Maps between globular types

Content created by Fredrik Bakke and Vojtěch Štěpančík.

Created on 2024-06-16.
Last modified on 2024-06-16.

{-# OPTIONS --guardedness #-}

module structured-types.maps-globular-types where
Imports
open import foundation.dependent-pair-types
open import foundation.function-types
open import foundation.identity-types
open import foundation.universe-levels

open import structured-types.globular-types

Idea

A map f between globular types A and B is a map F₀ of -cells, and for every pair of -cells x and y, a map of -cells

  Fₙ₊₁ : (𝑛+1)-Cell A x y → (𝑛+1)-Cell B (Fₙ x) (Fₙ y).

Definitions

Maps between globular types

record
  map-Globular-Type
  {l1 l2 l3 l4 : Level} (A : Globular-Type l1 l2) (B : Globular-Type l3 l4)
  : UU (l1  l2  l3  l4)
  where
  coinductive
  field
    0-cell-map-Globular-Type :
      0-cell-Globular-Type A  0-cell-Globular-Type B

    globular-type-1-cell-map-Globular-Type :
      {x y : 0-cell-Globular-Type A} 
      map-Globular-Type
        ( globular-type-1-cell-Globular-Type A x y)
        ( globular-type-1-cell-Globular-Type B
          ( 0-cell-map-Globular-Type x)
          ( 0-cell-map-Globular-Type y))

open map-Globular-Type public

module _
  {l1 l2 l3 l4 : Level}
  {A : Globular-Type l1 l2} {B : Globular-Type l3 l4}
  (F : map-Globular-Type A B)
  where

  1-cell-map-Globular-Type :
    {x y : 0-cell-Globular-Type A} 
    1-cell-Globular-Type A x y 
    1-cell-Globular-Type B
      ( 0-cell-map-Globular-Type F x)
      ( 0-cell-map-Globular-Type F y)
  1-cell-map-Globular-Type =
    0-cell-map-Globular-Type (globular-type-1-cell-map-Globular-Type F)

module _
  {l1 l2 l3 l4 : Level}
  {A : Globular-Type l1 l2} {B : Globular-Type l3 l4}
  (F : map-Globular-Type A B)
  where

  2-cell-map-Globular-Type :
    {x y : 0-cell-Globular-Type A}
    {f g : 1-cell-Globular-Type A x y} 
    2-cell-Globular-Type A f g 
    2-cell-Globular-Type B
      ( 1-cell-map-Globular-Type F f)
      ( 1-cell-map-Globular-Type F g)
  2-cell-map-Globular-Type =
    1-cell-map-Globular-Type (globular-type-1-cell-map-Globular-Type F)

module _
  {l1 l2 l3 l4 : Level}
  {A : Globular-Type l1 l2} {B : Globular-Type l3 l4}
  (F : map-Globular-Type A B)
  where

  3-cell-map-Globular-Type :
    {x y : 0-cell-Globular-Type A}
    {f g : 1-cell-Globular-Type A x y} 
    {H K : 2-cell-Globular-Type A f g} 
    3-cell-Globular-Type A H K 
    3-cell-Globular-Type B
      ( 2-cell-map-Globular-Type F H)
      ( 2-cell-map-Globular-Type F K)
  3-cell-map-Globular-Type =
    2-cell-map-Globular-Type (globular-type-1-cell-map-Globular-Type F)

The identity map on a globular type

id-map-Globular-Type :
  {l1 l2 : Level} (A : Globular-Type l1 l2)  map-Globular-Type A A
id-map-Globular-Type A =
  λ where
  .0-cell-map-Globular-Type  id
  .globular-type-1-cell-map-Globular-Type {x} {y} 
    id-map-Globular-Type (globular-type-1-cell-Globular-Type A x y)

Composition of maps of globular types

comp-map-Globular-Type :
  {l1 l2 l3 l4 l5 l6 : Level}
  {A : Globular-Type l1 l2}
  {B : Globular-Type l3 l4}
  {C : Globular-Type l5 l6} 
  map-Globular-Type B C  map-Globular-Type A B  map-Globular-Type A C
comp-map-Globular-Type g f =
  λ where
  .0-cell-map-Globular-Type 
    0-cell-map-Globular-Type g  0-cell-map-Globular-Type f
  .globular-type-1-cell-map-Globular-Type 
    comp-map-Globular-Type
      ( globular-type-1-cell-map-Globular-Type g)
      ( globular-type-1-cell-map-Globular-Type f)

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