Library HoTT.WildCat.Coproducts
Require Import Basics.Overture Basics.Tactics Basics.Decidable.
Require Import Types.Bool.
Require Import WildCat.Core WildCat.Equiv WildCat.Forall WildCat.NatTrans
WildCat.Opposite WildCat.Products WildCat.Universe
WildCat.Yoneda WildCat.ZeroGroupoid WildCat.PointedCat
WildCat.Monoidal WildCat.Bifunctor.
Require Import Types.Bool.
Require Import WildCat.Core WildCat.Equiv WildCat.Forall WildCat.NatTrans
WildCat.Opposite WildCat.Products WildCat.Universe
WildCat.Yoneda WildCat.ZeroGroupoid WildCat.PointedCat
WildCat.Monoidal WildCat.Bifunctor.
Definition cat_coprod_rec_inv {I A : Type} `{Is1Cat A}
(coprod : A) (x : I → A) (z : A) (inj : ∀ i, x i $-> coprod)
: yon_0gpd z coprod $-> prod_0gpd I (fun i ⇒ yon_0gpd z (x i))
:= cat_prod_corec_inv (coprod : A^op) x z inj.
Class Coproduct (I : Type) {A : Type} `{Is1Cat A} (x : I → A)
:= prod_co_coprod :: Product (A:=A^op) I x.
Definition cat_coprod (I : Type) {A : Type} (x : I → A) `{Coproduct I _ x} : A
:= cat_prod (A:=A^op) I x.
Definition cat_in {I : Type} {A : Type} {x : I → A} `{Coproduct I _ x}
: ∀ (i : I), x i $-> cat_coprod I x
:= cat_pr (A:=A^op) (x:=x).
Instance cat_isequiv_cat_coprod_rec_inv {I : Type} {A : Type}
{x : I → A} `{Coproduct I _ x}
: ∀ (z : A), CatIsEquiv (cat_coprod_rec_inv (cat_coprod I x) x z cat_in)
:= cat_isequiv_cat_prod_corec_inv (A:=A^op) (x:=x).
A convenience wrapper for building coproducts
Definition Build_Coproduct (I : Type) {A : Type} `{Is1Cat A} {x : I → A}
(cat_coprod : A) (cat_in : ∀ i : I, x i $-> cat_coprod)
(cat_coprod_rec : ∀ z : A,
(∀ i : I, x i $-> z) → (cat_coprod $-> z))
(cat_coprod_beta_in : ∀ (z : A) (f : ∀ i, x i $-> z) (i : I),
cat_coprod_rec z f $o cat_in i $== f i)
(cat_prod_eta_in : ∀ (z : A) (f g : cat_coprod $-> z),
(∀ i : I, f $o cat_in i $== g $o cat_in i) → f $== g)
: Coproduct I x
:= Build_Product I
(cat_coprod : A^op)
cat_in
cat_coprod_rec
cat_coprod_beta_in
cat_prod_eta_in.
Section Lemmata.
Context (I : Type) {A : Type} {x : I → A} `{Coproduct I _ x}.
Definition cate_cat_coprod_rec_inv {z : A}
: yon_0gpd z (cat_coprod I x) $<~> prod_0gpd I (fun i ⇒ yon_0gpd z (x i))
:= cate_cat_prod_corec_inv I (A:=A^op) (x:=x).
Definition cate_cat_coprod_rec {z : A}
: prod_0gpd I (fun i ⇒ yon_0gpd z (x i)) $<~> yon_0gpd z (cat_coprod I x)
:= cate_cat_prod_corec I (A:=A^op) (x:=x).
Definition cat_coprod_rec {z : A}
: (∀ i, x i $-> z) → cat_coprod I x $-> z
:= cat_prod_corec I (A:=A^op) (x:=x).
Definition cat_coprod_beta {z : A} (f : ∀ i, x i $-> z)
: ∀ i, cat_coprod_rec f $o cat_in i $== f i
:= cat_prod_beta I (A:=A^op) (x:=x) f.
