3.16.4 Tensor Covector Pullback Composition Rule

The Tensor Covector Pullback Composition Rule describes how covector fields transform under smooth mappings, preserving tensor structure through pullback operations.

Tensor Covector Pullback Composition Rule is the identity stating that the pullback of a composite linear map equals the composition of the individual pullbacks taken in reverse order, so that for maps f: U → V and g: V → W, the pullback of g ∘ f satisfies (g ∘ f)* = f* ∘ g*. The rule captures the contravariant nature of the pullback operation: because pullback reverses the direction of every map it acts on, a chain of maps composed in one order gives rise to a chain of pullbacks composed in the opposite order.

Statement of the Rule

The Composite Map and Its Pullback

Consider two linear maps arranged so that the output of the first feeds into the second:

f : U \to V, g : V \to W

Their composite g ∘ f maps U directly to W. Applying the pullback construction to this composite produces a map between dual spaces:

(g \circ f) * : W^{*} \to U^{*}

The Reversed Composition of Individual Pullbacks

Each individual map also has its own pullback, f*: V* → U* and g*: W* → V*. The composition rule asserts that composing these two pullbacks in the order f* ∘ g* — applying g* first, then f* — reproduces exactly the pullback of the composite map:

(g \circ f) * = f^{*} \circ g^{*}

The order of the factors on the right side is the reverse of the order on the left, which is the essential content of the rule.

Verification of the Identity

Direct Computation on a Covector

To confirm the identity, let ω be an arbitrary covector in W* and let u be an arbitrary vector in U. Evaluating the left side using the definition of the pullback applied to the composite map gives:

((g \circ f) * ω) (u) = ω (g (f (u)))

Evaluating the right side by applying g* to ω first, then f* to the result, gives the same expression:

(f^{*} (g^{*} ω)) (u) = (g^{*} ω) (f (u)) = ω (g (f (u)))

Since both sides agree on every covector ω and every vector u, the two maps (g ∘ f)* and f* ∘ g* are equal.

Matrix Perspective

If f and g are represented by matrices, the matrix of the composite g ∘ f is the product of the matrices in the order g times f. The pullback of a linear map corresponds to the transpose of its matrix, and the transpose of a product reverses the order of multiplication:

{(G F)}^{T} = F^{T} G^{T}

This matrix identity is the coordinate expression of the composition rule, with F^T and G^T playing the roles of f* and g* respectively.

Functorial Significance

Contravariant Functor Behavior

The composition rule is precisely the condition required for the pullback assignment to define a contravariant functor from the category of vector spaces to itself: it sends each space to its dual and each map to its pullback, while reversing the order of every composition. A covariant assignment would instead preserve the order, (g ∘ f)* = g* ∘ f*, which the pullback does not satisfy.

Identity Map Compatibility

The composition rule is consistent with, and generalizes, the fact that the pullback of the identity map is itself the identity map on the dual space. Setting g equal to the identity in the composition rule reduces the identity (id ∘ f)* = f* ∘ id* to the tautology f* = f* ∘ id*, confirming that id* acts as a two-sided identity for pullback composition, consistent with the functor preserving identity morphisms.

Extension to Chains of Several Maps

Associativity Across Multiple Factors

The composition rule extends inductively to any finite chain of linear maps. For three maps f: U → V, g: V → W, h: W → X, applying the rule twice gives:

(h \circ g \circ f) * = f^{*} \circ g^{*} \circ h^{*}

The order of the pullbacks on the right is the complete reversal of the order of the original maps, regardless of how many factors are present in the chain.

Relevance to Tensor Field Pullback

When the composition rule is applied to smooth maps between manifolds rather than linear maps between vector spaces, it governs how pulled-back covector fields and higher-rank covariant tensor fields behave under composite coordinate changes or composite embeddings, ensuring that pulling a tensor field back along a composite map gives the same result as pulling it back one map at a time in reverse order.