feat(category_theory/limits): add kernel pairs (#3925)

Add a subsingleton data structure expressing that a parallel pair of morphisms is a kernel pair of a given morphism.

Another PR from my topos project. A pretty minimal API since I didn't need much there - I also didn't introduce anything like `has_kernel_pairs` largely because I didn't need it, but also because I don't know that it's useful for anyone, and it might conflict with ideas in the prop-limits branch.

Author: b-mehta

src/category_theory/limits/shapes/kernel_pair.lean

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+/-
+Copyright (c) 2020 Bhavik Mehta. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Bhavik Mehta
+-/
+import category_theory.limits.shapes.equalizers
+import category_theory.limits.shapes.pullbacks
+import category_theory.limits.shapes.regular_mono
+
+/-!
+# Kernel pairs
+
+This file defines what it means for a parallel pair of morphisms `a b : R ⟶ X` to be the kernel pair
+for a morphism `f`.
+Some properties of kernel pairs are given, namely allowing one to transfer between
+the kernel pair of `f₁ ≫ f₂` to the kernel pair of `f₁`.
+It is also proved that if `f` is a coequalizer of some pair, and `a`,`b` is a kernel pair for `f` then
+it is a coequalizer of `a`,`b`.
+
+## Implementation
+
+The definition is essentially just a wrapper for `is_limit (pullback_cone.mk _ _ _)`, but the
+constructions given here are useful, yet awkward to present in that language, so a basic API
+is developed here.
+
+## TODO
+
+- Internal equivalence relations (or congruences) and the fact that every kernel pair induces one,
+  and the converse in an effective regular category (WIP by b-mehta).
+
+-/
+
+universes v u u₂
+
+namespace category_theory
+
+open category_theory category_theory.category category_theory.limits
+
+variables {C : Type u} [category.{v} C]
+
+variables {R X Y Z : C} (f : X ⟶ Y) (a b : R ⟶ X)
+
+/--
+`is_kernel_pair f a b` expresses that `(a, b)` is a kernel pair for `f`, i.e. `a ≫ f = b ≫ f`
+and the square
+  R → X
+  ↓   ↓
+  X → Y
+is a pullback square.
+This is essentially just a convenience wrapper over `is_limit (pullback_cone.mk _ _ _)`.
+-/
+structure is_kernel_pair :=
+(comm : a ≫ f = b ≫ f)
+(is_limit : is_limit (pullback_cone.mk _ _ comm))
+
+attribute [reassoc] is_kernel_pair.comm
+
+namespace is_kernel_pair
+
+/-- The data expressing that `(a, b)` is a kernel pair is subsingleton. -/
+instance : subsingleton (is_kernel_pair f a b) :=
+⟨λ P Q, by { cases P, cases Q, congr, }⟩
+
+/-- If `f` is a monomorphism, then `(𝟙 _, 𝟙 _)`  is a kernel pair for `f`. -/
+def id_of_mono [mono f] : is_kernel_pair f (𝟙 _) (𝟙 _) :=
+{ comm := rfl,
+  is_limit :=
+  pullback_cone.is_limit_aux' _ $ λ s,
+  begin
+    refine ⟨s.snd, _, comp_id _, λ m m₁ m₂, _⟩,
+    { rw [← cancel_mono f, s.condition, pullback_cone.mk_fst, cancel_mono f],
+      apply comp_id },
+    rw [← m₂],
+    apply (comp_id _).symm,
+  end }
+
+instance [mono f] : inhabited (is_kernel_pair f (𝟙 _) (𝟙 _)) := ⟨id_of_mono f⟩
+
+variables {f a b}
+
+/--
+Given a pair of morphisms `p`, `q` to `X` which factor through `f`, they factor through any kernel
+pair of `f`.
+-/
+def lift' {S : C} (k : is_kernel_pair f a b) (p q : S ⟶ X) (w : p ≫ f = q ≫ f) :
+  { t : S ⟶ R // t ≫ a = p ∧ t ≫ b = q } :=
+pullback_cone.is_limit.lift' k.is_limit _ _ w
+
+/--
+If `(a,b)` is a kernel pair for `f₁ ≫ f₂` and `a ≫ f₁ = b ≫ f₁`, then `(a,b)` is a kernel pair for
