algebraic_topology.simplicial_set ⟷ Mathlib.AlgebraicTopology.SimplicialSet

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -8,7 +8,7 @@ import AlgebraicTopology.TopologicalSimplex
 import CategoryTheory.Limits.Presheaf
 import CategoryTheory.Limits.Types
 import CategoryTheory.Yoneda
-import Topology.Category.Top.Limits.Basic
+import Topology.Category.TopCat.Limits.Basic
 
 #align_import algebraic_topology.simplicial_set from "leanprover-community/mathlib"@"f2b757fc5c341d88741b9c4630b1e8ba973c5726"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,12 +3,12 @@ Copyright (c) 2021 Scott Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin, Scott Morrison, Adam Topaz
 -/
-import Mathbin.AlgebraicTopology.SimplicialObject
-import Mathbin.AlgebraicTopology.TopologicalSimplex
-import Mathbin.CategoryTheory.Limits.Presheaf
-import Mathbin.CategoryTheory.Limits.Types
-import Mathbin.CategoryTheory.Yoneda
-import Mathbin.Topology.Category.Top.Limits.Basic
+import AlgebraicTopology.SimplicialObject
+import AlgebraicTopology.TopologicalSimplex
+import CategoryTheory.Limits.Presheaf
+import CategoryTheory.Limits.Types
+import CategoryTheory.Yoneda
+import Topology.Category.Top.Limits.Basic
 
 #align_import algebraic_topology.simplicial_set from "leanprover-community/mathlib"@"f2b757fc5c341d88741b9c4630b1e8ba973c5726"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,11 +2,6 @@
 Copyright (c) 2021 Scott Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin, Scott Morrison, Adam Topaz
-
-! This file was ported from Lean 3 source module algebraic_topology.simplicial_set
-! leanprover-community/mathlib commit f2b757fc5c341d88741b9c4630b1e8ba973c5726
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.AlgebraicTopology.SimplicialObject
 import Mathbin.AlgebraicTopology.TopologicalSimplex
@@ -15,6 +10,8 @@ import Mathbin.CategoryTheory.Limits.Types
 import Mathbin.CategoryTheory.Yoneda
 import Mathbin.Topology.Category.Top.Limits.Basic
 
+#align_import algebraic_topology.simplicial_set from "leanprover-community/mathlib"@"f2b757fc5c341d88741b9c4630b1e8ba973c5726"
+
 /-!
 A simplicial set is just a simplicial object in `Type`,
 i.e. a `Type`-valued presheaf on the simplex category.

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -65,7 +65,6 @@ def standardSimplex : SimplexCategory ⥤ SSet :=
 #align sSet.standard_simplex SSet.standardSimplex
 -/
 
--- mathport name: standard_simplex
 scoped[Simplicial] notation "Δ[" n "]" => SSet.standardSimplex.obj (SimplexCategory.mk n)
 
 instance : Inhabited SSet :=
@@ -73,11 +72,13 @@ instance : Inhabited SSet :=
 
 section
 
+#print SSet.asOrderHom /-
 /-- The `m`-simplices of the `n`-th standard simplex are
 the monotone maps from `fin (m+1)` to `fin (n+1)`. -/
 def asOrderHom {n} {m} (α : Δ[n].obj m) : OrderHom (Fin (m.unop.len + 1)) (Fin (n + 1)) :=
   α.toOrderHom
 #align sSet.as_order_hom SSet.asOrderHom
+-/
 
 end
 
@@ -93,12 +94,13 @@ def boundary (n : ℕ) : SSet
 #align sSet.boundary SSet.boundary
 -/
 
--- mathport name: sSet.boundary
 scoped[Simplicial] notation "∂Δ[" n "]" => SSet.boundary n
 
+#print SSet.boundaryInclusion /-
 /-- The inclusion of the boundary of the `n`-th standard simplex into that standard simplex. -/
 def boundaryInclusion (n : ℕ) : ∂Δ[n] ⟶ Δ[n] where app m (α : { α : Δ[n].obj m // _ }) := α
 #align sSet.boundary_inclusion SSet.boundaryInclusion
+-/
 
 #print SSet.horn /-
 /-- `horn n i` (or `Λ[n, i]`) is the `i`-th horn of the `n`-th standard simplex, where `i : n`.
@@ -118,13 +120,14 @@ def horn (n : ℕ) (i : Fin (n + 1)) : SSet
 #align sSet.horn SSet.horn
 -/
 
--- mathport name: sSet.horn
 scoped[Simplicial] notation "Λ[" n ", " i "]" => SSet.horn (n : ℕ) i
 
+#print SSet.hornInclusion /-
 /-- The inclusion of the `i`-th horn of the `n`-th standard simplex into that standard simplex. -/
 def hornInclusion (n : ℕ) (i : Fin (n + 1)) : Λ[n, i] ⟶ Δ[n]
     where app m (α : { α : Δ[n].obj m // _ }) := α
 #align sSet.horn_inclusion SSet.hornInclusion
+-/
 
 section Examples
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -50,7 +50,8 @@ open scoped Simplicial
 This is the category of contravariant functors from
 `simplex_category` to `Type u`. -/
 def SSet : Type (u + 1) :=
-  SimplicialObject (Type u)deriving LargeCategory, Limits.HasLimits, Limits.HasColimits
+  SimplicialObject (Type u)
+deriving LargeCategory, Limits.HasLimits, Limits.HasColimits
 #align sSet SSet
 -/
 
