14.14.4 Tensor Map Product Codomain Functoriality

Tensor Map Product Codomain Functoriality describes how tensor product maps interact with codomain structures through functorial properties in algebraic contexts.

Tensor Map Product Codomain Functoriality is the functorial behavior of the tensor product construction when the first factor space is held fixed and only the second factor varies, treating the assignment that sends a map on the second factor to its tensor product with a fixed identity on the first factor as a functor in its own right.

Fixing the First Factor

The Partial Assignment

With the first factor space and its identity map held fixed, the tensor product construction becomes an assignment that takes any linear map on the second factor space and produces a linear map on the tensor product space.

T \mapsto I_{V_{1}} \otimes T

Complement to Domain Functoriality

This assignment mirrors domain functoriality with the roles of the two factors exchanged, so every statement established for domain functoriality has a direct counterpart here, obtained simply by swapping which factor is held fixed and which one varies.

Verifying Functoriality in This Single Variable

Preservation of the Identity

Sending the identity map on the second factor through this partial assignment produces the identity map on the tensor product space, since the identity on the first factor combined with the identity on the second factor is the identity on the whole tensor product space.

I_{V_{1}} \otimes I_{V_{2}} = I_{V_{1} \otimes V_{2}}

Preservation of Composition

Sending a composition of two maps on the second factor through this partial assignment produces the same result as composing the images of the two individual maps under the assignment, since the first factor's identity contributes nothing to disrupt this composition.

I_{V_{1}} \otimes (S_{2} \circ T_{2}) = (I_{V_{1}} \otimes S_{2}) \circ (I_{V_{1}} \otimes T_{2})

Diagram of Codomain Functoriality

Varying the Second Factor While the First Stays Fixed

The diagram below shows a fixed first factor carrying only the identity, held alongside a single factor space varying while its map is applied, mapping to the resulting combined operator on the tensor product space.

Relationship to the Full Bifunctor

The Second of the Two Independent Directions

Codomain functoriality is the second of the two directions in which the tensor product construction behaves functorially in a single variable, complementing domain functoriality, which handles the case of the first factor varying instead.

Recombining Both Directions

The general two-variable combined operator can be recovered by combining the codomain functoriality result for the second factor with the domain functoriality result for the first factor, using the composition compatibility of combined operators to merge the two partial results together.

(T_{1} \otimes T_{2}) = (I_{V_{1}} \otimes T_{2}) \circ (T_{1} \otimes I_{V_{2}})

Consequences for the Second Factor Alone

Invertibility Transfers Through Codomain Functoriality

If a map on the second factor is invertible, its image under the codomain functoriality assignment is also invertible, with inverse equal to the tensor product of the identity on the first factor with the original map's inverse, following directly from the general rule for inverses of combined operators applied to this restricted case.

(I_{V_{1}} \otimes T)^{- 1} = I_{V_{1}} \otimes T^{- 1}

Kernel and Image Behavior

The kernel of the combined operator produced by codomain functoriality consists of all tensors formed from the entire first factor space tensored with the kernel of the second factor's map, since only the second component can be sent to zero while the first component, acted on by the identity, remains unchanged.

Matrix-Level View

Kronecker Product With a Fixed Identity Block

Relative to fixed bases, codomain functoriality corresponds to the matrix operation of forming the Kronecker product of a fixed identity matrix with a varying matrix, producing a block-diagonal composite matrix whose diagonal blocks are exact copies of the varying matrix.

Practical Use in Restricting Attention to One Factor

This matrix-level view is used whenever a computation needs to study how changes to the second factor alone propagate through the tensor product construction, without needing to account for simultaneous changes in the first factor.