Definition cat_coprod_eta {z : A} (f : cat_coprod I x $-> z)
: cat_coprod_rec (fun i ⇒ f $o cat_in i) $== f
:= cat_prod_eta I (A:=A^op) (x:=x) f.
Definition natequiv_cat_coprod_rec_inv
: NatEquiv (fun z ⇒ yon_0gpd z (cat_coprod I x))
(fun z : A ⇒ prod_0gpd I (fun i ⇒ yon_0gpd z (x i)))
:= natequiv_cat_prod_corec_inv I (A:=A^op) (x:=x).
Definition cat_coprod_rec_eta {z : A} {f g : ∀ i, x i $-> z}
: (∀ i, f i $== g i) → cat_coprod_rec f $== cat_coprod_rec g
:= cat_prod_corec_eta I (A:=A^op) (x:=x).
Definition cat_coprod_in_eta {z : A} {f g : cat_coprod I x $-> z}
: (∀ i, f $o cat_in i $== g $o cat_in i) → f $== g
:= cat_prod_pr_eta I (A:=A^op) (x:=x).
End Lemmata.
(cat_coprod : A) (cat_in : ∀ i : I, x i $-> cat_coprod)
(cat_coprod_rec : ∀ z : A,
(∀ i : I, x i $-> z) → (cat_coprod $-> z))
(cat_coprod_beta_in : ∀ (z : A) (f : ∀ i, x i $-> z) (i : I),
cat_coprod_rec z f $o cat_in i $== f i)
(cat_prod_eta_in : ∀ (z : A) (f g : cat_coprod $-> z),
(∀ i : I, f $o cat_in i $== g $o cat_in i) → f $== g)
: Coproduct I x
:= Build_Product I
(cat_coprod : A^op)
cat_in
cat_coprod_rec
cat_coprod_beta_in
cat_prod_eta_in.
Section Lemmata.
Context (I : Type) {A : Type} {x : I → A} `{Coproduct I _ x}.
Definition cate_cat_coprod_rec_inv {z : A}
: yon_0gpd z (cat_coprod I x) $<~> prod_0gpd I (fun i ⇒ yon_0gpd z (x i))
:= cate_cat_prod_corec_inv I (A:=A^op) (x:=x).
Definition cate_cat_coprod_rec {z : A}
: prod_0gpd I (fun i ⇒ yon_0gpd z (x i)) $<~> yon_0gpd z (cat_coprod I x)
:= cate_cat_prod_corec I (A:=A^op) (x:=x).
Definition cat_coprod_rec {z : A}
: (∀ i, x i $-> z) → cat_coprod I x $-> z
:= cat_prod_corec I (A:=A^op) (x:=x).
Definition cat_coprod_beta {z : A} (f : ∀ i, x i $-> z)
: ∀ i, cat_coprod_rec f $o cat_in i $== f i
:= cat_prod_beta I (A:=A^op) (x:=x) f.
Definition cat_coprod_eta {z : A} (f : cat_coprod I x $-> z)
: cat_coprod_rec (fun i ⇒ f $o cat_in i) $== f
:= cat_prod_eta I (A:=A^op) (x:=x) f.
Definition natequiv_cat_coprod_rec_inv
: NatEquiv (fun z ⇒ yon_0gpd z (cat_coprod I x))
(fun z : A ⇒ prod_0gpd I (fun i ⇒ yon_0gpd z (x i)))
:= natequiv_cat_prod_corec_inv I (A:=A^op) (x:=x).
Definition cat_coprod_rec_eta {z : A} {f g : ∀ i, x i $-> z}
: (∀ i, f i $== g i) → cat_coprod_rec f $== cat_coprod_rec g
:= cat_prod_corec_eta I (A:=A^op) (x:=x).
Definition cat_coprod_in_eta {z : A} {f g : cat_coprod I x $-> z}
: (∀ i, f $o cat_in i $== g $o cat_in i) → f $== g
:= cat_prod_pr_eta I (A:=A^op) (x:=x).
End Lemmata.
Definition cat_coprod_codiag {I : Type} {A : Type} (x : A) `{Coproduct I _ (fun _ ⇒ x)}
: cat_coprod I (fun _ ⇒ x) $-> x
:= cat_prod_diag (A:=A^op) x.