@@ -110,7 +111,7 @@ def horn (n : ℕ) (i : Fin (n + 1)) : SSet
   map m₁ m₂ f α :=
     ⟨f.unop ≫ (α : Δ[n].obj m₁), by
       intro h; apply α.property
-      rw [Set.eq_univ_iff_forall] at h⊢; intro j
+      rw [Set.eq_univ_iff_forall] at h ⊢; intro j
       apply Or.imp _ id (h j)
       intro hj
       exact Set.range_comp_subset_range _ _ hj⟩
@@ -143,7 +144,8 @@ end Examples
 #print SSet.Truncated /-
 /-- Truncated simplicial sets. -/
 def Truncated (n : ℕ) :=
-  SimplicialObject.Truncated (Type u) n deriving LargeCategory, Limits.HasLimits, Limits.HasColimits
+  SimplicialObject.Truncated (Type u) n
+deriving LargeCategory, Limits.HasLimits, Limits.HasColimits
 #align sSet.truncated SSet.Truncated
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -43,7 +43,7 @@ universe v u
 
 open CategoryTheory CategoryTheory.Limits
 
-open Simplicial
+open scoped Simplicial
 
 #print SSet /-
 /-- The category of simplicial sets.
@@ -127,7 +127,7 @@ def hornInclusion (n : ℕ) (i : Fin (n + 1)) : Λ[n, i] ⟶ Δ[n]
 
 section Examples
 
-open Simplicial
+open scoped Simplicial
 
 #print SSet.S1 /-
 /-- The simplicial circle. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -72,12 +72,6 @@ instance : Inhabited SSet :=
 
 section
 
-/- warning: sSet.as_order_hom -> SSet.asOrderHom is a dubious translation:
-lean 3 declaration is
-  forall {n : Nat} {m : Opposite.{1} SimplexCategory}, (CategoryTheory.Functor.obj.{0, 0, 0, 1} (Opposite.{1} SimplexCategory) (CategoryTheory.Category.opposite.{0, 0} SimplexCategory SimplexCategory.smallCategory) Type CategoryTheory.types.{0} (CategoryTheory.Functor.obj.{0, 0, 0, 1} SimplexCategory SimplexCategory.smallCategory SSet.{0} SSet.largeCategory.{0} SSet.standardSimplex (SimplexCategory.mk n)) m) -> (OrderHom.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (SimplexCategory.len (Opposite.unop.{1} SimplexCategory m)) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (SimplexCategory.len (Opposite.unop.{1} SimplexCategory m)) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (SimplexCategory.len (Opposite.unop.{1} SimplexCategory m)) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (SimplexCategory.len (SimplexCategory.mk n)) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (SimplexCategory.len (SimplexCategory.mk n)) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))
-but is expected to have type
-  forall {n : Nat} {m : Opposite.{1} SimplexCategory}, (Prefunctor.obj.{1, 1, 0, 1} (Opposite.{1} SimplexCategory) (CategoryTheory.CategoryStruct.toQuiver.{0, 0} (Opposite.{1} SimplexCategory) (CategoryTheory.Category.toCategoryStruct.{0, 0} (Opposite.{1} SimplexCategory) (CategoryTheory.Category.opposite.{0, 0} SimplexCategory SimplexCategory.smallCategory))) Type (CategoryTheory.CategoryStruct.toQuiver.{0, 1} Type (CategoryTheory.Category.toCategoryStruct.{0, 1} Type CategoryTheory.types.{0})) (CategoryTheory.Functor.toPrefunctor.{0, 0, 0, 1} (Opposite.{1} SimplexCategory) (CategoryTheory.Category.opposite.{0, 0} SimplexCategory SimplexCategory.smallCategory) Type CategoryTheory.types.{0} (Prefunctor.obj.{1, 1, 0, 1} SimplexCategory (CategoryTheory.CategoryStruct.toQuiver.{0, 0} SimplexCategory (CategoryTheory.Category.toCategoryStruct.{0, 0} SimplexCategory SimplexCategory.smallCategory)) SSet.{0} (CategoryTheory.CategoryStruct.toQuiver.{0, 1} SSet.{0} (CategoryTheory.Category.toCategoryStruct.{0, 1} SSet.{0} SSet.largeCategory.{0})) (CategoryTheory.Functor.toPrefunctor.{0, 0, 0, 1} SimplexCategory SimplexCategory.smallCategory SSet.{0} SSet.largeCategory.{0} SSet.standardSimplex) (SimplexCategory.mk n))) m) -> (OrderHom.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (SimplexCategory.len (Opposite.unop.{1} SimplexCategory m)) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (SimplexCategory.len (Opposite.unop.{1} SimplexCategory m)) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instPartialOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (SimplexCategory.len (Opposite.unop.{1} SimplexCategory m)) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instPartialOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))))
-Case conversion may be inaccurate. Consider using '#align sSet.as_order_hom SSet.asOrderHomₓ'. -/
 /-- The `m`-simplices of the `n`-th standard simplex are
 the monotone maps from `fin (m+1)` to `fin (n+1)`. -/
 def asOrderHom {n} {m} (α : Δ[n].obj m) : OrderHom (Fin (m.unop.len + 1)) (Fin (n + 1)) :=
@@ -101,12 +95,6 @@ def boundary (n : ℕ) : SSet
 -- mathport name: sSet.boundary
 scoped[Simplicial] notation "∂Δ[" n "]" => SSet.boundary n
 
