1.8.1 Tensorial Abstraction

Tensorial Abstraction is the recognition that "tensorial" names a general pattern of behavior, naturality with respect to structure-preserving maps of the underlying spaces, rather than a fixed recipe tied to vector spaces and the classical transformation law, so that the label "tensorial" can be applied consistently to constructions built from modules, from bundles, or from other structured objects, wherever that same naturality pattern is exhibited. It isolates the criterion that makes an object deserve the name tensor, independent of which specific stage of abstraction, component array, multilinear map, or universal-property object, is used to present it.

The Criterion Behind the Name

Naturality as the Defining Test

A construction assigning to each vector space V some derived object F(V), such as V ⊗ V or V^*, is tensorial precisely when every linear isomorphism f : V → W induces, in a canonically determined way, a corresponding isomorphism F(f) : F(V) → F(W), compatible with composition of maps.

Reading the Classical Transformation Law as Naturality

The ordinary tensor transformation law is exactly this naturality condition specialized to the case where f is a change of basis: demanding that components computed in W via f match the components obtained by applying F(f) to the components computed in V is the classical transformation law rewritten in the language of naturality.

Tensorial Constructions Versus Ad Hoc Constructions

What Fails the Test

Not every assignment of a derived object to a vector space is tensorial. A construction that depends on an arbitrary, unmotivated choice, such as selecting one distinguished basis vector without reference to any structure of V and forming a derived object from it alone, generally fails to respect isomorphisms of V and so is not tensorial, even if it produces a perfectly well-defined vector space at each individual V.

Distinguishing the Two Cases by Example

The dual space V^* is tensorial, since every linear isomorphism of V induces a canonical linear isomorphism of V^*. By contrast, an assignment that picks out "the third basis vector" of V, which requires an arbitrary ordered basis to even be stated, is not tensorial, since a different choice of basis produces a different, unrelated answer with no canonical way to match the two.

Tensorial Abstraction Across Different Base Categories

Beyond Vector Spaces

Because the naturality criterion is stated only in terms of maps and their composition, it applies unchanged when the base objects are modules over a ring, sheaves, or bundles: a construction on any of these types of objects is called tensorial exactly when it respects the structure-preserving maps native to that setting, in precise analogy with the vector space case.

Vector Bundles as an Illustration

Applying the tensor product construction fiberwise to a vector bundle produces a new vector bundle, because the fiberwise construction is tensorial: the transition functions relating overlapping local trivializations of the original bundle induce, automatically and canonically, transition functions for the new bundle, with no additional choice required.

Multilinearity as a Special Case of Tensoriality

Multilinear Maps Are Themselves Tensorial Data

The multilinear maps used to define tensors are themselves an instance of tensorial abstraction: the assignment sending V to the space of multilinear maps on V is natural in V, since an isomorphism of V transports a multilinear map on V to a corresponding multilinear map on the isomorphic space, with the transport rule given exactly by precomposition with the inverse isomorphism.

Why Tensor Operations Preserve Tensoriality

Operations such as the tensor product of two tensorial constructions, or the dual of a tensorial construction, are again tensorial, because naturality is preserved under composition and combination of natural assignments, which is why the entire apparatus of tensor algebra, built from these elementary operations, remains tensorial at every stage.

The Practical Value of the Abstraction

A Test for Legitimate Definitions

Tensorial abstraction supplies a discipline for proposing new constructions in tensor algebra and its applications: before accepting a newly proposed derived quantity as meaningful independent of arbitrary choices, one checks whether it is tensorial, that is, whether it transforms correctly under every relevant isomorphism of the underlying data.

Guiding the Search for Coordinate-Free Formulations

Recognizing which quantities are tensorial and which are not guides the reformulation of coordinate-dependent computations into coordinate-free ones, since only tensorial quantities can appear in statements intended to hold independent of the coordinate system or basis used to derive them.

Diagrammatic Summary

The diagram shows the commuting square that constitutes the test for tensorial abstraction: a construction F is tensorial exactly when this square commutes for every structure-preserving map f, meaning the induced map F(f) correctly carries the derived object over V to the derived object over W.