14.8.4 Tensor Map Product Codomain Compatibility

Tensor Map Product Codomain Compatibility is the requirement that the stated codomains of the two maps forming a factor map pair match, factor by factor, the actual spaces used to build the codomain tensor product, ensuring that the tensor product of maps can be regarded as landing correctly in that particular tensor product.

Statement of the Compatibility Requirement

Matching Codomains to Tensor Factors

For a tensor product of maps to be regarded as landing in a specific codomain tensor space W1 tensor W2, codomain compatibility requires that the map f assigned to the first factor have codomain exactly W1, and the map g assigned to the second factor have codomain exactly W2,

Unlike domain compatibility, a failure of codomain compatibility does not prevent the elementary output rule from being evaluated, since f(v) and g(w) are computed without reference to any particular codomain tensor space; instead, a failure of codomain compatibility means only that the resulting output cannot be regarded as an element of the specific tensor space W1 tensor W2 originally intended.

Compatibility Through Enlargement

Because a map's stated codomain need only contain its image rather than equal it exactly, codomain compatibility is more permissive than domain compatibility: a map with a smaller image can still be regarded as codomain-compatible with a larger stated codomain space, so long as that larger space is the one intended for the tensor product of maps to land in.

Consequences of Codomain Compatibility

Determining the Ambient Space of the Output

Codomain compatibility is what fixes the ambient tensor product space in which the output of f tensor g is considered to reside; without it, an output such as f(v) tensor g(w) could be regarded as lying in any tensor product built from spaces containing the images of f and g, rather than in one specific, agreed-upon codomain tensor space.

Necessity for Composability

Codomain compatibility of one tensor product of maps with the domain compatibility of a following tensor product of maps in a composition chain is what permits the two to be composed at all, since the intermediate tensor space must serve simultaneously as codomain for the first and as domain for the second.

Codomain Compatibility Under Restriction and Extension

Restoring Compatibility by Restriction to the Image

If codomain compatibility is desired with respect to the smallest possible codomain tensor space, the codomains of f and g can each be restricted to their respective images, producing a tensor product of maps that is codomain-compatible with the tensor product of these two image spaces rather than with any larger ambient tensor space.

Compatibility Preserved Under Passing to a Larger Codomain

If W1 and W2 are each replaced by larger containing spaces, codomain compatibility is preserved automatically provided the replacement is consistent on both factors, since enlarging the codomain of f to some space containing W1, and correspondingly enlarging the codomain of g, produces a tensor product of maps compatible with the tensor product of the two enlarged spaces without requiring any change to the underlying elementary output rule.

Codomain Compatibility Under Composition

Compatibility Required at the Junction of a Composition Chain

When composing a tensor product of maps f tensor g with a following tensor product of maps f prime tensor g prime, codomain compatibility of f tensor g with the tensor space W1 tensor W2 must coincide with domain compatibility of f prime tensor g prime with that same tensor space, since the composition identity for tensor products of maps is only meaningful once this junction is compatible on both sides.

Compatibility as a Precondition for the Composition Identity

Just as domain compatibility is a standing precondition for the composition identity of tensor products of maps, codomain compatibility at each stage of a composition chain is likewise a precondition rather than a consequence, ensuring that every intermediate tensor space in the chain is correctly identified before the composition identity is invoked to simplify the resulting composite map.