Library UniMath.CategoryTheory.limits.Opp
Duality between C and C^op
Contents
- From C^op to C
- Monics and Epis
- Initial, Terminal, and Zero
- Equalizers and Coequalizers
- Kernels and Cokernels
- Pullbacks and Pushouts
- BinProducts and BinCoproducts
- From C to C^op
- Monics and Epis
- Initial, Terminal, and Zero
- Equalizers and Coequalizers
- Kernels and Cokernels
- Pullbacks and Pushouts
- BinProducts and BinCoproducts
Require Import UniMath.Foundations.PartD.
Require Import UniMath.Foundations.Propositions.
Require Import UniMath.Foundations.Sets.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Monics.
Require Import UniMath.CategoryTheory.Epis.
Require Import UniMath.CategoryTheory.opp_precat.
Require Import UniMath.CategoryTheory.limits.initial.
Require Import UniMath.CategoryTheory.limits.terminal.
Require Import UniMath.CategoryTheory.limits.zero.
Require Import UniMath.CategoryTheory.limits.equalizers.
Require Import UniMath.CategoryTheory.limits.coequalizers.
Require Import UniMath.CategoryTheory.limits.kernels.
Require Import UniMath.CategoryTheory.limits.cokernels.
Require Import UniMath.CategoryTheory.limits.pushouts.
Require Import UniMath.CategoryTheory.limits.pullbacks.
Require Import UniMath.CategoryTheory.limits.binproducts.
Require Import UniMath.CategoryTheory.limits.bincoproducts.
Local Open Scope cat.
Local Notation "C '^op'" := (op_category C) (at level 3, format "C ^op") : cat.
Definition opp_isMonic {a b : C} (f : a --> b) (H : @isMonic (op_category C) _ _ f) : @isEpi C _ _ f := H.
Opaque opp_isMonic.
Definition opp_Monic {a b : C} (f : @Monic (op_category C) a b) : @Epi C b a :=
@make_Epi C _ _ f (opp_isMonic f (pr2 f)).
Definition opp_isEpi {a b : C} (f : a --> b) (H : @isEpi (C^op) _ _ f) : @isMonic C _ _ f := H.
Opaque opp_isEpi.
Definition opp_Epi {a b : C} (f : @Epi (C^op) a b) : @Monic C b a :=
@make_Monic C _ _ f (opp_isEpi f (pr2 f)).
Definition opp_isInitial {x : C} (H : @isInitial (C^op) x) : @isTerminal C x := H.
Definition opp_Initial (I : @Initial (C^op)) : @Terminal C :=
@make_Terminal C _ (opp_isInitial (pr2 I)).
Definition opp_isTerminal {x : C} (H : @isTerminal (C^op) x) : @isInitial C x := H.
Definition opp_Terminal (T : @Terminal (C^op)) : @Initial C :=
@make_Initial C _ (opp_isTerminal (pr2 T)).
Lemma opp_isZero {x : C} (H : @isZero (C^op) x) : @isZero C x.
Show proof.
Definition opp_Zero (Z : @Zero (C^op)) : @Zero C := @make_Zero C _ (opp_isZero (pr2 Z)).
Lemma opp_ZeroArrowTo {x : C} (Z : @Zero (C^op)) :
@ZeroArrowTo (C^op) Z x = @ZeroArrowFrom C (opp_Zero Z) x.
Show proof.
Lemma opp_ZeroArrowFrom {x : C} (Z : @Zero (C^op)) :
@ZeroArrowFrom (C^op) Z x = @ZeroArrowTo C (opp_Zero Z) x.
Show proof.
Lemma opp_ZeroArrow {x y : C} (Z : @Zero (C^op)) :
@ZeroArrow (C^op) Z x y = @ZeroArrow C (opp_Zero Z) y x.
Show proof.
Local Opaque ZeroArrow.
Lemma opp_isEqualizer {x y z : C} (f g : (C^op)⟦y, z⟧) (e : (C^op)⟦x, y⟧) (H : e · f = e · g)
(H' : @isEqualizer (op_category C) _ _ _ f g e H) : @isCoequalizer C _ _ _ f g e H.
Show proof.
exact H'.
Lemma opp_isCoequalizer {x y z : C} (f g : (C^op)⟦x, y⟧) (e : (C^op)⟦y, z⟧)
(H : f · e = g · e) (H' : @isCoequalizer (C^op) _ _ _ f g e H) :
@isEqualizer C _ _ _ f g e H.
