14.13.2 Tensor Map Product Associative Isomorphism

The Tensor Map Product Associative Isomorphism establishes a structural equivalence between tensor products under associative mappings in algebraic frameworks.

Tensor Map Product Associative Isomorphism is the specific natural isomorphism between two differently parenthesized tensor product spaces that underlies and justifies the associativity behavior of tensor products of maps, matching each elementary tensor formed under one grouping to the corresponding elementary tensor formed under the other grouping.

The Isomorphism Itself

Domain and Codomain of the Isomorphism

The associative isomorphism is a linear bijection between the tensor product space formed by grouping the first two factors first and the tensor product space formed by grouping the last two factors first.

α : (V_{1} \otimes V_{2}) \otimes V_{3} \to V_{1} \otimes (V_{2} \otimes V_{3})

Action on Elementary Tensors

The isomorphism acts on elementary tensors by simply regrouping the three vectors involved, without changing the vectors themselves or their order.

α ((v_{1} \otimes v_{2}) \otimes v_{3}) = v_{1} \otimes (v_{2} \otimes v_{3})

Naturality of the Isomorphism

Commuting With Tensor Products of Maps

The isomorphism commutes with the tensor product of any three maps in the sense that applying the maps under one grouping and then reassociating gives the same result as reassociating first and then applying the maps under the other grouping.

α \circ [(T_{1} \otimes T_{2}) \otimes T_{3}] = [T_{1} \otimes (T_{2} \otimes T_{3})] \circ α

Consequence of Naturality

This commuting relationship is precisely what allows the associativity statement for tensor products of maps to be phrased as an equality rather than merely a correspondence, since the natural isomorphism identifies the two codomain spaces so tightly that the two combined operators can be regarded as literally the same map.

Diagram of the Isomorphism

Square Diagram Commuting

The diagram below shows the associative isomorphism connecting the two groupings both before and after applying the tensor product of three maps, with the resulting square commuting.

Invertibility of the Isomorphism

Inverse Isomorphism

The associative isomorphism has an inverse that reverses the regrouping, sending an elementary tensor grouped one way back to the elementary tensor grouped the other way, and this inverse is itself a natural isomorphism of the same kind.

α^{- 1} (v_{1} \otimes (v_{2} \otimes v_{3})) = (v_{1} \otimes v_{2}) \otimes v_{3}

Dimension Preservation

Because the isomorphism is a linear bijection, the two tensor product spaces it connects have exactly the same dimension, consistent with the fact that both are built from the same three underlying factor spaces regardless of how they are grouped.

Extension Beyond Three Factors

Isomorphisms for Longer Chains

For four or more factors, a network of associative isomorphisms connects every possible grouping of the factors to every other grouping, with each individual isomorphism in the network built from the same basic three-factor regrouping applied at different positions within the longer chain.

Coherence of the Network

A key feature of this network of isomorphisms is coherence: any two paths of individual regrouping steps that connect the same starting grouping to the same ending grouping produce the same overall isomorphism, which is what allows parentheses to be suppressed entirely without ambiguity in the general case.