an associative algebra A, the multiplication map μ: A × A → A is bilinear and therefore factors through A ⊗ A as a linear map μ̃: A ⊗ A → A. This factored form of multiplication is the standard way multiplication is expressed in the axiomatic definition of an algebra as a vector space equipped with a linear map out of its own tensor square.

Factoring Tensor Contraction

A contraction pairing a vector space V with its dual V*, given by evaluation (φ, v) ↦ φ(v), is bilinear and factors through V* ⊗ V as the canonical linear evaluation map V* ⊗ V → F. Repeated contractions on higher tensor powers are built by applying this same factorization to each pair of dual slots in turn.

Why Factorization Matters Structurally

Reduces Multilinear Problems to Linear Ones

Once a multilinear map is known to factor through the tensor product, every question about it, injectivity, surjectivity, rank, invariant subspaces, can be studied using the tools of linear algebra applied to the single linear map f̃, rather than the comparatively awkward machinery required to analyze multilinearity directly in several variables.

Basis for Iterated Constructions

Factorization through a tensor product is applied repeatedly in the construction of exterior powers, symmetric powers, and general tensor algebras, where multilinear maps that are additionally alternating or symmetric are shown to factor further through the corresponding quotient of the tensor product, refining the same underlying factorization argument to enforce the additional symmetry or antisymmetry constraint.