Definition cate_cat_coprod {I J : Type} (ie : I <~> J) {A : Type} `{HasEquivs A}
(x : I → A) `{!Coproduct I x} (y : J → A) `{!Coproduct J y}
(e : ∀ (i : I), y (ie i) $<~> x i)
: cat_coprod J y $<~> cat_coprod I x
:= cate_cat_prod (A:=A^op) ie x y e.
(x : I → A) `{!Coproduct I x} (y : J → A) `{!Coproduct J y}
(e : ∀ (i : I), y (ie i) $<~> x i)
: cat_coprod J y $<~> cat_coprod I x
:= cate_cat_prod (A:=A^op) ie x y e.
Class HasCoproducts (I A : Type) `{Is1Cat A}
:= has_coproducts :: ∀ x : I → A, Coproduct I x.
Class HasAllCoproducts (A : Type) `{Is1Cat A}
:= has_all_coproducts :: ∀ I : Type, HasCoproducts I A.
Local Instance hasproductsop_hascoproducts {I A : Type} `{HasCoproducts I A}
: HasProducts I A^op
:= fun x : I → A^op ⇒ @has_coproducts I A _ _ _ _ _ x.
Instance is0functor_cat_coprod (I : Type) `{IsGraph I}
(A : Type) `{HasCoproducts I A}
: @Is0Functor (I → A) A (isgraph_forall I (fun _ ⇒ A)) _
(fun x : I → A ⇒ cat_coprod I x).
Proof.
apply is0functor_op'.
exact (is0functor_cat_prod I A^op).
Defined.
Instance is1functor_cat_coprod (I : Type) `{IsGraph I}
(A : Type) `{HasCoproducts I A}
: @Is1Functor (I → A) A _ _ _ (is1cat_forall I (fun _ ⇒ A)) _ _ _ _
(fun x : I → A ⇒ cat_coprod I x) _.
Proof.
apply is1functor_op'.
exact (is1functor_cat_prod I A^op).
Defined.
Definition isinitial_coprodempty {A : Type} {z : A}
`{Coproduct Empty A (fun _ ⇒ z)}
: IsInitial (cat_coprod Empty (fun _ ⇒ z)).
Proof.
intros a.
snrefine (cat_coprod_rec _ _; fun f ⇒ cat_coprod_in_eta _ _); intros [].
Defined.
Class BinaryCoproduct {A : Type} `{Is1Cat A} (x y : A)
:= prod_co_bincoprod :: BinaryProduct (A:=A^op) x y.
Definition cat_bincoprod' {A : Type} `{Is1Cat A} (x y : A) `{!BinaryCoproduct x y} : A
:= cat_binprod' (x : A^op) y.
Definition cat_inl {A : Type} `{Is1Cat A} {x y : A} `{!BinaryCoproduct x y}
: x $-> cat_bincoprod' x y
:= cat_pr1 (A:=A^op) (x:=x) (y:=y).
Definition cat_inr {A : Type} `{Is1Cat A} {x y : A} `{!BinaryCoproduct x y}
: y $-> cat_bincoprod' x y
:= cat_pr2 (A:=A^op) (x:=x) (y:=y).
A category with binary coproducts is a category with a binary coproduct for each pair of objects.
Class HasBinaryCoproducts (A : Type) `{Is1Cat A}
:= binary_coproducts :: ∀ x y : A, BinaryCoproduct x y.
Instance hasbinarycoproducts_hascoproductsbool {A : Type}
`{HasCoproducts Bool A}
: HasBinaryCoproducts A
:= fun x y ⇒ has_coproducts (fun b : Bool ⇒ if b then x else y).
:= binary_coproducts :: ∀ x y : A, BinaryCoproduct x y.
Instance hasbinarycoproducts_hascoproductsbool {A : Type}
`{HasCoproducts Bool A}
: HasBinaryCoproducts A
:= fun x y ⇒ has_coproducts (fun b : Bool ⇒ if b then x else y).