Show proof.
exact H'.
Definition opp_Equalizer {y z : C} (f g : (C^op)⟦y, z⟧) (E : @Equalizer (op_category C) y z f g) :
@Coequalizer C z y f g := @make_Coequalizer C _ _ _ f g (EqualizerArrow E) (EqualizerEqAr E)
(opp_isEqualizer f g (EqualizerArrow E)
(EqualizerEqAr E)
(isEqualizer_Equalizer E)).
Definition opp_Coequalizer {y z : C} (f g : (C^op)⟦y, z⟧) (E : @Coequalizer (C^op) y z f g) :
@Equalizer C z y f g := @make_Equalizer C _ _ _ f g (CoequalizerArrow E) (CoequalizerEqAr E)
(opp_isCoequalizer f g (CoequalizerArrow E)
(CoequalizerEqAr E)
(isCoequalizer_Coequalizer E)).
Definition opp_Equalizers (E : @Equalizers (op_category C)) : @Coequalizers C.
Show proof.
Definition opp_Coequalizers (E : @Coequalizers (C^op)) : @Equalizers C.
Show proof.
Local Lemma opp_isCokernel_eq {x y z : C^op} (f : (C^op)⟦x, y⟧) (g : C^op⟦y, z⟧) (Z : Zero (C^op))
(H : f · g = ZeroArrow Z _ _) (Z' : Zero C) :
(g : C⟦z, y⟧) · (f : C⟦y, x⟧) = ZeroArrow Z' _ _.
Show proof.
Lemma opp_isCokernel {x y z : C^op} {f : (C^op)⟦x, y⟧} {g : C^op⟦y, z⟧} {Z : Zero (C^op)}
{H : f · g = ZeroArrow Z _ _} (K' : isKernel (C:=op_category C) Z f g H) {Z' : Zero C} :
isCokernel Z' (g : C⟦z, y⟧) (f : C⟦y, x⟧) (opp_isCokernel_eq f g Z H Z').
Show proof.
set (K := make_Kernel _ _ _ _ K').
use make_isCokernel.
- intros w h H'.
rewrite <- (ZerosArrowEq C (opp_Zero Z) Z' z w) in H'.
rewrite <- opp_ZeroArrow in H'.
use unique_exists.
+ exact (KernelIn (C:=op_category _ )Z K w h H').
+ use (KernelCommutes (C:=op_category _) Z K).
+ intros y0. apply hs.
+ cbn. intros y0 X. use (KernelInsEq (C:=op_category _) Z K). rewrite (KernelCommutes (C:=op_category _) Z K). cbn. rewrite X.
apply idpath.
use make_isCokernel.
- intros w h H'.
rewrite <- (ZerosArrowEq C (opp_Zero Z) Z' z w) in H'.
rewrite <- opp_ZeroArrow in H'.
use unique_exists.
+ exact (KernelIn (C:=op_category _ )Z K w h H').
+ use (KernelCommutes (C:=op_category _) Z K).
+ intros y0. apply hs.
+ cbn. intros y0 X. use (KernelInsEq (C:=op_category _) Z K). rewrite (KernelCommutes (C:=op_category _) Z K). cbn. rewrite X.
apply idpath.
Local Lemma opp_Kernel_eq {y z : C} (f : (C^op)⟦y, z⟧) (Z : Zero (C^op))
(K : @Kernel (op_category C) Z y z f) :
@compose C^op _ _ _ (KernelArrow K) f = ZeroArrow (opp_Zero Z) z K.
Show proof.
Lemma opp_Kernel_isCokernel {y z : C} (f : (C^op)⟦y, z⟧) (Z : Zero (C^op))
(K : @Kernel (C^op) Z y z f) :
isCokernel (opp_Zero Z) f (KernelArrow K) (opp_Kernel_eq f Z K).
Show proof.
use make_isCokernel.
- intros w h H'. rewrite <- opp_ZeroArrow in H'.
use unique_exists.
+ exact (KernelIn Z K w h H').
+ use (KernelCommutes Z K).
+ intros y0. apply hs.
+ cbn. intros y0 X. use (@KernelInsEq C^op). rewrite (KernelCommutes Z K). cbn. rewrite X.
apply idpath.
