category_theory.limits.small_complete
⟷
Mathlib.CategoryTheory.Limits.SmallComplete
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -46,7 +46,36 @@ in Lean, a preorder category is one where the morphisms are in Prop, which is we
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `preorder C` instance.
-/
-instance (priority := 100) : Quiver.IsThin C := fun X Y => ⟨fun r s => by classical⟩
+instance (priority := 100) : Quiver.IsThin C := fun X Y =>
+ ⟨fun r s => by
+ classical
+ by_contra r_ne_s
+ have z : (2 : Cardinal) ≤ (#X ⟶ Y) :=
+ by
+ rw [Cardinal.two_le_iff]
+ exact ⟨_, _, r_ne_s⟩
+ let md := Σ Z W : C, Z ⟶ W
+ let α := #md
+ apply not_le_of_lt (Cardinal.cantor α)
+ let yp : C := ∏ fun f : md => Y
+ trans #X ⟶ yp
+ · apply le_trans (Cardinal.power_le_power_right z)
+ rw [Cardinal.power_def]
+ apply le_of_eq
+ rw [Cardinal.eq]
+ refine' ⟨⟨pi.lift, fun f k => f ≫ pi.π _ k, _, _⟩⟩
+ · intro f
+ ext k
+ simp
+ · intro f
+ ext ⟨j⟩
+ simp
+ · apply Cardinal.mk_le_of_injective _
+ · intro f
+ exact ⟨_, _, f⟩
+ · rintro f g k
+ cases k
+ rfl⟩
end CategoryTheory
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -46,36 +46,7 @@ in Lean, a preorder category is one where the morphisms are in Prop, which is we
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `preorder C` instance.
-/
-instance (priority := 100) : Quiver.IsThin C := fun X Y =>
- ⟨fun r s => by
- classical
- by_contra r_ne_s
- have z : (2 : Cardinal) ≤ (#X ⟶ Y) :=
- by
- rw [Cardinal.two_le_iff]
- exact ⟨_, _, r_ne_s⟩
- let md := Σ Z W : C, Z ⟶ W
- let α := #md
- apply not_le_of_lt (Cardinal.cantor α)
- let yp : C := ∏ fun f : md => Y
- trans #X ⟶ yp
- · apply le_trans (Cardinal.power_le_power_right z)
- rw [Cardinal.power_def]
- apply le_of_eq
- rw [Cardinal.eq]
- refine' ⟨⟨pi.lift, fun f k => f ≫ pi.π _ k, _, _⟩⟩
- · intro f
- ext k
- simp
- · intro f
- ext ⟨j⟩
- simp
- · apply Cardinal.mk_le_of_injective _
- · intro f
- exact ⟨_, _, f⟩
- · rintro f g k
- cases k
- rfl⟩
+instance (priority := 100) : Quiver.IsThin C := fun X Y => ⟨fun r s => by classical⟩
end CategoryTheory
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,8 +3,8 @@ Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
-import Mathbin.CategoryTheory.Limits.Shapes.Products
-import Mathbin.SetTheory.Cardinal.Basic
+import CategoryTheory.Limits.Shapes.Products
+import SetTheory.Cardinal.Basic
#align_import category_theory.limits.small_complete from "leanprover-community/mathlib"@"69c6a5a12d8a2b159f20933e60115a4f2de62b58"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,15 +2,12 @@
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-
-! This file was ported from Lean 3 source module category_theory.limits.small_complete
-! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.CategoryTheory.Limits.Shapes.Products
import Mathbin.SetTheory.Cardinal.Basic
+#align_import category_theory.limits.small_complete from "leanprover-community/mathlib"@"69c6a5a12d8a2b159f20933e60115a4f2de62b58"
+
/-!
# Any small complete category is a preorder
mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297
@@ -52,33 +52,33 @@ rather than providing a `preorder C` instance.
