14.14 Tensor Map Product Functorial Behavior

Tensor Map Product Functorial Behavior is the property that forming the tensor product of maps respects the identity map and the composition of maps in a manner consistent with the general notion of a functor, meaning that the tensor product construction sends identity maps to identity maps and sends a composition of maps to the composition of the corresponding tensor products.

Preservation of Identities

Sending Identity to Identity

The tensor product of the identity map on the first factor with the identity map on the second factor produces the identity map on the tensor product of the two factor spaces, matching the requirement that a functor must send identity morphisms to identity morphisms.

Significance for the Functorial Reading

This preservation is the first of two conditions required for a construction to qualify as a functor, and it holds directly as a consequence of the tensor identity map product preservation property established for combined operators.

Preservation of Composition

Sending Composition to Composition

The tensor product of two composed maps in each factor equals the composition of the tensor products taken factor by factor, matching the second requirement of a functor, that composition of morphisms is preserved.

Relationship to the Composition Compatibility Property

This preservation of composition is the same relationship already established under the composition compatibility of combined operators, here reframed explicitly as one of the two defining conditions of functorial behavior.

Diagram of the Functorial Picture

Two Categories Connected by the Tensor Product Construction

The diagram below depicts the tensor product construction as an arrow-preserving assignment from pairs of vector spaces and maps to single vector spaces and maps.

Bifunctoriality

Two Independent Slots

Because the tensor product accepts a pair of maps, one for each factor, and preserves identities and composition independently in each slot, the construction is described as a bifunctor: fixing the second factor and letting the first vary behaves functorially, and fixing the first factor and letting the second vary behaves functorially as well.

Consistency Between the Two Slots

The two independent functorial behaviors combine consistently, since fixing one factor to the identity and varying the other, then fixing the other factor to the identity and varying the first, produces the same overall combined operator as directly combining both nontrivial maps at once, exactly as shown earlier in the composition compatibility discussion.

Consequences of Functorial Behavior

Predictable Behavior Under Substitution

Because the construction is functorial, replacing a factor map with a composition of two simpler maps and then forming the tensor product produces a predictable, decomposable result, matching the composition of the tensor products of the simpler pieces.

Compatibility With Invertible Maps

Functorial behavior guarantees that if both factor maps are invertible, their tensor product is invertible with inverse equal to the tensor product of the individual inverses, since a functor sends invertible morphisms to invertible morphisms whenever composition and identities are both preserved.

Extension to Several Factors

Functorial Behavior With Many Simultaneous Slots

When the tensor product involves three or more factor spaces, the construction remains functorial in each individual slot simultaneously, preserving identities and composition independently for every factor while the other factors are held fixed.

Iterated Application Consistent With Associativity

Applying the functorial behavior repeatedly across several factors is consistent with the associativity of the tensor product of maps, since regrouping the factors before or after applying the functorial preservation of identities and composition produces the same final combined operator either way.