- intros w h H'. rewrite <- opp_ZeroArrow in H'.
use unique_exists.
+ exact (KernelIn Z K w h H').
+ use (KernelCommutes Z K).
+ intros y0. apply hs.
+ cbn. intros y0 X. use (@KernelInsEq C^op). rewrite (KernelCommutes Z K). cbn. rewrite X.
apply idpath.
Definition opp_Kernel {y z : C} (f : (C^op)⟦y, z⟧) (Z : Zero (C^op))
(K : @Kernel (C^op) Z y z f) : @Cokernel C (opp_Zero Z) z y f.
Show proof.
use make_Cokernel.
- exact K.
- exact (KernelArrow K).
- exact (opp_Kernel_eq f Z K).
- exact (opp_Kernel_isCokernel f Z K).
- exact K.
- exact (KernelArrow K).
- exact (opp_Kernel_eq f Z K).
- exact (opp_Kernel_isCokernel f Z K).
Lemma opp_isKernel {x y z : op_category C} {f : (C^op)⟦x, y⟧} {g : C^op⟦y, z⟧} {Z : Zero (C^op)}
{H : f · g = ZeroArrow Z _ _} (CK' : isCokernel Z f g H) {Z' : Zero C} :
isKernel Z' (g : C⟦z, y⟧) (f : C⟦y, x⟧) (opp_isCokernel_eq f g Z H Z').
Show proof.
set (CK := make_Cokernel _ _ _ _ CK').
use make_isKernel.
intros w h H'.
rewrite <- (ZerosArrowEq C (opp_Zero Z) Z' w x) in H'.
rewrite <- opp_ZeroArrow in H'.
use unique_exists.
+ exact (CokernelOut Z CK w h H').
+ use (CokernelCommutes Z CK).
+ intros y0. apply hs.
+ cbn. intros y0 X. use (CokernelOutsEq _ CK). rewrite (CokernelCommutes Z CK). cbn. rewrite X.
apply idpath.
use make_isKernel.
intros w h H'.
rewrite <- (ZerosArrowEq C (opp_Zero Z) Z' w x) in H'.
rewrite <- opp_ZeroArrow in H'.
use unique_exists.
+ exact (CokernelOut Z CK w h H').
+ use (CokernelCommutes Z CK).
+ intros y0. apply hs.
+ cbn. intros y0 X. use (CokernelOutsEq _ CK). rewrite (CokernelCommutes Z CK). cbn. rewrite X.
apply idpath.
Local Lemma opp_Cokernel_eq {y z : C} (f : (C^op)⟦y, z⟧) (Z : Zero (C^op))
(CK : @Cokernel (C^op) Z y z f) :
@compose (C^op) _ _ _ f (CokernelArrow CK) = ZeroArrow (opp_Zero Z) CK y.
Show proof.
Lemma opp_Cokernel_isKernel {y z : C} (f : (C^op)⟦y, z⟧) (Z : Zero (C^op))
(CK : @Cokernel (C^op) Z y z f) :
isKernel (opp_Zero Z) (CokernelArrow CK) f (opp_Cokernel_eq f Z CK).
Show proof.
use make_isKernel.
intros w h H'. rewrite <- opp_ZeroArrow in H'.
use unique_exists.
+ exact (CokernelOut Z CK w h H').
+ use (CokernelCommutes Z CK).
+ intros y0. apply hs.
+ cbn. intros y0 X. use (@CokernelOutsEq C^op). rewrite (CokernelCommutes Z CK). cbn. rewrite X.
apply idpath.
intros w h H'. rewrite <- opp_ZeroArrow in H'.
use unique_exists.
+ exact (CokernelOut Z CK w h H').
+ use (CokernelCommutes Z CK).
+ intros y0. apply hs.
+ cbn. intros y0 X. use (@CokernelOutsEq C^op). rewrite (CokernelCommutes Z CK). cbn. rewrite X.
apply idpath.
Definition opp_Cokernel {y z : C} (f : (C^op)⟦y, z⟧) (Z : Zero (C^op))
(CK : @Cokernel (C^op) Z y z f) : @Kernel C (opp_Zero Z) z y f.
Show proof.
use make_Kernel.
- exact CK.
- exact (CokernelArrow CK).
- exact (opp_Cokernel_eq f Z CK).