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
⟨fun r s => by
classical
- by_contra r_ne_s
- have z : (2 : Cardinal) ≤ (#X ⟶ Y) :=
- by
- rw [Cardinal.two_le_iff]
- exact ⟨_, _, r_ne_s⟩
- let md := Σ Z W : C, Z ⟶ W
- let α := #md
- apply not_le_of_lt (Cardinal.cantor α)
- let yp : C := ∏ fun f : md => Y
- trans #X ⟶ yp
- · apply le_trans (Cardinal.power_le_power_right z)
- rw [Cardinal.power_def]
- apply le_of_eq
- rw [Cardinal.eq]
- refine' ⟨⟨pi.lift, fun f k => f ≫ pi.π _ k, _, _⟩⟩
- · intro f
- ext k
- simp
- · intro f
- ext ⟨j⟩
- simp
- · apply Cardinal.mk_le_of_injective _
- · intro f
- exact ⟨_, _, f⟩
- · rintro f g k
- cases k
- rfl⟩
+ by_contra r_ne_s
+ have z : (2 : Cardinal) ≤ (#X ⟶ Y) :=
+ by
+ rw [Cardinal.two_le_iff]
+ exact ⟨_, _, r_ne_s⟩
+ let md := Σ Z W : C, Z ⟶ W
+ let α := #md
+ apply not_le_of_lt (Cardinal.cantor α)
+ let yp : C := ∏ fun f : md => Y
+ trans #X ⟶ yp
+ · apply le_trans (Cardinal.power_le_power_right z)
+ rw [Cardinal.power_def]
+ apply le_of_eq
+ rw [Cardinal.eq]
+ refine' ⟨⟨pi.lift, fun f k => f ≫ pi.π _ k, _, _⟩⟩
+ · intro f
+ ext k
+ simp
+ · intro f
+ ext ⟨j⟩
+ simp
+ · apply Cardinal.mk_le_of_injective _
+ · intro f
+ exact ⟨_, _, f⟩
+ · rintro f g k
+ cases k
+ rfl⟩
end CategoryTheory
mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb
@@ -57,7 +57,7 @@ instance (priority := 100) : Quiver.IsThin C := fun X Y =>
by
rw [Cardinal.two_le_iff]
exact ⟨_, _, r_ne_s⟩
- let md := ΣZ W : C, Z ⟶ W
+ let md := Σ Z W : C, Z ⟶ W
let α := #md
apply not_le_of_lt (Cardinal.cantor α)
let yp : C := ∏ fun f : md => Y
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -37,7 +37,7 @@ namespace CategoryTheory
open Category Limits
-open Cardinal
+open scoped Cardinal
universe u
mathlib commit https://github.com/leanprover-community/mathlib/commit/2196ab363eb097c008d4497125e0dde23fb36db2
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
! This file was ported from Lean 3 source module category_theory.limits.small_complete
-! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a
+! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -14,6 +14,9 @@ import Mathbin.SetTheory.Cardinal.Basic
/-!
# Any small complete category is a preorder
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
We show that any small category which has all (small) limits is a preorder: In particular, we show
that if a small category `C` in universe `u` has products of size `u`, then for any `X Y : C`
there is at most one morphism `X ⟶ Y`.
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
@@ -2,15 +2,12 @@
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-
-! This file was ported from Lean 3 source module category_theory.limits.small_complete
-! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.SetTheory.Cardinal.Basic
+#align_import category_theory.limits.small_complete from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
+
/-!
# Any small complete category is a preorder
@@ -50,7 +50,7 @@ instance (priority := 100) : Quiver.IsThin C := fun X Y =>
⟨fun r s => by
classical
by_contra r_ne_s
- have z : (2 : Cardinal) ≤ (#X ⟶ Y) := by
+ have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
rw [Cardinal.two_le_iff]
exact ⟨_, _, r_ne_s⟩
let md := ΣZ W : C, Z ⟶ W
@@ -58,7 +58,7 @@ instance (priority := 100) : Quiver.IsThin C := fun X Y =>
apply not_le_of_lt (Cardinal.cantor α)
let yp : C := ∏ fun _ : md => Y
refine' _root_.trans _ _
- · exact #X ⟶ yp
+ · exact #(X ⟶ yp)
· apply le_trans (Cardinal.power_le_power_right z)
rw [Cardinal.power_def]
apply le_of_eq
The unported dependencies are