A convenience wrapper for building binary coproducts
Definition Build_BinaryCoproduct {A : Type} `{Is1Cat A} {x y : A}
(cat_coprod : A) (cat_inl : x $-> cat_coprod) (cat_inr : y $-> cat_coprod)
(cat_coprod_rec : ∀ z : A, (x $-> z) → (y $-> z) → cat_coprod $-> z)
(cat_coprod_beta_inl : ∀ (z : A) (f : x $-> z) (g : y $-> z),
cat_coprod_rec z f g $o cat_inl $== f)
(cat_coprod_beta_inr : ∀ (z : A) (f : x $-> z) (g : y $-> z),
cat_coprod_rec z f g $o cat_inr $== g)
(cat_coprod_in_eta : ∀ (z : A) (f g : cat_coprod $-> z),
f $o cat_inl $== g $o cat_inl → f $o cat_inr $== g $o cat_inr → f $== g)
: BinaryCoproduct x y
:= Build_BinaryProduct
(cat_coprod : A^op)
cat_inl
cat_inr
cat_coprod_rec
cat_coprod_beta_inl
cat_coprod_beta_inr
cat_coprod_in_eta.
Section Lemmata.
Context {A : Type} {x y z : A} `{BinaryCoproduct _ x y}.
Definition cat_bincoprod_rec (f : x $-> z) (g : y $-> z)
: cat_bincoprod' x y $-> z
:= @cat_binprod_corec A^op _ _ _ _ x y _ _ f g.
Definition cat_bincoprod_beta_inl (f : x $-> z) (g : y $-> z)
: cat_bincoprod_rec f g $o cat_inl $== f
:= @cat_binprod_beta_pr1 A^op _ _ _ _ x y _ _ f g.
Definition cat_bincoprod_beta_inr (f : x $-> z) (g : y $-> z)
: cat_bincoprod_rec f g $o cat_inr $== g
:= @cat_binprod_beta_pr2 A^op _ _ _ _ x y _ _ f g.
Definition cat_bincoprod_eta (f : cat_bincoprod' x y $-> z)
: cat_bincoprod_rec (f $o cat_inl) (f $o cat_inr) $== f
:= @cat_binprod_eta A^op _ _ _ _ x y _ _ f.
Definition cat_bincoprod_eta_in {f g : cat_bincoprod' x y $-> z}
: f $o cat_inl $== g $o cat_inl → f $o cat_inr $== g $o cat_inr → f $== g
:= @cat_binprod_eta_pr A^op _ _ _ _ x y _ _ f g.
Definition cat_bincoprod_rec_eta {f f' : x $-> z} {g g' : y $-> z}
: f $== f' → g $== g' → cat_bincoprod_rec f g $== cat_bincoprod_rec f' g'
:= @cat_binprod_corec_eta A^op _ _ _ _ x y _ _ f f' g g'.
End Lemmata.
Definition cat_bincoprod {A: Type} `{HasBinaryCoproducts A} (x y : A) := cat_bincoprod' x y.
(cat_coprod : A) (cat_inl : x $-> cat_coprod) (cat_inr : y $-> cat_coprod)
(cat_coprod_rec : ∀ z : A, (x $-> z) → (y $-> z) → cat_coprod $-> z)
(cat_coprod_beta_inl : ∀ (z : A) (f : x $-> z) (g : y $-> z),
cat_coprod_rec z f g $o cat_inl $== f)
(cat_coprod_beta_inr : ∀ (z : A) (f : x $-> z) (g : y $-> z),
cat_coprod_rec z f g $o cat_inr $== g)
(cat_coprod_in_eta : ∀ (z : A) (f g : cat_coprod $-> z),
f $o cat_inl $== g $o cat_inl → f $o cat_inr $== g $o cat_inr → f $== g)
: BinaryCoproduct x y
:= Build_BinaryProduct
(cat_coprod : A^op)
cat_inl
cat_inr
cat_coprod_rec
cat_coprod_beta_inl
cat_coprod_beta_inr
cat_coprod_in_eta.
Section Lemmata.
Context {A : Type} {x y z : A} `{BinaryCoproduct _ x y}.