+just `f₁`.
+That is, to show that `(a,b)` is a kernel pair for `f₁` it suffices to only show the square
+commutes, rather than to additionally show it's a pullback.
+-/
+def cancel_right {f₁ : X ⟶ Y} {f₂ : Y ⟶ Z} (comm : a ≫ f₁ = b ≫ f₁) (big_k : is_kernel_pair (f₁ ≫ f₂) a b) :
+  is_kernel_pair f₁ a b :=
+{ comm := comm,
+  is_limit := pullback_cone.is_limit_aux' _ $ λ s,
+  begin
+    let s' : pullback_cone (f₁ ≫ f₂) (f₁ ≫ f₂) := pullback_cone.mk s.fst s.snd (s.condition_assoc _),
+    refine ⟨big_k.is_limit.lift s',
+            big_k.is_limit.fac _ walking_cospan.left,
+            big_k.is_limit.fac _ walking_cospan.right,
+            λ m m₁ m₂, _⟩,
+    apply big_k.is_limit.hom_ext,
+    refine ((pullback_cone.mk a b _) : pullback_cone (f₁ ≫ f₂) _).equalizer_ext _ _,
+    apply m₁.trans (big_k.is_limit.fac s' walking_cospan.left).symm,
+    apply m₂.trans (big_k.is_limit.fac s' walking_cospan.right).symm,
+  end }
+
+/--
+If `(a,b)` is a kernel pair for `f₁ ≫ f₂` and `f₂` is mono, then `(a,b)` is a kernel pair for
+just `f₁`.
+The converse of `comp_of_mono`.
+-/
+def cancel_right_of_mono {f₁ : X ⟶ Y} {f₂ : Y ⟶ Z} [mono f₂] (big_k : is_kernel_pair (f₁ ≫ f₂) a b) :
+  is_kernel_pair f₁ a b :=
+cancel_right (begin rw [← cancel_mono f₂, assoc, assoc, big_k.comm] end) big_k
+
+/--
+If `(a,b)` is a kernel pair for `f₁` and `f₂` is mono, then `(a,b)` is a kernel pair for `f₁ ≫ f₂`.
+The converse of `cancel_right_of_mono`.
+-/
+def comp_of_mono {f₁ : X ⟶ Y} {f₂ : Y ⟶ Z} [mono f₂] (small_k : is_kernel_pair f₁ a b) :
+  is_kernel_pair (f₁ ≫ f₂) a b :=
+{ comm := by rw [small_k.comm_assoc],
+  is_limit := pullback_cone.is_limit_aux' _ $ λ s,
+  begin
+    refine ⟨_, _, _, _⟩,
+    apply (pullback_cone.is_limit.lift' small_k.is_limit s.fst s.snd _).1,
+    rw [← cancel_mono f₂, assoc, s.condition, assoc],
+    apply (pullback_cone.is_limit.lift' small_k.is_limit s.fst s.snd _).2.1,
+    apply (pullback_cone.is_limit.lift' small_k.is_limit s.fst s.snd _).2.2,
+    intros m m₁ m₂,
+    apply small_k.is_limit.hom_ext,
+    refine ((pullback_cone.mk a b _) : pullback_cone f₁ _).equalizer_ext _ _,
+    rwa (pullback_cone.is_limit.lift' small_k.is_limit s.fst s.snd _).2.1,
+    rwa (pullback_cone.is_limit.lift' small_k.is_limit s.fst s.snd _).2.2,
+  end }
+
+/--
+If `(a,b)` is the kernel pair of `f`, and `f` is a coequalizer morphism for some parallel pair, then
+`f` is a coequalizer morphism of `a` and `b`.
+-/
+def to_coequalizer (k : is_kernel_pair f a b) [r : regular_epi f] :
+  is_colimit (cofork.of_π f k.comm) :=
+begin
+  let t := k.is_limit.lift (pullback_cone.mk _ _ r.w),
+  have ht : t ≫ a = r.left := k.is_limit.fac _ walking_cospan.left,
+  have kt : t ≫ b = r.right := k.is_limit.fac _ walking_cospan.right,
+  apply cofork.is_colimit.mk _ _ _ _,
+  { intro s,
+    apply (cofork.is_colimit.desc' r.is_colimit s.π _).1,
+    rw [← ht, assoc, s.condition, reassoc_of kt] },
+  { intro s,
+    apply (cofork.is_colimit.desc' r.is_colimit s.π _).2 },
+  { intros s m w,
+    apply r.is_colimit.hom_ext,
+    rintro ⟨⟩,
+    change (r.left ≫ f) ≫ m = (r.left ≫ f) ≫ _,
+    rw [assoc, assoc],
+    congr' 1,
+    erw (cofork.is_colimit.desc' r.is_colimit s.π _).2,
+    apply w walking_parallel_pair.one,
+    erw (cofork.is_colimit.desc' r.is_colimit s.π _).2,
+    apply w walking_parallel_pair.one }
+end
+
+end is_kernel_pair
+end category_theory