12.17.4 Tensor Pushforward Composition Compatibility

Tensor Pushforward Composition Compatibility ensures consistent tensor transformation under multiple mappings, preserving structural integrity.

Tensor Pushforward Composition Compatibility is the property that pushing a contravariant tensor forward along a composite source map agrees exactly with pushing it forward in stages, one factor of the composite at a time, in the same order as the composition: (g ∘ f)* = g* ∘ f* for source maps f : V → W and g : W → U. This compatibility mirrors the pullback's composition compatibility but preserves, rather than reverses, the order of the maps involved, which is the defining structural signature separating the covariant pushforward from the contravariant pullback.

Statement and Direct Verification

The Composition Law

(g \circ f)_{*} T = g_{*} (f_{*} T)

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

Verification on Simple Tensors

(g \circ f)_{*} (v_{1} \otimes \dots \otimes v_{k}) = g (f (v_{1})) \otimes \dots \otimes g (f (v_{k}))

Each factor of a simple tensor transforms by the composite map g ∘ f, which by definition of function composition equals applying f first and then g to that same factor. Since this holds factor by factor, and the pushforward is defined by applying the map to each factor independently, the identity holds on every simple tensor, and extends to all contravariant tensors by linearity.

Diagram of the Two Equivalent Routes

Consequences of the Composition Compatibility

Associativity of Chained Pushforwards

(h \circ g \circ f)_{*} = h_{*} \circ g_{*} \circ f_{*}

For a chain of three or more composable source maps, the composition compatibility extends by repeated application, and the grouping of the intermediate pushforwards does not affect the final answer, exactly mirroring the associativity argument used for the pullback but with every arrow, and every composition, kept in forward order throughout.

Independence from the Chosen Factorization

If a linear map h : V → U admits two different factorizations h = g₁ ∘ f₁ = g₂ ∘ f₂ through possibly different intermediate spaces, the pushforward h*T computed directly agrees with the result computed via either factorization, since g₁*(f₁*T) = h*T = g₂*(f₂*T). The pushforward of T along h therefore depends only on h itself, never on how h happens to be expressed as a composite of simpler maps.

Identity Map as the Compatibility's Base Case

Setting g = id_W in the composition law gives (id_W ∘ f)* = (id_W)* ∘ f* = id ∘ f* = f*, confirming the identity source map behaves as expected under composition and anchoring the general law in its simplest instance, exactly paralleling the corresponding base case for the pullback.

Role in Building the Pushforward into a Functor

Covariant Functoriality

Composition compatibility, together with (id_V)* = id on contravariant tensors, establishes the pushforward assignment V ↦ Tₖ(V), f ↦ f* as a covariant functor from the category of vector spaces and linear maps to itself (restricted to rank-k contravariant tensor spaces). This is the direct structural counterpart to the pullback's contravariant functoriality, distinguished only by preserving rather than reversing the order in which morphisms compose.

Practical Use in Multi-Stage Coordinate or Frame Changes

When a contravariant tensor such as a vector or multivector is transported through several successive linear transformations — for instance, a sequence of frame changes in a computational pipeline — composition compatibility guarantees that computing the cumulative pushforward via the single overall composite map gives the same result as applying each transformation's pushforward one after another, which is what justifies treating a chain of transformations as interchangeable with their single composed equivalent throughout tensor computations.