-/- warning: sSet.boundary_inclusion -> SSet.boundaryInclusion is a dubious translation:
-lean 3 declaration is
-  forall (n : Nat), Quiver.Hom.{1, 1} SSet.{0} (CategoryTheory.CategoryStruct.toQuiver.{0, 1} SSet.{0} (CategoryTheory.Category.toCategoryStruct.{0, 1} SSet.{0} SSet.largeCategory.{0})) (SSet.boundary n) (CategoryTheory.Functor.obj.{0, 0, 0, 1} SimplexCategory SimplexCategory.smallCategory SSet.{0} SSet.largeCategory.{0} SSet.standardSimplex (SimplexCategory.mk n))
-but is expected to have type
-  forall (n : Nat), Quiver.Hom.{1, 1} SSet.{0} (CategoryTheory.CategoryStruct.toQuiver.{0, 1} SSet.{0} (CategoryTheory.Category.toCategoryStruct.{0, 1} SSet.{0} SSet.largeCategory.{0})) (SSet.boundary n) (Prefunctor.obj.{1, 1, 0, 1} SimplexCategory (CategoryTheory.CategoryStruct.toQuiver.{0, 0} SimplexCategory (CategoryTheory.Category.toCategoryStruct.{0, 0} SimplexCategory SimplexCategory.smallCategory)) SSet.{0} (CategoryTheory.CategoryStruct.toQuiver.{0, 1} SSet.{0} (CategoryTheory.Category.toCategoryStruct.{0, 1} SSet.{0} SSet.largeCategory.{0})) (CategoryTheory.Functor.toPrefunctor.{0, 0, 0, 1} SimplexCategory SimplexCategory.smallCategory SSet.{0} SSet.largeCategory.{0} SSet.standardSimplex) (SimplexCategory.mk n))
-Case conversion may be inaccurate. Consider using '#align sSet.boundary_inclusion SSet.boundaryInclusionₓ'. -/
 /-- The inclusion of the boundary of the `n`-th standard simplex into that standard simplex. -/
 def boundaryInclusion (n : ℕ) : ∂Δ[n] ⟶ Δ[n] where app m (α : { α : Δ[n].obj m // _ }) := α
 #align sSet.boundary_inclusion SSet.boundaryInclusion
@@ -132,12 +120,6 @@ def horn (n : ℕ) (i : Fin (n + 1)) : SSet
 -- mathport name: sSet.horn
 scoped[Simplicial] notation "Λ[" n ", " i "]" => SSet.horn (n : ℕ) i
 
-/- warning: sSet.horn_inclusion -> SSet.hornInclusion is a dubious translation:
-lean 3 declaration is
-  forall (n : Nat) (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), Quiver.Hom.{1, 1} SSet.{0} (CategoryTheory.CategoryStruct.toQuiver.{0, 1} SSet.{0} (CategoryTheory.Category.toCategoryStruct.{0, 1} SSet.{0} SSet.largeCategory.{0})) (SSet.horn n i) (CategoryTheory.Functor.obj.{0, 0, 0, 1} SimplexCategory SimplexCategory.smallCategory SSet.{0} SSet.largeCategory.{0} SSet.standardSimplex (SimplexCategory.mk n))
-but is expected to have type
-  forall (n : Nat) (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), Quiver.Hom.{1, 1} SSet.{0} (CategoryTheory.CategoryStruct.toQuiver.{0, 1} SSet.{0} (CategoryTheory.Category.toCategoryStruct.{0, 1} SSet.{0} SSet.largeCategory.{0})) (SSet.horn n i) (Prefunctor.obj.{1, 1, 0, 1} SimplexCategory (CategoryTheory.CategoryStruct.toQuiver.{0, 0} SimplexCategory (CategoryTheory.Category.toCategoryStruct.{0, 0} SimplexCategory SimplexCategory.smallCategory)) SSet.{0} (CategoryTheory.CategoryStruct.toQuiver.{0, 1} SSet.{0} (CategoryTheory.Category.toCategoryStruct.{0, 1} SSet.{0} SSet.largeCategory.{0})) (CategoryTheory.Functor.toPrefunctor.{0, 0, 0, 1} SimplexCategory SimplexCategory.smallCategory SSet.{0} SSet.largeCategory.{0} SSet.standardSimplex) (SimplexCategory.mk n))
-Case conversion may be inaccurate. Consider using '#align sSet.horn_inclusion SSet.hornInclusionₓ'. -/
 /-- The inclusion of the `i`-th horn of the `n`-th standard simplex into that standard simplex. -/
 def hornInclusion (n : ℕ) (i : Fin (n + 1)) : Λ[n, i] ⟶ Δ[n]
     where app m (α : { α : Δ[n].obj m // _ }) := α

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -94,10 +94,7 @@ def boundary (n : ℕ) : SSet
     where
   obj m := { α : Δ[n].obj m // ¬Function.Surjective (asOrderHom α) }
   map m₁ m₂ f α :=
-    ⟨f.unop ≫ (α : Δ[n].obj m₁), by
-      intro h
-      apply α.property
-      exact Function.Surjective.of_comp h⟩
+    ⟨f.unop ≫ (α : Δ[n].obj m₁), by intro h; apply α.property; exact Function.Surjective.of_comp h⟩
 #align sSet.boundary SSet.boundary
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/86d04064ca33ee3d3405fbfc497d494fd2dd4796

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin, Scott Morrison, Adam Topaz
 
 ! This file was ported from Lean 3 source module algebraic_topology.simplicial_set
-! leanprover-community/mathlib commit 178a32653e369dce2da68dc6b2694e385d484ef1
+! leanprover-community/mathlib commit f2b757fc5c341d88741b9c4630b1e8ba973c5726
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -29,6 +29,9 @@ and their boundaries `∂Δ[n]` and horns `Λ[n, i]`.
 
 ## Future work
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 There isn't yet a complete API for simplices, boundaries, and horns.
 As an example, we should have a function that constructs
 from a non-surjective order preserving function `fin n → fin n`

mathlib commit https://github.com/leanprover-community/mathlib/commit/fa78268d4d77cb2b2fbc89f0527e2e7807763780

@@ -42,20 +42,24 @@ open CategoryTheory CategoryTheory.Limits
 
 open Simplicial
 
+#print SSet /-
 /-- The category of simplicial sets.
 This is the category of contravariant functors from
 `simplex_category` to `Type u`. -/
 def SSet : Type (u + 1) :=
   SimplicialObject (Type u)deriving LargeCategory, Limits.HasLimits, Limits.HasColimits
 #align sSet SSet
+-/
 
 namespace SSet
 
+#print SSet.standardSimplex /-
 /-- The `n`-th standard simplex `Δ[n]` associated with a nonempty finite linear order `n`
 is the Yoneda embedding of `n`. -/
 def standardSimplex : SimplexCategory ⥤ SSet :=
   yoneda
 #align sSet.standard_simplex SSet.standardSimplex
+-/
 