- exact (opp_Cokernel_isKernel f Z CK).
- exact CK.
- exact (CokernelArrow CK).
- exact (opp_Cokernel_eq f Z CK).
- exact (opp_Cokernel_isKernel f Z CK).
Definition opp_Kernels (Z : Zero (C^op)) (K : @Kernels (C^op) Z) : @Cokernels C (opp_Zero Z).
Show proof.
Definition opp_Cokernels (Z : Zero (C^op)) (CK : @Cokernels (C^op) Z) : @Kernels C (opp_Zero Z).
Show proof.
Lemma opp_isPushout {a b c d : C} (f : (C^op)⟦a, b⟧) (g : (C^op)⟦a, c⟧)
(in1 : (C^op)⟦b, d⟧) (in2 : (C^op)⟦c, d⟧) (H : f · in1 = g · in2)
(iPo : @isPushout (C^op) a b c d f g in1 in2 H) : @isPullback C a b c d f g in1 in2 H.
Show proof.
exact iPo.
Lemma opp_isPullback {a b c d : C} (f : (C^op)⟦b, a⟧) (g : (C^op)⟦c, a⟧)
(p1 : (C^op)⟦d, b⟧) (p2 : (C^op)⟦d, c⟧) (H : p1 · f = p2 · g)
(iPb : @isPullback (C^op) a b c d f g p1 p2 H) : @isPushout C a b c d f g p1 p2 H.
Show proof.
exact iPb.
Definition opp_Pushout {a b c : C} (f : (C^op)⟦a, b⟧) (g : (C^op)⟦a, c⟧)
(Po : @Pushout (C^op) a b c f g) : @Pullback C a b c f g.
Show proof.
exact Po.
Definition opp_Pullback {a b c : C} (f : (C^op)⟦b, a⟧) (g : (C^op)⟦c, a⟧)
(Pb : @Pullback (C^op) a b c f g) : @Pushout C a b c f g.
Show proof.
exact Pb.
Definition opp_Pushouts (Pos : @Pushouts (C^op)) : @Pullbacks C.
Show proof.
exact Pos.
Definition opp_Pullbacks (Pbs : @Pushouts (C^op)) : @Pullbacks C.
Show proof.
exact Pbs.
Definition opp_isBinProduct (c d p : C) (p1 : (C^op)⟦p, c⟧) (p2 : (C^op)⟦p, d⟧)
(iBPC : @isBinProduct (C^op) c d p p1 p2) : @isBinCoproduct C c d p p1 p2 :=
iBPC.
Definition opp_isBinCoproduct (a b co : C) (ia : (C^op)⟦a, co⟧) (ib : (C^op)⟦b, co⟧)
(iBCC : @isBinCoproduct (C^op) a b co ia ib) :
@isBinProduct C a b co ia ib := iBCC.
Definition opp_BinProduct (c d : C) (BPC : @BinProduct (C^op) c d) :
@BinCoproduct C c d := BPC.
Definition opp_BinCoproduct (c d : C) (BCC : @BinCoproduct (C^op) c d) :
@BinProduct C c d := BCC.
Definition opp_BinProducts (BP : @BinProducts (C^op)) : @BinCoproducts C := BP.
Definition opp_BinCoproducts (BC : @BinCoproducts (C^op)) : @BinProducts C := BC.
End def_opposites.
Definition isMonic_opp {a b : C} {f : C⟦a, b⟧} (H : @isMonic C a b f) : @isEpi (C^op) b a f := H.
Opaque isMonic_opp.
Definition Monic_opp {a b : C} (f : @Monic C a b) : @Epi (C^op) b a :=
@make_Epi (C^op) b a f (isMonic_opp (pr2 f)).
Definition isEpi_opp {a b : C} {f : C⟦a, b⟧} (H : @isEpi C a b f) : @isMonic (C^op) b a f := H.
Opaque isEpi_opp.
Definition Epi_opp {a b : C} (f : @Epi C a b) : @Monic (C^op) b a :=
@make_Monic (C^op) b a f (isEpi_opp (pr2 f)).
Definition isInitial_opp {x : C} (H : @isInitial C x) : @isTerminal (C^op) x := H.
Definition Initial_opp (I : @Initial C) : @Terminal (C^op) :=
@make_Terminal (C^op) _ (isInitial_opp (pr2 I)).