Definition cat_bincoprod_rec (f : x $-> z) (g : y $-> z)
: cat_bincoprod' x y $-> z
:= @cat_binprod_corec A^op _ _ _ _ x y _ _ f g.
Definition cat_bincoprod_beta_inl (f : x $-> z) (g : y $-> z)
: cat_bincoprod_rec f g $o cat_inl $== f
:= @cat_binprod_beta_pr1 A^op _ _ _ _ x y _ _ f g.
Definition cat_bincoprod_beta_inr (f : x $-> z) (g : y $-> z)
: cat_bincoprod_rec f g $o cat_inr $== g
:= @cat_binprod_beta_pr2 A^op _ _ _ _ x y _ _ f g.
Definition cat_bincoprod_eta (f : cat_bincoprod' x y $-> z)
: cat_bincoprod_rec (f $o cat_inl) (f $o cat_inr) $== f
:= @cat_binprod_eta A^op _ _ _ _ x y _ _ f.
Definition cat_bincoprod_eta_in {f g : cat_bincoprod' x y $-> z}
: f $o cat_inl $== g $o cat_inl → f $o cat_inr $== g $o cat_inr → f $== g
:= @cat_binprod_eta_pr A^op _ _ _ _ x y _ _ f g.
Definition cat_bincoprod_rec_eta {f f' : x $-> z} {g g' : y $-> z}
: f $== f' → g $== g' → cat_bincoprod_rec f g $== cat_bincoprod_rec f' g'
:= @cat_binprod_corec_eta A^op _ _ _ _ x y _ _ f f' g g'.
End Lemmata.
Definition cat_bincoprod {A: Type} `{HasBinaryCoproducts A} (x y : A) := cat_bincoprod' x y.
Binary coproduct functor
Instance is0functor_cat_bincoprod_l {A : Type}
`{hbc : HasBinaryCoproducts A} y
: Is0Functor (A:=A) (fun x ⇒ cat_bincoprod x y).
Proof.
rapply is0functor_op'.
exact (is0functor_cat_binprod_l (A:=A^op) (H0:=hbc) y).
Defined.
Instance is1functor_cat_bincoprod_l {A : Type}
`{hbc : HasBinaryCoproducts A} y
: Is1Functor (fun x ⇒ cat_bincoprod x y).
Proof.
rapply is1functor_op'.
exact (is1functor_cat_binprod_l (A:=A^op) (H0:=hbc) y).
Defined.
Instance is0functor_cat_bincoprod_r {A : Type}
`{hbc : HasBinaryCoproducts A} x
: Is0Functor (cat_bincoprod x).
Proof.
rapply is0functor_op'.
exact (is0functor_cat_binprod_r (A:=A^op) (H0:=hbc) x).
Defined.
Instance is1functor_cat_bincoprod_r {A : Type}
`{hbc : HasBinaryCoproducts A} x
: Is1Functor (cat_bincoprod x).
Proof.
rapply is1functor_op'.
exact (is1functor_cat_binprod_r (A:=A^op) (H0:=hbc) x).
Defined.
Instance is0bifunctor_cat_bincoprod {A : Type}
`{hbc : HasBinaryCoproducts A}
: Is0Bifunctor cat_bincoprod.
Proof.
napply is0bifunctor_op'.
exact (is0bifunctor_cat_binprod (A:=A^op) (H0:=hbc)).
Defined.
Instance is1bifunctor_cat_bincoprod {A : Type}
`{hbc : HasBinaryCoproducts A}
: Is1Bifunctor cat_bincoprod.
Proof.
napply is1bifunctor_op'.
exact (is1bifunctor_cat_binprod (A:=A^op) (H0:=hbc)).
Defined.
Definition hasbinarycoproducts_op_hasbinaryproducts {A : Type}
`{hbp : HasBinaryProducts A}
: HasBinaryCoproducts A^op
:= hbp.
Hint Immediate hasbinarycoproducts_op_hasbinaryproducts : typeclass_instances.
Definition hasbinarycoproducts_hasbinaryproducts_op {A : Type}
`{Is1Cat A, hbp : !HasBinaryProducts A^op}
: HasBinaryCoproducts A
:= hbp.