 -- mathport name: standard_simplex
 scoped[Simplicial] notation "Δ[" n "]" => SSet.standardSimplex.obj (SimplexCategory.mk n)
@@ -65,6 +69,12 @@ instance : Inhabited SSet :=
 
 section
 
+/- warning: sSet.as_order_hom -> SSet.asOrderHom is a dubious translation:
+lean 3 declaration is
+  forall {n : Nat} {m : Opposite.{1} SimplexCategory}, (CategoryTheory.Functor.obj.{0, 0, 0, 1} (Opposite.{1} SimplexCategory) (CategoryTheory.Category.opposite.{0, 0} SimplexCategory SimplexCategory.smallCategory) Type CategoryTheory.types.{0} (CategoryTheory.Functor.obj.{0, 0, 0, 1} SimplexCategory SimplexCategory.smallCategory SSet.{0} SSet.largeCategory.{0} SSet.standardSimplex (SimplexCategory.mk n)) m) -> (OrderHom.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (SimplexCategory.len (Opposite.unop.{1} SimplexCategory m)) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (SimplexCategory.len (Opposite.unop.{1} SimplexCategory m)) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (SimplexCategory.len (Opposite.unop.{1} SimplexCategory m)) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (SimplexCategory.len (SimplexCategory.mk n)) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (SimplexCategory.len (SimplexCategory.mk n)) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))
+but is expected to have type
+  forall {n : Nat} {m : Opposite.{1} SimplexCategory}, (Prefunctor.obj.{1, 1, 0, 1} (Opposite.{1} SimplexCategory) (CategoryTheory.CategoryStruct.toQuiver.{0, 0} (Opposite.{1} SimplexCategory) (CategoryTheory.Category.toCategoryStruct.{0, 0} (Opposite.{1} SimplexCategory) (CategoryTheory.Category.opposite.{0, 0} SimplexCategory SimplexCategory.smallCategory))) Type (CategoryTheory.CategoryStruct.toQuiver.{0, 1} Type (CategoryTheory.Category.toCategoryStruct.{0, 1} Type CategoryTheory.types.{0})) (CategoryTheory.Functor.toPrefunctor.{0, 0, 0, 1} (Opposite.{1} SimplexCategory) (CategoryTheory.Category.opposite.{0, 0} SimplexCategory SimplexCategory.smallCategory) Type CategoryTheory.types.{0} (Prefunctor.obj.{1, 1, 0, 1} SimplexCategory (CategoryTheory.CategoryStruct.toQuiver.{0, 0} SimplexCategory (CategoryTheory.Category.toCategoryStruct.{0, 0} SimplexCategory SimplexCategory.smallCategory)) SSet.{0} (CategoryTheory.CategoryStruct.toQuiver.{0, 1} SSet.{0} (CategoryTheory.Category.toCategoryStruct.{0, 1} SSet.{0} SSet.largeCategory.{0})) (CategoryTheory.Functor.toPrefunctor.{0, 0, 0, 1} SimplexCategory SimplexCategory.smallCategory SSet.{0} SSet.largeCategory.{0} SSet.standardSimplex) (SimplexCategory.mk n))) m) -> (OrderHom.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (SimplexCategory.len (Opposite.unop.{1} SimplexCategory m)) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (SimplexCategory.len (Opposite.unop.{1} SimplexCategory m)) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instPartialOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (SimplexCategory.len (Opposite.unop.{1} SimplexCategory m)) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instPartialOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))))
+Case conversion may be inaccurate. Consider using '#align sSet.as_order_hom SSet.asOrderHomₓ'. -/
 /-- The `m`-simplices of the `n`-th standard simplex are
 the monotone maps from `fin (m+1)` to `fin (n+1)`. -/
 def asOrderHom {n} {m} (α : Δ[n].obj m) : OrderHom (Fin (m.unop.len + 1)) (Fin (n + 1)) :=
@@ -73,6 +83,7 @@ def asOrderHom {n} {m} (α : Δ[n].obj m) : OrderHom (Fin (m.unop.len + 1)) (Fin
 
 end
 
+#print SSet.boundary /-
 /-- The boundary `∂Δ[n]` of the `n`-th standard simplex consists of
 all `m`-simplices of `standard_simplex n` that are not surjective
 (when viewed as monotone function `m → n`). -/
@@ -85,14 +96,22 @@ def boundary (n : ℕ) : SSet
       apply α.property
       exact Function.Surjective.of_comp h⟩
 #align sSet.boundary SSet.boundary
+-/
 
 -- mathport name: sSet.boundary
 scoped[Simplicial] notation "∂Δ[" n "]" => SSet.boundary n
 
+/- warning: sSet.boundary_inclusion -> SSet.boundaryInclusion is a dubious translation:
+lean 3 declaration is
+  forall (n : Nat), Quiver.Hom.{1, 1} SSet.{0} (CategoryTheory.CategoryStruct.toQuiver.{0, 1} SSet.{0} (CategoryTheory.Category.toCategoryStruct.{0, 1} SSet.{0} SSet.largeCategory.{0})) (SSet.boundary n) (CategoryTheory.Functor.obj.{0, 0, 0, 1} SimplexCategory SimplexCategory.smallCategory SSet.{0} SSet.largeCategory.{0} SSet.standardSimplex (SimplexCategory.mk n))
+but is expected to have type
+  forall (n : Nat), Quiver.Hom.{1, 1} SSet.{0} (CategoryTheory.CategoryStruct.toQuiver.{0, 1} SSet.{0} (CategoryTheory.Category.toCategoryStruct.{0, 1} SSet.{0} SSet.largeCategory.{0})) (SSet.boundary n) (Prefunctor.obj.{1, 1, 0, 1} SimplexCategory (CategoryTheory.CategoryStruct.toQuiver.{0, 0} SimplexCategory (CategoryTheory.Category.toCategoryStruct.{0, 0} SimplexCategory SimplexCategory.smallCategory)) SSet.{0} (CategoryTheory.CategoryStruct.toQuiver.{0, 1} SSet.{0} (CategoryTheory.Category.toCategoryStruct.{0, 1} SSet.{0} SSet.largeCategory.{0})) (CategoryTheory.Functor.toPrefunctor.{0, 0, 0, 1} SimplexCategory SimplexCategory.smallCategory SSet.{0} SSet.largeCategory.{0} SSet.standardSimplex) (SimplexCategory.mk n))
+Case conversion may be inaccurate. Consider using '#align sSet.boundary_inclusion SSet.boundaryInclusionₓ'. -/
 /-- The inclusion of the boundary of the `n`-th standard simplex into that standard simplex. -/
 def boundaryInclusion (n : ℕ) : ∂Δ[n] ⟶ Δ[n] where app m (α : { α : Δ[n].obj m // _ }) := α
 #align sSet.boundary_inclusion SSet.boundaryInclusion
 
