Theorems · Definition · linear algebra

TensorProduct.tensorTensorTensorComm

(R : Type u_1) →
  [inst : CommSemiring R] →
    (M : Type u_5) →
      (N : Type u_6) →
        (P : Type u_7) →
          (Q : Type u_8) →
            [inst_1 : AddCommMonoid M] →
              [inst_2 : AddCommMonoid N] →
                [inst_3 : AddCommMonoid P] →
                  [inst_4 : AddCommMonoid Q] →
                    [inst_5 : Module R M] →
                      [inst_6 : Module R N] →
                        [inst_7 : Module R Q] →
                          [inst_8 : Module R P] →
                            TensorProduct R (TensorProduct R M N) (TensorProduct R P Q) ≃ₗ[R]
                              TensorProduct R (TensorProduct R M P) (TensorProduct R N Q)

This special case is worth defining explicitly since it is useful for defining multiplication on tensor products of modules carrying multiplications (e.g., associative rings, Lie rings, ...). E.g., suppose M = P and N = Q and that M and N carry bilinear multiplications: M ⊗ M → M and N ⊗ N → N. Using map, we can define (M ⊗ M) ⊗ (N ⊗ N) → M ⊗ N which, when combined with this definition, yields a bilinear multiplication on M ⊗ N: (M ⊗ N) ⊗ (M ⊗ N) → M ⊗ N. In particular we could use this to define the multiplication in the TensorProduct.semiring instance (currently defined "by hand" using TensorProduct.mul). See also mul_mul_mul_comm.

Defined in

Mathlib.LinearAlgebra.TensorProduct.Associator

Cited by

15 results in Mathlib

Foundations

Depth 70 from the axioms · uses propext, Quot.sound

Assumes

CommSemiringAddCommMonoidAddCommMonoidAddCommMonoidAddCommMonoidModuleModuleModuleModule