3.16.3 Tensor Covector Pullback Domain Reversal

Tensor Covector Pullback Domain Reversal is the specific instance of dual map domain reversal that occurs within the pullback operation, whereby the pullback map f*: W* → V* associated with a source map f: V → W has as its domain the dual of f's codomain rather than the dual of f's domain. Because the pullback is defined by precomposition, (f*φ)(v) = φ(f(v)), the covector φ being pulled back must already live in W*, the dual of the space f maps into, and the result lands in V*, the dual of the space f maps out of, producing the same reversal of domain and codomain that characterizes dual maps in general, specialized to the pullback context.

The Reversal Specific to Pullback

Domain of the Pullback Is Dual to the Codomain of f

For the pullback operation f*: W* → V* built from f: V → W, the domain of f* is W*, which is dual not to the domain of f but to its codomain:

This is the reason the pullback operation is said to reverse domain: the covector inputs it accepts correspond to the far end of f, not the near end, because a covector must be defined on whatever space f actually lands in before it can be composed with f.

Why the Pullback Cannot Take Inputs from V*

A covector ψ ∈ V* cannot serve as input to the pullback formula ψ(f(v)), because f(v) is an element of W, not V, and ψ is only defined to accept elements of V. The domain reversal is therefore forced by a basic type mismatch: only covectors defined on the codomain W are composable with f in the manner the pullback requires.

Consequences for the Pullback's Structure

The Pullback as a Map Out of W*

Domain reversal means the pullback operation should be understood as originating at W*, moving toward V*. Every property of the pullback as a linear map, such as its kernel, its image, and its rank, must be analyzed with W* playing the role of domain and V* playing the role of codomain, exactly reversed from the roles V and W play for the original map f.

Kernel of the Pullback

The kernel of the pullback operation, viewed as a subset of its domain W*, consists of all covectors φ ∈ W* that vanish on the entire image of f:

Because this kernel is a subset of W*, domain reversal is what makes it meaningful to speak of the kernel of the pullback as living inside the dual of f's codomain rather than inside the dual of its domain.

Interaction with Rank-Nullity

Rank-Nullity Applied to the Reversed Domain

Since f* is a linear map with domain W*, the rank-nullity theorem applies to f* using the dimension of W* as the relevant total:

Domain reversal is essential to applying this theorem correctly: without recognizing that W*, not V*, is the domain of f*, one would use the wrong dimension on the left-hand side of the identity.

Matching Codomain of f to Domain of f*

Domain reversal in the pullback context matches precisely codomain reversal for the same construction: the codomain W of f becomes, upon dualization, the domain W* of f*. The two phenomena, domain reversal and codomain reversal, are two names for the same single exchange of roles applied to the pullback operation specifically.

Reversal Across Chains of Pullbacks

Composite Source Maps

When pullbacks are chained through a composite source map g ∘ f: V → U, with f: V → W and g: W → U, the domain of the composite pullback (g ∘ f)* is U*, the dual of the final codomain in the chain, consistent with domain reversal applied to the composite as a whole rather than to each factor separately. The intermediate space W* appears only as a stepping stone, entered when applying g* and exited when applying f*.

Domain Reversal Preserved at Every Stage

At each stage of a chain of pullbacks, domain reversal holds individually: the domain of g* is U*, dual to the codomain of g, and the domain of f* is W*, dual to the codomain of f. The consistency of domain reversal at every individual stage is what guarantees the composite pullback also satisfies domain reversal relative to the composite source map.

Diagrammatic Summary

The diagram labels W* explicitly as the domain of the pullback operation, positioned at the same end as the codomain W of the source map f, making visible that the pullback's domain sits at the opposite end of the diagram from the domain of f itself.