+#print SSet.horn /-
 /-- `horn n i` (or `Λ[n, i]`) is the `i`-th horn of the `n`-th standard simplex, where `i : n`.
 It consists of all `m`-simplices `α` of `Δ[n]`
 for which the union of `{i}` and the range of `α` is not all of `n`
@@ -108,10 +127,17 @@ def horn (n : ℕ) (i : Fin (n + 1)) : SSet
       intro hj
       exact Set.range_comp_subset_range _ _ hj⟩
 #align sSet.horn SSet.horn
+-/
 
 -- mathport name: sSet.horn
 scoped[Simplicial] notation "Λ[" n ", " i "]" => SSet.horn (n : ℕ) i
 
+/- warning: sSet.horn_inclusion -> SSet.hornInclusion is a dubious translation:
+lean 3 declaration is
+  forall (n : Nat) (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), Quiver.Hom.{1, 1} SSet.{0} (CategoryTheory.CategoryStruct.toQuiver.{0, 1} SSet.{0} (CategoryTheory.Category.toCategoryStruct.{0, 1} SSet.{0} SSet.largeCategory.{0})) (SSet.horn n i) (CategoryTheory.Functor.obj.{0, 0, 0, 1} SimplexCategory SimplexCategory.smallCategory SSet.{0} SSet.largeCategory.{0} SSet.standardSimplex (SimplexCategory.mk n))
+but is expected to have type
+  forall (n : Nat) (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), Quiver.Hom.{1, 1} SSet.{0} (CategoryTheory.CategoryStruct.toQuiver.{0, 1} SSet.{0} (CategoryTheory.Category.toCategoryStruct.{0, 1} SSet.{0} SSet.largeCategory.{0})) (SSet.horn n i) (Prefunctor.obj.{1, 1, 0, 1} SimplexCategory (CategoryTheory.CategoryStruct.toQuiver.{0, 0} SimplexCategory (CategoryTheory.Category.toCategoryStruct.{0, 0} SimplexCategory SimplexCategory.smallCategory)) SSet.{0} (CategoryTheory.CategoryStruct.toQuiver.{0, 1} SSet.{0} (CategoryTheory.Category.toCategoryStruct.{0, 1} SSet.{0} SSet.largeCategory.{0})) (CategoryTheory.Functor.toPrefunctor.{0, 0, 0, 1} SimplexCategory SimplexCategory.smallCategory SSet.{0} SSet.largeCategory.{0} SSet.standardSimplex) (SimplexCategory.mk n))
+Case conversion may be inaccurate. Consider using '#align sSet.horn_inclusion SSet.hornInclusionₓ'. -/
 /-- The inclusion of the `i`-th horn of the `n`-th standard simplex into that standard simplex. -/
 def hornInclusion (n : ℕ) (i : Fin (n + 1)) : Λ[n, i] ⟶ Δ[n]
     where app m (α : { α : Δ[n].obj m // _ }) := α
@@ -121,36 +147,45 @@ section Examples
 
 open Simplicial
 
+#print SSet.S1 /-
 /-- The simplicial circle. -/
-noncomputable def s1 : SSet :=
+noncomputable def S1 : SSet :=
   Limits.colimit <|
     Limits.parallelPair (standardSimplex.map <| SimplexCategory.δ 0 : Δ[0] ⟶ Δ[1])
       (standardSimplex.map <| SimplexCategory.δ 1)
-#align sSet.S1 SSet.s1
+#align sSet.S1 SSet.S1
+-/
 
 end Examples
 
+#print SSet.Truncated /-
 /-- Truncated simplicial sets. -/
 def Truncated (n : ℕ) :=
   SimplicialObject.Truncated (Type u) n deriving LargeCategory, Limits.HasLimits, Limits.HasColimits
 #align sSet.truncated SSet.Truncated
+-/
 
+#print SSet.sk /-
 /-- The skeleton functor on simplicial sets. -/
 def sk (n : ℕ) : SSet ⥤ SSet.Truncated n :=
   SimplicialObject.sk n
 #align sSet.sk SSet.sk
+-/
 
 instance {n} : Inhabited (SSet.Truncated n) :=
   ⟨(sk n).obj <| Δ[0]⟩
 
+#print SSet.Augmented /-
 /-- The category of augmented simplicial sets, as a particular case of
 augmented simplicial objects. -/
 abbrev Augmented :=
   SimplicialObject.Augmented (Type u)
 #align sSet.augmented SSet.Augmented
+-/
 
 namespace Augmented
 
+#print SSet.Augmented.standardSimplex /-
 /-- The functor which sends `[n]` to the simplicial set `Δ[n]` equipped by
 the obvious augmentation towards the terminal object of the category of sets. -/
 @[simps]
@@ -164,30 +199,39 @@ noncomputable def standardSimplex : SimplexCategory ⥤ SSet.Augmented
     { left := SSet.standardSimplex.map θ
       right := terminal.from _ }
 #align sSet.augmented.standard_simplex SSet.Augmented.standardSimplex
+-/
 
 end Augmented
 
 end SSet
 
+#print TopCat.toSSet /-
 /-- The functor associating the singular simplicial set to a topological space. -/
 def TopCat.toSSet : TopCat ⥤ SSet :=
   ColimitAdj.restrictedYoneda SimplexCategory.toTop
 #align Top.to_sSet TopCat.toSSet
+-/
 