Definition isTerminal_opp {x : C} (H : @isTerminal C x) : @isInitial (C^op) x := H.
Definition Terminal_opp (T : @Terminal C) : @Initial (C^op) :=
@make_Initial (C^op) _ (isTerminal_opp (pr2 T)).
Lemma isZero_opp {x : C} (H : @isZero C x) : @isZero (C^op) x.
Show proof.
Definition Zero_opp (T : @Zero C) : @Zero (C^op) := @make_Zero (C^op) _ (isZero_opp (pr2 T)).
Lemma ZeroArrowTo_opp {x : C} (Z : @Zero C) :
@ZeroArrowTo C Z x = @ZeroArrowFrom (C^op) (Zero_opp Z) x.
Show proof.
Lemma ZeroArrowFrom_opp {x : C} (Z : @Zero C) :
@ZeroArrowFrom C Z x = @ZeroArrowTo (C^op) (Zero_opp Z) x.
Show proof.
Lemma ZeroArrow_opp {x y : C} (Z : @Zero C) :
@ZeroArrow C Z x y = @ZeroArrow (C^op) (Zero_opp Z) y x.
Show proof.
Local Opaque ZeroArrow.
Definition isEqualizer_opp {x y z : C} (f g : C⟦y, z⟧) (e : C⟦x, y⟧) (H : e · f = e · g)
(isE : @isEqualizer C _ _ _ f g e H) : @isCoequalizer (C^op) _ _ _ f g e H := isE.
Definition isCoequalizer_opp {x y z : C} (f g : C⟦x, y⟧) (e : C⟦y, z⟧) (H : f · e = g · e)
(isC : @isCoequalizer C _ _ _ f g e H) : @isEqualizer (C^op) _ _ _ f g e H := isC.
Definition Equalizer_opp {y z : C} (f g : C⟦y, z⟧) (E : @Equalizer C y z f g) :
@Coequalizer (C^op) z y f g := @make_Coequalizer (C^op) _ _ _ f g (EqualizerArrow E)
(EqualizerEqAr E)
(isEqualizer_opp f g (EqualizerArrow E)
(EqualizerEqAr E)
(isEqualizer_Equalizer E)).
Definition Coequalizer_opp {y z : C} (f g : C⟦y, z⟧) (CE : @Coequalizer C y z f g) :
@Equalizer (C^op) z y f g := @make_Equalizer (C^op) _ _ _ f g (CoequalizerArrow CE)
(CoequalizerEqAr CE)
(isCoequalizer_opp f g (CoequalizerArrow CE)
(CoequalizerEqAr CE)
(isCoequalizer_Coequalizer CE)).
Definition Equalizers_opp (E : @Equalizers C) : @Coequalizers (C^op).
Show proof.
Definition Coequalizers_opp (CE : @Coequalizers C) : @Equalizers (C^op).
Show proof.
Local Lemma isCokernel_opp_eq {x y z : C} (f : C⟦x, y⟧) (g : C⟦y, z⟧) (Z : Zero C)
(H : f · g = ZeroArrow Z _ _) (Z' : Zero C^op) :
(g : C^op⟦z, y⟧) · (f : C^op⟦y, x⟧) = ZeroArrow Z' _ _.
Show proof.
Lemma isCokernel_opp {x y z : C} {f : C⟦x, y⟧} {g : C⟦y, z⟧} {Z : Zero C}
{H : f · g = ZeroArrow Z _ _} (K' : isKernel Z f g H) {Z' : Zero C^op} :
isCokernel Z' (g : C^op⟦z, y⟧) (f : C^op⟦y, x⟧) (isCokernel_opp_eq f g Z H Z').
Show proof.
set (K := make_Kernel _ _ _ _ K').
use make_isCokernel.
- intros w h H'. cbn in H'.
set (XXX := (ZerosArrowEq C^op (Zero_opp Z) Z' z w)).
use unique_exists.
+ use (KernelIn Z K w h _).
rewrite ZeroArrow_opp.
rewrite XXX.
apply H'.
+ cbn. use (KernelCommutes Z K).
+ intros y0. apply (has_homsets_opp hs).
+ cbn. intros y0 X. use (KernelInsEq Z K). rewrite KernelCommutes. exact X.
use make_isCokernel.
- intros w h H'. cbn in H'.
set (XXX := (ZerosArrowEq C^op (Zero_opp Z) Z' z w)).
use unique_exists.