Hint Immediate hasbinarycoproducts_hasbinaryproducts_op : typeclass_instances.
Definition hasbinaryproducts_op_hasbinarycoproducts {A : Type}
`{hbc : HasBinaryCoproducts A}
: HasBinaryProducts A^op
:= hbc.
Hint Immediate hasbinarycoproducts_op_hasbinaryproducts : typeclass_instances.
Definition hasbinaryproducts_hasbinarycoproducts_op {A : Type}
`{Is1Cat A, hbc : !HasBinaryCoproducts A^op}
: HasBinaryProducts A
:= hbc.
Hint Immediate hasbinaryproducts_hasbinarycoproducts_op : typeclass_instances.
Definition cat_bincoprod_swap {A : Type} `{Is1Cat A}
{hbc : HasBinaryCoproducts A} (x y : A)
: cat_bincoprod x y $-> cat_bincoprod y x.
Proof.
exact (@cat_binprod_swap A^op _ _ _ _ hbc _ _).
Defined.
Definition cate_bincoprod_swap {A : Type} `{HasEquivs A}
{hbc : HasBinaryCoproducts A} (x y : A)
: cat_bincoprod x y $<~> cat_bincoprod y x.
Proof.
exact (@cate_binprod_swap A^op _ _ _ _ _ hbc _ _).
Defined.
Lemma cate_coprod_assoc {A : Type} `{HasEquivs A}
{hbc : HasBinaryCoproducts A} (x y z : A)
: cat_bincoprod x (cat_bincoprod y z)
$<~> cat_bincoprod (cat_bincoprod x y) z.
Proof.
exact (@associator_cat_binprod A^op _ _ _ _ _ hbc x y z)^-1$.
Defined.
Definition associator_cat_bincoprod {A : Type} `{HasEquivs A}
`{!HasBinaryCoproducts A}
: Associator (fun x y ⇒ cat_bincoprod x y).
Proof.
unfold Associator.
snapply associator_op'.
1: exact _.
exact associator_cat_binprod.
Defined.
Definition cat_bincoprod_codiag {A : Type}
`{Is1Cat A} (x : A) `{!BinaryCoproduct x x}
: cat_bincoprod' x x $-> x
:= cat_binprod_diag (A:=A^op) x.
Lemmas about cat_bincoprod_rec
Definition cat_bincoprod_fmap01_rec {A : Type}
`{Is1Cat A, !HasBinaryCoproducts A} {w x y z : A}
(f : z $-> w) (g : y $-> x) (h : x $-> w)
: cat_bincoprod_rec f h $o fmap01 (fun x y ⇒ cat_bincoprod x y) z g
$== cat_bincoprod_rec f (h $o g)
:= @cat_binprod_fmap01_corec A^op _ _ _ _
hasbinaryproducts_op_hasbinarycoproducts _ _ _ _ f g h.
Definition cat_bincoprod_fmap10_rec {A : Type}
`{Is1Cat A, !HasBinaryCoproducts A} {w x y z : A}
(f : y $-> x) (g : x $-> w) (h : z $-> w)
: cat_bincoprod_rec g h $o fmap10 (fun x y ⇒ cat_bincoprod x y) f z
$== cat_bincoprod_rec (g $o f) h
:= @cat_binprod_fmap10_corec A^op _ _ _ _
hasbinaryproducts_op_hasbinarycoproducts _ _ _ _ f g h.
Definition cat_bincoprod_fmap11_rec {A : Type}
`{Is1Cat A, !HasBinaryCoproducts A} {v w x y z : A}
(f : y $-> w) (g : z $-> x) (h : w $-> v) (i : x $-> v)
: cat_bincoprod_rec h i $o fmap11 cat_bincoprod f g
$== cat_bincoprod_rec (h $o f) (i $o g)
:= @cat_binprod_fmap11_corec A^op _ _ _ _
hasbinaryproducts_op_hasbinarycoproducts _ _ _ _ _ f g h i.
