12.16.4 Tensor Pullback Composition Compatibility

Tensor Pullback Composition Compatibility is the property that pulling a tensor back along a composite source map agrees exactly with pulling it back in stages, one factor of the composite at a time, in the reverse order of composition: (g ∘ f)* = f* ∘ g* for source maps f : V → W and g : W → U. This compatibility is what allows a long chain of linear maps to be processed either all at once as a single composite pullback or piecewise as a sequence of simpler pullbacks, with both routes guaranteed to produce identical results.

Statement and Direct Verification

The Composition Law

For a covariant tensor T on U, pulling T back along the single composite map g ∘ f : V → U produces the same tensor on V as first pulling T back along g to get a tensor on W, and then pulling that result back along f to get a tensor on V.

Verification by Direct Substitution

Written out for a single argument v (the argument for all k slots when T has covariant rank k), the identity follows from nothing more than the associativity of ordinary function composition: g(f(v)) can be grouped either as evaluating T at the fully composed point, or as evaluating g*T at f(v). Both groupings describe the identical underlying computation T(g(f(v))), which is why the composition compatibility holds unconditionally, for every linear f, g, and every covariant tensor T, with no additional hypotheses required.

Diagram of the Two Equivalent Routes

Consequences of the Composition Compatibility

Associativity of Chained Pullbacks

For a chain of three or more composable maps, the composition compatibility extends by repeated application, and the order in which the intermediate pullbacks are grouped does not affect the final answer, since ordinary function composition is associative and the pullback identity reduces entirely to this associativity, slot by slot.

Independence from the Chosen Factorization

If a single linear map h : V → U happens to admit two different factorizations, h = g₁ ∘ f₁ = g₂ ∘ f₂, through possibly different intermediate spaces, the pullback h*T computed as a single step agrees with the result computed via either factorization, since composition compatibility guarantees f₁*(g₁*T) = h*T = f₂*(g₂*T). This means the pullback of T along h depends only on h itself, never on how h happens to be expressed as a composite.

Identity Map as the Compatibility's Base Case

Setting g = id_W (or f = id_V) in the composition law recovers the fact that (id_W ∘ f)* = f* ∘ (id_W)* = f* ∘ id = f*, confirming that the identity source map acts as expected under composition and anchoring the general law in the simplest possible case.

Role in Building the Pullback into a Functor

Contravariant Functoriality

Composition compatibility, together with the fact that (id_V)* = id on tensors, is precisely what is required to establish the pullback assignment V ↦ T⁰ₖ(V), f ↦ f* as a contravariant functor from the category of vector spaces and linear maps to the category of vector spaces (of covariant k-tensors) and linear maps. Composition compatibility is the half of the functor laws governing how composite morphisms behave; the identity law is the other half, and together they are exactly the two conditions a contravariant functor must satisfy.

Practical Use in Multi-Stage Coordinate Changes

When a tensor is transported through several successive changes of coordinates, each represented by its own linear map, composition compatibility guarantees that computing the cumulative pullback via the single overall composite map gives the same coordinate expression as applying each coordinate change's pullback one after another — a fact routinely relied upon without restating it explicitly whenever tensors are converted through a sequence of intermediate bases.