+ use (KernelIn Z K w h _).
rewrite ZeroArrow_opp.
rewrite XXX.
apply H'.
+ cbn. use (KernelCommutes Z K).
+ intros y0. apply (has_homsets_opp hs).
+ cbn. intros y0 X. use (KernelInsEq Z K). rewrite KernelCommutes. exact X.
Local Lemma Kernel_opp_eq {y z : C} (f : C⟦y, z⟧) (Z : Zero C) (K : @Kernel C Z y z f) :
@compose C^op _ _ _ f (KernelArrow K) = ZeroArrow (Zero_opp Z) z K.
Show proof.
Lemma Kernel_opp_isCokernel {y z : C} (f : C⟦y, z⟧) (Z : Zero C) (K : @Kernel C Z y z f) :
isCokernel (Zero_opp Z) f (KernelArrow K) (Kernel_opp_eq f Z K).
Show proof.
use make_isCokernel.
- intros w h H'. cbn in H'.
use unique_exists.
+ rewrite <- ZeroArrow_opp in H'. exact (KernelIn Z K w h H').
+ cbn. use KernelCommutes.
+ intros y0. apply (has_homsets_opp hs).
+ cbn. intros y0 X. use KernelInsEq. rewrite KernelCommutes. exact X.
- intros w h H'. cbn in H'.
use unique_exists.
+ rewrite <- ZeroArrow_opp in H'. exact (KernelIn Z K w h H').
+ cbn. use KernelCommutes.
+ intros y0. apply (has_homsets_opp hs).
+ cbn. intros y0 X. use KernelInsEq. rewrite KernelCommutes. exact X.
Definition Kernel_opp {y z : C} (f : C⟦y, z⟧) (Z : Zero C) (K : @Kernel C Z y z f) :
@Cokernel (C^op) (Zero_opp Z) z y f.
Show proof.
use make_Cokernel.
- exact K.
- exact (KernelArrow K).
- exact (Kernel_opp_eq f Z K).
- exact (Kernel_opp_isCokernel f Z K).
- exact K.
- exact (KernelArrow K).
- exact (Kernel_opp_eq f Z K).
- exact (Kernel_opp_isCokernel f Z K).
Lemma isKernel_opp {x y z : C^op} {f : C⟦x, y⟧} {g : C⟦y, z⟧} {Z : Zero C}
{H : f · g = ZeroArrow Z _ _} (CK' : isCokernel Z f g H) {Z' : Zero C^op} :
isKernel Z' (g : C^op⟦z, y⟧) (f : C^op⟦y, x⟧) (isCokernel_opp_eq f g Z H Z').
Show proof.
set (CK := make_Cokernel _ _ _ _ CK').
use make_isKernel.
- intros w h H'.
rewrite <- (ZerosArrowEq C^op (Zero_opp Z) Z' w x) in H'.
rewrite <- ZeroArrow_opp in H'.
use unique_exists.
+ exact (CokernelOut Z CK w h H').
+ use (CokernelCommutes Z CK).
+ intros y0. apply hs.
+ cbn. intros y0 X. use (CokernelOutsEq _ CK). rewrite (CokernelCommutes Z CK). cbn. rewrite X.
apply idpath.
use make_isKernel.
- intros w h H'.
rewrite <- (ZerosArrowEq C^op (Zero_opp Z) Z' w x) in H'.
rewrite <- ZeroArrow_opp in H'.
use unique_exists.
+ exact (CokernelOut Z CK w h H').
+ use (CokernelCommutes Z CK).
+ intros y0. apply hs.
+ cbn. intros y0 X. use (CokernelOutsEq _ CK). rewrite (CokernelCommutes Z CK). cbn. rewrite X.
apply idpath.
Local Lemma Cokernel_opp_eq {y z : C} (f : C⟦y, z⟧) (Z : Zero C) (CK : @Cokernel C Z y z f) :
@compose C^op _ _ _ (CokernelArrow CK) f = ZeroArrow (Zero_opp Z) CK y.
Show proof.
Lemma Cokernel_opp_isKernel {y z : C} (f : C⟦y, z⟧) (Z : Zero C)
(CK : @Cokernel C Z y z f) :
isKernel (Zero_opp Z) (CokernelArrow CK) f (Cokernel_opp_eq f Z CK).