+#print SSet.toTop /-
 /-- The geometric realization functor. -/
 noncomputable def SSet.toTop : SSet ⥤ TopCat :=
   ColimitAdj.extendAlongYoneda SimplexCategory.toTop
 #align sSet.to_Top SSet.toTop
+-/
 
+#print sSetTopAdj /-
 /-- Geometric realization is left adjoint to the singular simplicial set construction. -/
 noncomputable def sSetTopAdj : SSet.toTop ⊣ TopCat.toSSet :=
   ColimitAdj.yonedaAdjunction _
 #align sSet_Top_adj sSetTopAdj
+-/
 
+#print SSet.toTopSimplex /-
 /-- The geometric realization of the representable simplicial sets agree
   with the usual topological simplices. -/
 noncomputable def SSet.toTopSimplex :
     (yoneda : SimplexCategory ⥤ _) ⋙ SSet.toTop ≅ SimplexCategory.toTop :=
   ColimitAdj.isExtensionAlongYoneda _
 #align sSet.to_Top_simplex SSet.toTopSimplex
+-/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9b2b58d6b14b895b2f375108e765cb47de71aebd

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin, Scott Morrison, Adam Topaz
 
 ! This file was ported from Lean 3 source module algebraic_topology.simplicial_set
-! leanprover-community/mathlib commit d79c3671549ace83a93d2a97f316d12ab42232ae
+! leanprover-community/mathlib commit 178a32653e369dce2da68dc6b2694e385d484ef1
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -13,7 +13,7 @@ import Mathbin.AlgebraicTopology.TopologicalSimplex
 import Mathbin.CategoryTheory.Limits.Presheaf
 import Mathbin.CategoryTheory.Limits.Types
 import Mathbin.CategoryTheory.Yoneda
-import Mathbin.Topology.Category.Top.Limits
+import Mathbin.Topology.Category.Top.Limits.Basic
 
 /-!
 A simplicial set is just a simplicial object in `Type`,

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

Scatter the content of Data.Nat.Basic across:

• Data.Nat.Defs for the lemmas having no dependencies
• Algebra.Group.Nat for the monoid instances and the few miscellaneous lemmas needing them.
• Algebra.Ring.Nat for the semiring instance and the few miscellaneous lemmas following it.

Similarly, scatter

• Data.Int.Basic across Data.Int.Defs, Algebra.Group.Int, Algebra.Ring.Int
• Data.Nat.Order.Basic across Data.Nat.Defs, Algebra.Order.Group.Nat, Algebra.Order.Ring.Nat
• Data.Int.Order.Basic across Data.Int.Defs, Algebra.Order.Group.Int, Algebra.Order.Ring.Int

Also move a few lemmas from Data.Nat.Order.Lemmas to Data.Nat.Defs.

Before

After

@@ -7,6 +7,7 @@ import Mathlib.AlgebraicTopology.SimplicialObject
 import Mathlib.CategoryTheory.Limits.Shapes.Types
 import Mathlib.CategoryTheory.Yoneda
 import Mathlib.Data.Fin.VecNotation
+import Mathlib.Data.Nat.Units
 import Mathlib.Tactic.FinCases
 
 #align_import algebraic_topology.simplicial_set from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d484ef1"

We cannot literally use @[inherit_doc] in these cases, but we can slightly modify the underlying docstring or a turn a regular comment into a doc comment.

@@ -78,7 +78,6 @@ def standardSimplex : SimplexCategory ⥤ SSet.{u} :=
 set_option linter.uppercaseLean3 false in
 #align sSet.standard_simplex SSet.standardSimplex
 
--- mathport name: standard_simplex
 @[inherit_doc SSet.standardSimplex]
 scoped[Simplicial] notation3 "Δ[" n "]" => SSet.standardSimplex.obj (SimplexCategory.mk n)
 
@@ -173,7 +172,7 @@ def boundary (n : ℕ) : SSet.{u} where
 set_option linter.uppercaseLean3 false in
 #align sSet.boundary SSet.boundary
 
--- mathport name: sSet.boundary
+/-- The boundary `∂Δ[n]` of the `n`-th standard simplex -/
 scoped[Simplicial] notation3 "∂Δ[" n "]" => SSet.boundary n
 
 /-- The inclusion of the boundary of the `n`-th standard simplex into that standard simplex. -/
@@ -197,7 +196,7 @@ def horn (n : ℕ) (i : Fin (n + 1)) : SSet where
 set_option linter.uppercaseLean3 false in
 #align sSet.horn SSet.horn
 
--- mathport name: sSet.horn
+/-- The `i`-th horn `Λ[n, i]` of the standard `n`-simplex -/
 scoped[Simplicial] notation3 "Λ[" n ", " i "]" => SSet.horn (n : ℕ) i
 
 /-- The inclusion of the `i`-th horn of the `n`-th standard simplex into that standard simplex. -/

... or reduce its scope (the full removal is not as obvious).

