Functoriality of the type of vectors

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

Created on 2022-03-15.
Last modified on 2024-02-06.

module linear-algebra.functoriality-vectors where
Imports
open import elementary-number-theory.natural-numbers

open import foundation.action-on-identifications-binary-functions
open import foundation.dependent-pair-types
open import foundation.function-extensionality
open import foundation.function-types
open import foundation.homotopies
open import foundation.identity-types
open import foundation.postcomposition-functions
open import foundation.universe-levels
open import foundation.whiskering-homotopies-composition

open import linear-algebra.vectors

open import univalent-combinatorics.standard-finite-types

Idea

Any map f : A → B determines a map vec n A → vec n B for every n.

Definition

Functoriality of the type of listed vectors

module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2}
  where

  map-vec : {n : }  (A  B)  vec A n  vec B n
  map-vec _ empty-vec = empty-vec
  map-vec f (x  xs) = f x  map-vec f xs

  htpy-vec :
    {n : } {f g : A  B}  (f ~ g)  map-vec {n = n} f ~ map-vec {n = n} g
  htpy-vec H empty-vec = refl
  htpy-vec H (x  v) = ap-binary _∷_ (H x) (htpy-vec H v)

Binary functoriality of the type of listed vectors

module _
  {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3}
  where

  binary-map-vec :
    {n : }  (A  B  C)  vec A n  vec B n  vec C n
  binary-map-vec f empty-vec empty-vec = empty-vec
  binary-map-vec f (x  v) (y  w) = f x y  binary-map-vec f v w

Functoriality of the type of functional vectors

module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2}
  where

  map-functional-vec :
    (n : )  (A  B)  functional-vec A n  functional-vec B n
  map-functional-vec n f v = f  v

  htpy-functional-vec :
    (n : ) {f g : A  B}  (f ~ g) 
    map-functional-vec n f ~ map-functional-vec n g
  htpy-functional-vec n = htpy-postcomp (Fin n)

Binary functoriality of the type of functional vectors

module _
  {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3}
  where

  binary-map-functional-vec :
    (n : )  (A  B  C) 
    functional-vec A n  functional-vec B n  functional-vec C n
  binary-map-functional-vec n f v w i = f (v i) (w i)
module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2}
  (f : A  B)
  where

  map-vec-map-functional-vec :
    (n : ) (v : vec A n) 
    listed-vec-functional-vec
      ( n)
      ( map-functional-vec n f (functional-vec-vec n v)) 
    map-vec f v
  map-vec-map-functional-vec zero-ℕ empty-vec = refl
  map-vec-map-functional-vec (succ-ℕ n) (x  v) =
    eq-Eq-vec
      ( succ-ℕ n)
      ( listed-vec-functional-vec
        ( succ-ℕ n)
        ( map-functional-vec
          ( succ-ℕ n)
          ( f)
          ( functional-vec-vec (succ-ℕ n) (x  v))))
      ( map-vec f (x  v))
      ( refl ,
        Eq-eq-vec
          ( n)
          ( listed-vec-functional-vec
            ( n)
            ( map-functional-vec n f (functional-vec-vec n v)))
          ( map-vec f v)
          ( map-vec-map-functional-vec n v))

Recent changes