Show proof.
use make_isKernel.
- intros w h H'. cbn in H'.
use unique_exists.
+ rewrite <- ZeroArrow_opp in H'. exact (CokernelOut Z CK w h H').
+ cbn. use CokernelCommutes.
+ intros y0. apply (has_homsets_opp hs).
+ cbn. intros y0 X. use CokernelOutsEq. rewrite CokernelCommutes. exact X.
- intros w h H'. cbn in H'.
use unique_exists.
+ rewrite <- ZeroArrow_opp in H'. exact (CokernelOut Z CK w h H').
+ cbn. use CokernelCommutes.
+ intros y0. apply (has_homsets_opp hs).
+ cbn. intros y0 X. use CokernelOutsEq. rewrite CokernelCommutes. exact X.
Definition Cokernel_opp {y z : C} (f : C⟦y, z⟧) (Z : Zero C) (CK : @Cokernel C Z y z f) :
@Kernel (C^op) (Zero_opp Z) z y f.
Show proof.
use make_Kernel.
- exact CK.
- exact (CokernelArrow CK).
- exact (Cokernel_opp_eq f Z CK).
- exact (Cokernel_opp_isKernel f Z CK).
- exact CK.
- exact (CokernelArrow CK).
- exact (Cokernel_opp_eq f Z CK).
- exact (Cokernel_opp_isKernel f Z CK).
Definition Kernels_opp (Z : Zero C) (K : @Kernels C Z) : @Cokernels (C^op) (Zero_opp Z).
Show proof.
Definition Cokernels_opp (Z : Zero C) (CK : @Cokernels C Z) : @Kernels (C^op) (Zero_opp Z).
Show proof.
Definition isPushout_opp {a b c d : C} (f : C⟦a, b⟧) (g : C⟦a, c⟧) (in1 : C⟦b, d⟧) (in2 : C⟦c, d⟧)
(H : f · in1 = g · in2) (iPo : @isPushout C a b c d f g in1 in2 H) :
@isPullback (C^op) a b c d f g in1 in2 H := iPo.
Definition isPullback_opp {a b c d : C} (f : C⟦b, a⟧) (g : C⟦c, a⟧) (p1 : C⟦d, b⟧) (p2 : C⟦d, c⟧)
(H : p1 · f = p2 · g) (iPb : @isPullback C a b c d f g p1 p2 H) :
@isPushout (C^op) a b c d f g p1 p2 H := iPb.
Definition Pushout_opp {a b c : C} (f : C⟦a, b⟧) (g : C⟦a, c⟧) (Po : @Pushout C a b c f g) :
@Pullback (C^op) a b c f g := Po.
Definition Pullback_opp {a b c : C} (f : C⟦b, a⟧) (g : C⟦c, a⟧) (Pb : @Pullback C a b c f g) :
@Pushout (C^op) a b c f g := Pb.
Definition Pushouts_opp (Pos : @Pushouts C) : @Pullbacks (C^op) := Pos.
Definition Pullbacks_opp (Pbs : @Pushouts C) : @Pullbacks (C^op) := Pbs.
Definition isBinProduct_opp (c d p : C) (p1 : C⟦p, c⟧) (p2 : C⟦p, d⟧)
(iBPC : @isBinProduct C c d p p1 p2) :
@isBinCoproduct (C^op) c d p p1 p2 := iBPC.
Definition isBinCoproduct_opp (a b co : C) (ia : C⟦a, co⟧) (ib : C⟦b, co⟧)
(iBCC : @isBinCoproduct C a b co ia ib) :
@isBinProduct (C^op) a b co ia ib := iBCC.
Definition BinProduct_opp (c d : C) (iBPC : @BinProduct C c d) :
@BinCoproduct (C^op) c d := iBPC.
Definition BinCoproduct_opp (c d : C) (iBCC : @BinCoproduct C c d) :
@BinProduct (C^op) c d := iBCC.
Definition BinProducts_opp (BP : @BinProducts C) : @BinCoproducts (C^op) := BP.
Definition BinCoproducts_opp (BC : @BinCoproducts C) : @BinProducts (C^op) := BC.
End def_opposites'.
Definition opp_zero_lifts {C:category} {X:Type} (j : X -> ob C) :
zero_lifts C j -> zero_lifts C^op j.
Show proof.