@@ -33,9 +33,6 @@ from a non-surjective order preserving function `Fin n → Fin n`
 a morphism `Δ[n] ⟶ ∂Δ[n]`.
 -/
 
-set_option autoImplicit true
-
-
 universe v u
 
 open CategoryTheory CategoryTheory.Limits
@@ -352,18 +349,19 @@ instance Truncated.largeCategory (n : ℕ) : LargeCategory (Truncated n) := by
   dsimp only [Truncated]
   infer_instance
 
-instance Truncated.hasLimits : HasLimits (Truncated n) := by
+instance Truncated.hasLimits {n : ℕ} : HasLimits (Truncated n) := by
   dsimp only [Truncated]
   infer_instance
 
-instance Truncated.hasColimits : HasColimits (Truncated n) := by
+instance Truncated.hasColimits {n : ℕ} : HasColimits (Truncated n) := by
   dsimp only [Truncated]
   infer_instance
 
 -- Porting note (#10756): added an `ext` lemma.
 -- See https://github.com/leanprover-community/mathlib4/issues/5229
 @[ext]
-lemma Truncated.hom_ext {X Y : Truncated n} {f g : X ⟶ Y} (w : ∀ n, f.app n = g.app n) : f = g :=
+lemma Truncated.hom_ext {n : ℕ} {X Y : Truncated n} {f g : X ⟶ Y} (w : ∀ n, f.app n = g.app n) :
+    f = g :=
   NatTrans.ext _ _ (funext w)
 
 /-- The skeleton functor on simplicial sets. -/

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

• converts "--porting note" into "-- Porting note";
• converts "porting note" into "Porting note".

@@ -384,7 +384,7 @@ set_option linter.uppercaseLean3 false in
 
 namespace Augmented
 
--- porting note: an instance of `Subsingleton (⊤_ (Type u))` was added in
+-- Porting note: an instance of `Subsingleton (⊤_ (Type u))` was added in
 -- `CategoryTheory.Limits.Types` to ease the automation in this definition
 /-- The functor which sends `[n]` to the simplicial set `Δ[n]` equipped by
 the obvious augmentation towards the terminal object of the category of sets. -/

Classifies by adding issue number #10756 to porting notes claiming added lemma.

@@ -64,7 +64,7 @@ instance hasColimits : HasColimits SSet := by
   dsimp only [SSet]
   infer_instance
 
--- Porting note: added an `ext` lemma.
+-- Porting note (#10756): added an `ext` lemma.
 -- See https://github.com/leanprover-community/mathlib4/issues/5229
 @[ext]
 lemma hom_ext {X Y : SSet} {f g : X ⟶ Y} (w : ∀ n, f.app n = g.app n) : f = g :=
@@ -360,7 +360,7 @@ instance Truncated.hasColimits : HasColimits (Truncated n) := by
   dsimp only [Truncated]
   infer_instance
 
--- Porting note: added an `ext` lemma.
+-- Porting note (#10756): added an `ext` lemma.
 -- See https://github.com/leanprover-community/mathlib4/issues/5229
 @[ext]
 lemma Truncated.hom_ext {X Y : Truncated n} {f g : X ⟶ Y} (w : ∀ n, f.app n = g.app n) : f = g :=

Make all the notations that unambiguously should inherit the docstring of their definition actually inherit it.

Also write a few docstrings by hand. I only wrote the ones I was competent to write and which I was sure of. Some docstrings come from mathlib3 as they were lost during the early port.

This PR is only intended as a first pass There are many more docstrings to add.

@@ -82,6 +82,7 @@ set_option linter.uppercaseLean3 false in
 #align sSet.standard_simplex SSet.standardSimplex
 
 -- mathport name: standard_simplex
+@[inherit_doc SSet.standardSimplex]
 scoped[Simplicial] notation3 "Δ[" n "]" => SSet.standardSimplex.obj (SimplexCategory.mk n)
 
 instance : Inhabited SSet :=

This uses the improved shake script from #9772 to reduce imports across mathlib. The corresponding noshake.json file has been added to #9772.

Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin, Scott Morrison, Adam Topaz
 -/
 import Mathlib.AlgebraicTopology.SimplicialObject
-import Mathlib.CategoryTheory.Limits.Presheaf
 import Mathlib.CategoryTheory.Limits.Shapes.Types
 import Mathlib.CategoryTheory.Yoneda
 import Mathlib.Data.Fin.VecNotation

The PR generalizes universes for standardSimplex, which is now a functor SimplexCategory ⥤ SSet.{u} for any universe u.

@@ -71,10 +71,14 @@ instance hasColimits : HasColimits SSet := by
 lemma hom_ext {X Y : SSet} {f g : X ⟶ Y} (w : ∀ n, f.app n = g.app n) : f = g :=
   SimplicialObject.hom_ext _ _ w
 
+/-- The ulift functor `SSet.{u} ⥤ SSet.{max u v}` on simplicial sets. -/
+def uliftFunctor : SSet.{u} ⥤ SSet.{max u v} :=
+  (SimplicialObject.whiskering _ _).obj CategoryTheory.uliftFunctor.{v, u}
+
 /-- The `n`-th standard simplex `Δ[n]` associated with a nonempty finite linear order `n`
 is the Yoneda embedding of `n`. -/
-def standardSimplex : SimplexCategory ⥤ SSet :=
-  yoneda
+def standardSimplex : SimplexCategory ⥤ SSet.{u} :=
+  yoneda ⋙ uliftFunctor
 set_option linter.uppercaseLean3 false in
 #align sSet.standard_simplex SSet.standardSimplex
 