Definition cat_bincoprod_rec_associator {A : Type} `{HasEquivs A}
{hbc : HasBinaryCoproducts A}
{w x y z : A} (f : w $-> z) (g : x $-> z) (h : y $-> z)
: cat_bincoprod_rec (cat_bincoprod_rec f g) h $o associator_cat_bincoprod w x y
$== cat_bincoprod_rec f (cat_bincoprod_rec g h).
Proof.
napply cate_moveR_eV.
symmetry.
exact (cat_binprod_associator_corec
(HasBinaryProducts0:=hasbinaryproducts_op_hasbinarycoproducts (hbc:=hbc))
f g h).
Defined.
Definition cat_bincoprod_swap_rec {A : Type} `{Is1Cat A}
`{!HasBinaryCoproducts A} {a b c : A} (f : a $-> c) (g : b $-> c)
: cat_bincoprod_rec f g $o cat_bincoprod_swap b a $== cat_bincoprod_rec g f
:= @cat_binprod_swap_corec A^op _ _ _ _
hasbinaryproducts_op_hasbinarycoproducts _ _ _ _ _.
Instance ismonoidal_cat_bincoprod {A : Type} `{HasEquivs A}
`{!HasBinaryCoproducts A} (zero : A) `{!IsInitial zero}
: IsMonoidal A cat_bincoprod zero | 10.
Proof.
napply ismonoidal_op'.
napply (ismonoidal_cat_binprod (A:=A^op) zero).
by napply isterminal_op_isinitial.
Defined.
Instance issymmetricmonoidal_cat_bincoprod {A : Type} `{HasEquivs A}
`{!HasBinaryCoproducts A} (zero : A) `{!IsInitial zero}
: IsSymmetricMonoidal A cat_bincoprod zero | 10.
Proof.
napply issymmetricmonoidal_op'.
napply (issymmetricmonoidal_cat_binprod (A:=A^op) zero).
by napply isterminal_op_isinitial.
Defined.
Instance hasallcoproducts_type : HasAllCoproducts Type.
Proof.
intros I x.
snapply Build_Coproduct.
- exact (sig (fun i : I ⇒ x i)).
- exact (exist x).
- intros A f [i xi].
exact (f i xi).
- intros A f i xi; reflexivity.
- intros A f g p [i xi].
exact (p i xi).
Defined.
Proof.
intros I x.
snapply Build_Coproduct.
- exact (sig (fun i : I ⇒ x i)).
- exact (exist x).
- intros A f [i xi].
exact (f i xi).
- intros A f i xi; reflexivity.
- intros A f g p [i xi].
exact (p i xi).
Defined.
In particular, Type has all binary coproducts.
Canonical coproduct-product map
Definition cat_coprod_prod {I : Type} `{DecidablePaths I} {A : Type}
`{Is1Cat A, !IsPointedCat A}
(x : I → A) `{!Coproduct I x, !Product I x}
: cat_coprod I x $-> cat_prod I x.
Proof.
apply cat_coprod_rec.
intros i.
apply cat_prod_corec.
intros a.
destruct (dec_paths i a) as [p|].
- destruct p.
exact (Id _).
- exact zero_morphism.
Defined.
Definition cat_bincoprod_binprod {A : Type} `{Is1Cat A, !IsPointedCat A}
(x y : A) `{!BinaryCoproduct x y, !BinaryProduct x y}
: cat_bincoprod' x y $-> cat_binprod' x y.
Proof.
napply cat_coprod_prod; exact _.
Defined.
`{Is1Cat A, !IsPointedCat A}
(x : I → A) `{!Coproduct I x, !Product I x}
: cat_coprod I x $-> cat_prod I x.
Proof.
apply cat_coprod_rec.
intros i.
apply cat_prod_corec.
intros a.
destruct (dec_paths i a) as [p|].
- destruct p.
exact (Id _).
- exact zero_morphism.
Defined.
Definition cat_bincoprod_binprod {A : Type} `{Is1Cat A, !IsPointedCat A}
(x y : A) `{!BinaryCoproduct x y, !BinaryProduct x y}
: cat_bincoprod' x y $-> cat_binprod' x y.
Proof.
napply cat_coprod_prod; exact _.
Defined.