14.14.3 Tensor Map Product Domain Functoriality

Tensor Map Product Domain Functoriality explores how tensor maps interact with product domains through structured transformations in algebraic structures.

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

Fixing the Second Factor

The Partial Assignment

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

T \mapsto T \otimes I_{V_{2}}

Why This Deserves Separate Attention

Although the full tensor product construction takes two maps as input, fixing one of them and studying how the result depends on the other in isolation reveals a functor of a single variable, which is easier to analyze and serves as a building block for understanding the full two-variable construction.

Verifying Functoriality in This Single Variable

Preservation of the Identity

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

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

Diagram of Domain Functoriality

Varying the First Factor While the Second Stays Fixed

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

Relationship to the Full Bifunctor

One Half of the Two Independent Functorial Directions

Domain functoriality is one of the two directions in which the tensor product construction behaves functorially in a single variable, the other being the analogous statement obtained by fixing the first factor and varying the second instead.

Recombining Both Directions

Any statement about the full two-variable functorial behavior can be recovered by combining domain functoriality, applied to the first factor, with the corresponding statement for the second factor, using the composition compatibility of combined operators to merge the two partial results into the general two-factor combined operator.

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

Consequences for the First Factor Alone

Invertibility Transfers Through Domain Functoriality

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

Kernel and Image Behavior

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

Matrix-Level View

Kronecker Product With a Fixed Identity Block

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

Practical Use in Restricting Attention to One Factor

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