4.12.5 Tensor Partial Evaluation Tensor Role

Tensor Partial Evaluation Tensor Role is the formal identification of the object produced by partial evaluation, the tensor reduced arity result, as itself a genuine tensor belonging to an appropriate tensor product space, rather than merely a function that happens to behave in a tensor-like way. It establishes that partial evaluation stays entirely within the category of tensors: fixing some arguments of a tensor never produces something outside the tensorial framework, but only a smaller tensor built from fewer copies of the underlying vector space and its dual.

From Multilinear Behavior to Tensor Identity

The Universal Property Behind the Identification

Any multilinear map on a finite collection of vector spaces and their duals corresponds to a unique linear map out of the tensor product of those spaces, by the universal property defining the tensor product; since the tensor remaining slot map obtained by partial evaluation is multilinear on the open slots, this universal property applies directly to it, yielding a unique element of the tensor product space matching those open slots.

Constructing the Corresponding Tensor Product Element

Concretely, if partial evaluation of a type (p, q) tensor T fixes a subset S of slots and leaves p - k contravariant and q - (m - k) covariant slots open, the reduced arity result corresponds to a unique element

so that the reduced arity result is not merely function-like but is literally a tensor of type (p - k, q - (m - k)), occupying exactly the tensor product space that the classification of multilinear maps by type would predict.

Partial Evaluation as a Map Between Tensor Spaces

Fixing Slots as a Linear Operation on Tensors

Viewed at the level of tensor product spaces, partial evaluation with a given fixed slot evaluation is itself a linear map, sending the original tensor T, or more generally any tensor of the same type, to its corresponding reduced arity result:

where T^p_q(V) denotes the space of type (p, q) tensors on V. This map φ_S, determined entirely by the fixed slot evaluation, is linear because it is built from contracting the fixed slots of its input against the chosen fixed values, and contraction against fixed values is a linear operation on the input tensor.

Consistency with the Tensor Contraction Operation

The map φ_S realizing partial evaluation at the level of tensor spaces coincides exactly with performing tensor contraction against the fixed values in each of the slots in S, confirming that partial evaluation is not a separate operation invented alongside contraction but a specific instance of contraction applied to a subset of the available slots.

Functorial Behavior of the Tensor Role

Composability Reflected at the Tensor Product Level

Because partial evaluation over a subset S_1 followed by partial evaluation over a disjoint subset S_2 agrees with a single partial evaluation over S_1 ∪ S_2, the corresponding linear maps between tensor spaces compose in the same way:

confirming that the tensor role assigned to partial evaluation is consistent across every way of building up the same overall fixed slot evaluation from smaller steps.

The Identity Case and the Full Evaluation Case

When S is empty, φ_S is the identity map on T^p_q(V), since no slots have been fixed; when S includes every slot, φ_S reduces to the ordinary tensor multilinear evaluation operation, sending T to a single scalar, matching the type (0, 0) tensor space, which is naturally identified with the base field itself.

Why the Tensor Role Matters

Preserving the Algebraic Structure of Tensors

Establishing that reduced arity results are genuine tensors, rather than merely tensor-like functions, ensures that all of the ordinary algebraic operations available to tensors, such as addition, scalar multiplication, further contraction, and further partial evaluation, apply equally well to the results of partial evaluation, without requiring any special-case treatment.

Enabling Iterated Tensor Constructions

Because a reduced arity result is itself a tensor occupying its own tensor product space, it can serve as the starting point for further tensor operations, including additional partial evaluations, tensor products with other tensors, or further identification via the tensor vector valued tensor role if the reduced result is later reinterpreted as a vector-valued map; the tensor role of partial evaluation is what makes such iterated constructions well-defined.

Diagrammatic Summary

The diagram shows partial evaluation as a linear map φ_S sending the original tensor space T^p_q(V) to a strictly smaller tensor product space, confirming that the reduced arity result always occupies a genuine tensor role within the algebra of tensors.