5.21 Tensor Product Associativity Structure

Tensor Product Associativity Structure is the natural isomorphism (U ⊗ V) ⊗ W ≅ U ⊗ (V ⊗ W) relating the two possible ways of forming an iterated tensor product of three vector spaces, together with the coherence conditions ensuring that this isomorphism is canonical, basis-independent, and compatible with tensoring further factors. This associativity structure justifies writing an unparenthesized triple (or higher) tensor product U ⊗ V ⊗ W without ambiguity, and it is a prerequisite for treating the tensor product as a genuinely associative binary operation on vector spaces up to canonical isomorphism.

Statement of Associativity

The Two Parenthesizations

Given vector spaces U, V, W over a field F, there are two evident ways to iterate the tensor product: first combining U and V and then tensoring with W, or first combining V and W and then tensoring with U:

The Associativity Isomorphism

There is a canonical linear isomorphism

determined uniquely by its action on simple tensors, α((u ⊗ v) ⊗ w) = u ⊗ (v ⊗ w), identifying the two parenthesizations as the same object up to this canonical relabeling.

Construction of the Isomorphism

Building the Map via the Universal Property

For fixed w ∈ W, the map (u, v) ↦ u ⊗ (v ⊗ w) is bilinear in (u, v), so it induces a linear map (U ⊗ V) → U ⊗ (V ⊗ W) sending u ⊗ v ↦ u ⊗ (v ⊗ w). Varying w bilinearly and invoking the universal property once more produces the full map α : (U ⊗ V) ⊗ W → U ⊗ (V ⊗ W) on simple tensors of the triple product, with the analogous construction in the opposite direction producing an inverse map, confirming α is an isomorphism.

Verifying the Inverse

The map β : U ⊗ (V ⊗ W) → (U ⊗ V) ⊗ W defined symmetrically by β(u ⊗ (v ⊗ w)) = (u ⊗ v) ⊗ w satisfies β ∘ α = id and α ∘ β = id on simple tensors of each respective space, and since simple tensors span, these identities extend to the whole spaces, establishing α as a genuine isomorphism rather than merely an injective or surjective map.

Diagram of the Associativity Isomorphism

Coherence and Naturality

Naturality with Respect to Linear Maps

The isomorphism α is natural: for linear maps f : U → U′, g : V → V′, h : W → W′, the diagram formed by α, (f ⊗ g) ⊗ h, and f ⊗ (g ⊗ h) commutes,

meaning the identification of the two parenthesizations does not depend on which representative spaces U, V, W are used, only on the tensor product structure itself.

The Pentagon Coherence Condition

For four factors, there are five ways to parenthesize U ⊗ V ⊗ W ⊗ X, connected by instances of α; Mac Lane's pentagon coherence condition states that all paths of associativity isomorphisms between any two parenthesizations agree, so no ambiguity arises from choosing a particular sequence of reassociations, however many factors are involved. This coherence is what licenses writing multi-factor tensor products without parentheses at all.

Consequences of Associativity

Well-Defined Multi-Factor Tensor Products

Because of the associativity isomorphism and its coherence, the unparenthesized expression V₁ ⊗ V₂ ⊗ ... ⊗ Vₙ denotes a single, unambiguous vector space up to canonical isomorphism, regardless of the order in which the pairwise tensor products are actually formed to construct it.

Compatibility with Dimension and Bases

Associativity is consistent with the dimension relation, since dim((U ⊗ V) ⊗ W) = dim(U)dim(V)dim(W) = dim(U ⊗ (V ⊗ W)), and the basis {(eᵢ ⊗ fⱼ) ⊗ gₖ} of the left parenthesization corresponds under α exactly to the basis {eᵢ ⊗ (fⱼ ⊗ gₖ)} of the right parenthesization, matching term by term.

Significance of Associativity Structure

Foundation for Symmetric Monoidal Category Structure

Associativity, together with the unit isomorphisms F ⊗ V ≅ V and the symmetry isomorphism V ⊗ W ≅ W ⊗ V, equips the category of vector spaces with the structure of a symmetric monoidal category, the categorical framework in which tensor products, and their coherence properties, are studied in full generality.

Practical License for Multi-Index Tensor Notation

Associativity structure is what ultimately justifies the informal but ubiquitous practice, throughout multilinear algebra and its applications, of treating an n-fold tensor product as a single object with n independent indices rather than as a nested, parenthesis-dependent construction, since the coherent associativity isomorphisms guarantee all parenthesizations agree.