12.19.3 Tensor Composition Closure Context

Tensor Composition Closure Context is the specific setting — a designated collection of maps or operations acting on a fixed tensor space, or a fixed chain of spaces connected by source maps — within which composing two or more tensor operations is guaranteed to produce another operation still belonging to that same designated collection, as opposed to producing a composite that falls outside it. Where the general tensor operation closure property asks whether a single operation's output stays within its input's space, composition closure context asks the analogous question one level up: whether chaining several operations together stays within whatever restricted family of operations was originally being considered.

The Context-Dependence of Composition Closure

Composition Is Always Defined, Closure Is Not Automatic

Any two composable linear maps, or any two composable tensor operations with matching domain and codomain, can always be composed as a raw operation. Whether the resulting composite belongs to some further-restricted collection of interest — endomorphisms of a single space, structure-preserving maps, maps of a bounded rank — is a separate question, answered only relative to a specifically named context.

Naming the Context Precisely

Composition closure is never asked in the abstract; it is always asked relative to an explicitly identified collection, such as "linear endomorphisms of V," "inclusions into a fixed ambient space T," or "source maps used in a fixed chain of pullbacks." The composition closure context is exactly this named collection, and the closure question only has content once the context has been fixed.

Contexts in Which Composition Is Closed

Endomorphisms of a Fixed Space

If both f and g are linear maps from V to itself, their composite g ∘ f is again a linear map from V to itself. This is the paradigm case of a composition closure context: the collection of endomorphisms of a single fixed space V is closed under composition, which is precisely what allows this collection to be studied as an algebraic structure (a monoid under composition, or a group when restricted to invertible endomorphisms) in its own right.

Idempotent Projections Are Not Automatically Closed

Composing two projections π₁ and π₂ on the same space produces another linear endomorphism, but that composite is not automatically idempotent unless π₁ and π₂ commute; the context "linear endomorphisms of V" is closed under composition, but the narrower context "idempotent endomorphisms (projections) of V" is closed under composition only under this additional commutativity condition, illustrating that composition closure context must be checked at exactly the level of restriction actually intended.

Chained Pullbacks Along a Fixed Sequence of Spaces

Within the context of a fixed sequence of vector spaces connected by source maps, the pullback operations f* and g* compose to give f* ∘ g* = (g∘f)*, a pullback again associated with a map in the same sequence (now the composite map g∘f : V → U). This is a composition closure context specific to the pullback operation, distinct from the endomorphism context, since it tracks composability of the source maps themselves rather than composability of a fixed space's self-maps.

Contexts in Which Composition Is Not Closed

Inclusions Compose to Inclusions, But Only Within a Compatible Chain

If ι₁ : U → W and ι₂ : W → T are both inclusions, their composite ι₂ ∘ ι₁ is again an inclusion (linear and injective), so the context "inclusions along a fixed chain U → W → T" is closed under this specific composition. However, this closure depends on the two inclusions sharing a common intermediate space W; two arbitrary inclusions into unrelated target spaces, with no shared intermediate, are simply not composable in the first place, so the question of closure does not even arise outside a properly chained context.

Projections onto Different Summands Generally Fail to Compose Within the Same Restricted Context

Composing the projection π₁ onto summand V₁ of a direct sum with the projection π₂ onto a different summand V₂ produces the zero map, which is technically still an endomorphism but no longer belongs to the narrower context of "nonzero projections onto a designated summand," showing that even a composition landing formally within a larger context can still exit a more narrowly specified one.

Diagram of Composition Closure Within a Fixed Context

Establishing a Composition Closure Context

Verifying Closure Requires Checking Every Defining Property

To confirm a given collection is a valid composition closure context, every property defining membership in that collection — linearity, injectivity, idempotency, a fixed domain and codomain, symmetry preservation — must be individually verified to survive composition, since a collection closed with respect to one defining property is not automatically closed with respect to another imposed alongside it.

Practical Use in Restricting Attention to a Manageable Family

Identifying a valid composition closure context allows a computation involving many successive tensor operations to be analyzed entirely within that fixed, closed family, guaranteeing in advance that no intermediate step in a long chain of compositions will unexpectedly produce an object falling outside the family under study, and thereby justifying treating the whole family as a single coherent algebraic structure for the purposes of that computation.