{-# OPTIONS --safe #-}
module Cubical.Categories.Category.Properties where

open import Cubical.Foundations.Prelude
open import Cubical.Foundations.HLevels

open import Cubical.Categories.Category.Base

open Category

private
  variable
     ℓ' : Level

module _ {C : Category  ℓ'} where

  -- isSet where your allowed to compare paths where one side is only
  -- equal up to path. Is there a generalization of this?
  isSetHomP1 :  {x y : C .ob} {n : C [ x , y ]}
               isOfHLevelDep 1  m  m  n)
  isSetHomP1 {x = x} {y} {n} = isOfHLevel→isOfHLevelDep 1  m  isSetHom C m n)

  -- isSet where the arrows can be between non-definitionally equal obs
  isSetHomP2l :  {y : C .ob}
               isOfHLevelDep 2  x  C [ x , y ])
  isSetHomP2l = isOfHLevel→isOfHLevelDep 2  a  isSetHom C {x = a})

  isSetHomP2r :  {x : C .ob}
               isOfHLevelDep 2  y  C [ x , y ])
  isSetHomP2r = isOfHLevel→isOfHLevelDep 2  a  isSetHom C {y = a})


-- opposite of opposite is definitionally equal to itself
involutiveOp :  {C : Category  ℓ'}  C ^op ^op  C
involutiveOp = refl

module _ {C : Category  ℓ'} where
  -- Other useful operations on categories

  -- whisker the parallel morphisms g and g' with f
  lCatWhisker : {x y z : C .ob} (f : C [ x , y ]) (g g' : C [ y , z ]) (p : g  g')
                  f ⋆⟨ C  g  f ⋆⟨ C  g'
  lCatWhisker f _ _ p = cong (_⋆_ C f) p

  rCatWhisker : {x y z : C .ob} (f f' : C [ x , y ]) (g : C [ y , z ]) (p : f  f')
                  f ⋆⟨ C  g  f' ⋆⟨ C  g
  rCatWhisker _ _ g p = cong  v  v ⋆⟨ C  g) p

  -- working with equal objects
  idP :  {x x'} {p : x  x'}  C [ x , x' ]
  idP {x} {x'} {p} = subst  v  C [ x , v ]) p (C .id)

  -- heterogeneous seq
  seqP :  {x y y' z} {p : y  y'}
        (f : C [ x , y ]) (g : C [ y' , z ])
        C [ x , z ]
  seqP {x} {_} {_} {z} {p} f g = f ⋆⟨ C  (subst  a  C [ a , z ]) (sym p) g)

  -- also heterogeneous seq, but substituting on the left
  seqP' :  {x y y' z} {p : y  y'}
         (f : C [ x , y ]) (g : C [ y' , z ])
         C [ x , z ]
  seqP' {x} {_} {_} {z} {p} f g = subst  a  C [ x , a ]) p f ⋆⟨ C  g

  -- show that they're equal
  seqP≡seqP' :  {x y y' z} {p : y  y'}
              (f : C [ x , y ]) (g : C [ y' , z ])
              seqP {p = p} f g  seqP' {p = p} f g
  seqP≡seqP' {x = x} {z = z} {p = p} f g i =
    (toPathP {A = λ i'  C [ x , p i' ]} {f} refl i)
      ⋆⟨ C 
    (toPathP {A = λ i'  C [ p (~ i') , z ]} {x = g} (sym refl) (~ i))

  -- seqP is equal to normal seq when y ≡ y'
  seqP≡seq :  {x y z}
            (f : C [ x , y ]) (g : C [ y , z ])
            seqP {p = refl} f g  f ⋆⟨ C  g
  seqP≡seq {y = y} {z} f g i = f ⋆⟨ C  toPathP {A = λ _  C [ y , z ]} {x = g} refl (~ i)


  -- whiskering with heterogenous seq -- (maybe should let z be heterogeneous too)
  lCatWhiskerP : {x y z y' : C .ob} {p : y  y'}
                   (f : C [ x , y ]) (g : C [ y , z ]) (g' : C [ y' , z ])
                   (r : PathP  i  C [ p i , z ]) g g')
                   f ⋆⟨ C  g  seqP {p = p} f g'
  lCatWhiskerP {z = z} {p = p} f g g' r =
    cong  v  f ⋆⟨ C  v) (sym (fromPathP (symP {A = λ i  C [ p (~ i) , z ]} r)))

  rCatWhiskerP : {x y' y z : C .ob} {p : y'  y}
                   (f' : C [ x , y' ]) (f : C [ x , y ]) (g : C [ y , z ])
                   (r : PathP  i  C [ x , p i ]) f' f)
                   f ⋆⟨ C  g  seqP' {p = p} f' g
  rCatWhiskerP f' f g r = cong  v  v ⋆⟨ C  g) (sym (fromPathP r))


  AssocCong₂⋆L : {x y' y z w : C .ob} 
          {f' : C [ x , y' ]} {f : C [ x , y ]}
          {g' : C [ y' , z ]} {g : C [ y , z ]}
           f ⋆⟨ C  g  f' ⋆⟨ C  g'  (h : C [ z , w ])
           f ⋆⟨ C  (g ⋆⟨ C  h) 
              f' ⋆⟨ C  (g' ⋆⟨ C  h)
  AssocCong₂⋆L p h =
    sym (⋆Assoc C _ _ h)
      ∙∙  i  p i ⋆⟨ C  h) ∙∙
    ⋆Assoc C _ _ h

  AssocCong₂⋆R : {x y z z' w : C .ob} 
          (f : C [ x , y ])
          {g' : C [ y , z' ]} {g : C [ y , z ]}
          {h' : C [ z' , w ]} {h : C [ z , w ]}
           g ⋆⟨ C  h  g' ⋆⟨ C  h'
           (f ⋆⟨ C  g) ⋆⟨ C  h 
              (f ⋆⟨ C  g') ⋆⟨ C  h'
  AssocCong₂⋆R f p =
    ⋆Assoc C f _ _
      ∙∙  i  f ⋆⟨ C  p i) ∙∙
    sym (⋆Assoc C _ _ _)