The interchange law

Content created by Fredrik Bakke, Egbert Rijke and Jonathan Prieto-Cubides.

Created on 2022-01-27.
Last modified on 2023-06-10.

module foundation.interchange-law where
open import foundation.action-on-identifications-functions
open import foundation.universe-levels

open import foundation-core.identity-types


The interchange law shows up in many variations in type theory. The interchange law for Σ-types is useful in characterizations of identity types; the interchange law for higher identity types is used in the Eckmann-Hilton argument to show that the higher homotopy groups of a type are abelian, and the interchange law for binary operations in rings are useful in computations. We first define a fully generalized interchange law, and then we specialize it to make it more directly applicable.


The fully generalized interchange law

module _
  {l1 l2 l3 l4 l5 l6 l7 l8 l9 l10 : Level}
  {A : UU l1} {B : A  UU l2} {C : A  UU l3} {D : (a : A)  B a  C a  UU l4}
  {X1 : UU l5} {Y1 : X1  UU l6} {X2 : UU l7} {Y2 : X2  UU l8}
  {Z : UU l9} (R : Z  Z  UU l10)
  (μ1 : (a : A)  B a  X1)
  (μ2 : (a : A) (b : B a) (c : C a)  D a b c  Y1 (μ1 a b))
  (μ3 : (x : X1)  Y1 x  Z)
  (μ4 : (a : A)  C a  X2)
  (μ5 : (a : A) (c : C a) (b : B a)  D a b c  Y2 (μ4 a c))
  (μ6 : (x : X2)  Y2 x  Z)

  fully-generalized-interchange-law : UU (l1  l2  l3  l4  l10)
  fully-generalized-interchange-law =
    (a : A) (b : B a) (c : C a) (d : D a b c) 
    R (μ3 (μ1 a b) (μ2 a b c d)) (μ6 (μ4 a c) (μ5 a c b d))

The interchange law for two binary operations on a type

module _
  {l : Level} {X : UU l} (μ : X  X  X)

  interchange-law : (X  X  X)  UU l
  interchange-law ν = (x y u v : X)  μ (ν x y) (ν u v)  ν (μ x u) (μ y v)

  interchange-law-commutative-and-associative :
    ((x y : X)  μ x y  μ y x)  ((x y z : X)  μ (μ x y) z  μ x (μ y z)) 
    interchange-law μ
  interchange-law-commutative-and-associative C A x y u v =
    ( A x y (μ u v)) 
    ( ( ap
        ( μ x)
        ( (inv (A y u v))  ((ap  z  μ z v) (C y u))  (A u y v)))) 
      ( inv (A x u (μ y v))))

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