@@ -88,36 +92,62 @@ namespace standardSimplex
 
 open Finset Opposite SimplexCategory
 
+@[simp]
+lemma map_id (n : SimplexCategory) :
+    (SSet.standardSimplex.map (SimplexCategory.Hom.mk OrderHom.id : n ⟶ n)) = 𝟙 _ :=
+  CategoryTheory.Functor.map_id _ _
+
+/-- Simplices of the standard simplex identify to morphisms in `SimplexCategory`. -/
+def objEquiv (n : SimplexCategory) (m : SimplexCategoryᵒᵖ) :
+    (standardSimplex.{u}.obj n).obj m ≃ (m.unop ⟶ n) :=
+  Equiv.ulift.{u, 0}
+
+/-- Constructor for simplices of the standard simplex which takes a `OrderHom` as an input. -/
+abbrev objMk {n : SimplexCategory} {m : SimplexCategoryᵒᵖ}
+    (f : Fin (len m.unop + 1) →o Fin (n.len + 1)) :
+    (standardSimplex.{u}.obj n).obj m :=
+  (objEquiv _ _).symm (Hom.mk f)
+
+lemma map_apply {m₁ m₂ : SimplexCategoryᵒᵖ} (f : m₁ ⟶ m₂) {n : SimplexCategory}
+    (x : (standardSimplex.{u}.obj n).obj m₁) :
+    (standardSimplex.{u}.obj n).map f x = (objEquiv _ _).symm (f.unop ≫ (objEquiv _ _) x) := by
+  rfl
+
+/-- The canonical bijection `(standardSimplex.obj n ⟶ X) ≃ X.obj (op n)`. -/
+def _root_.SSet.yonedaEquiv (X : SSet.{u}) (n : SimplexCategory) :
+    (standardSimplex.obj n ⟶ X) ≃ X.obj (op n) :=
+  yonedaCompUliftFunctorEquiv X n
+
 /-- The (degenerate) `m`-simplex in the standard simplex concentrated in vertex `k`. -/
 def const (n : ℕ) (k : Fin (n+1)) (m : SimplexCategoryᵒᵖ) : Δ[n].obj m :=
-  Hom.mk <| OrderHom.const _ k
+  objMk (OrderHom.const _ k )
 
 @[simp]
-lemma const_toOrderHom (n : ℕ) (k : Fin (n+1)) (m : SimplexCategoryᵒᵖ) :
-    (const n k m).toOrderHom = OrderHom.const _ k :=
+lemma const_down_toOrderHom (n : ℕ) (k : Fin (n+1)) (m : SimplexCategoryᵒᵖ) :
+    (const n k m).down.toOrderHom = OrderHom.const _ k :=
   rfl
 
 /-- The edge of the standard simplex with endpoints `a` and `b`. -/
 def edge (n : ℕ) (a b : Fin (n+1)) (hab : a ≤ b) : Δ[n] _[1] := by
-  refine Hom.mk ⟨![a, b], ?_⟩
+  refine objMk ⟨![a, b], ?_⟩
   rw [Fin.monotone_iff_le_succ]
   simp only [unop_op, len_mk, Fin.forall_fin_one]
   apply Fin.mk_le_mk.mpr hab
 
-lemma coe_edge_toOrderHom (n : ℕ) (a b : Fin (n+1)) (hab : a ≤ b) :
-    ↑(edge n a b hab).toOrderHom = ![a, b] :=
+lemma coe_edge_down_toOrderHom (n : ℕ) (a b : Fin (n+1)) (hab : a ≤ b) :
+    ↑(edge n a b hab).down.toOrderHom = ![a, b] :=
   rfl
 
 /-- The triangle in the standard simplex with vertices `a`, `b`, and `c`. -/
 def triangle {n : ℕ} (a b c : Fin (n+1)) (hab : a ≤ b) (hbc : b ≤ c) : Δ[n] _[2] := by
-  refine Hom.mk ⟨![a, b, c], ?_⟩
+  refine objMk ⟨![a, b, c], ?_⟩
   rw [Fin.monotone_iff_le_succ]
   simp only [unop_op, len_mk, Fin.forall_fin_two]
   dsimp
   simp only [*, Matrix.tail_cons, Matrix.head_cons, true_and]
 
-lemma coe_triangle_toOrderHom {n : ℕ} (a b c : Fin (n+1)) (hab : a ≤ b) (hbc : b ≤ c) :
-    ↑(triangle a b c hab hbc).toOrderHom = ![a, b, c] :=
+lemma coe_triangle_down_toOrderHom {n : ℕ} (a b c : Fin (n+1)) (hab : a ≤ b) (hbc : b ≤ c) :
+    ↑(triangle a b c hab hbc).down.toOrderHom = ![a, b, c] :=
   rfl
 
 end standardSimplex
@@ -127,7 +157,7 @@ section
 /-- The `m`-simplices of the `n`-th standard simplex are
 the monotone maps from `Fin (m+1)` to `Fin (n+1)`. -/
 def asOrderHom {n} {m} (α : Δ[n].obj m) : OrderHom (Fin (m.unop.len + 1)) (Fin (n + 1)) :=
-  α.toOrderHom
+  α.down.toOrderHom
 set_option linter.uppercaseLean3 false in
 #align sSet.as_order_hom SSet.asOrderHom
 
@@ -136,10 +166,10 @@ end
 /-- The boundary `∂Δ[n]` of the `n`-th standard simplex consists of
 all `m`-simplices of `standardSimplex n` that are not surjective
 (when viewed as monotone function `m → n`). -/
-def boundary (n : ℕ) : SSet where
+def boundary (n : ℕ) : SSet.{u} where
   obj m := { α : Δ[n].obj m // ¬Function.Surjective (asOrderHom α) }
   map {m₁ m₂} f α :=
-    ⟨f.unop ≫ (α : Δ[n].obj m₁), by
+    ⟨Δ[n].map f α.1, by
       intro h
       apply α.property
       exact Function.Surjective.of_comp h⟩
@@ -161,7 +191,7 @@ for which the union of `{i}` and the range of `α` is not all of `n`
 def horn (n : ℕ) (i : Fin (n + 1)) : SSet where
   obj m := { α : Δ[n].obj m // Set.range (asOrderHom α) ∪ {i} ≠ Set.univ }
   map {m₁ m₂} f α :=
-    ⟨f.unop ≫ (α : Δ[n].obj m₁), by
+    ⟨Δ[n].map f α.1, by
       intro h; apply α.property
       rw [Set.eq_univ_iff_forall] at h ⊢; intro j
       apply Or.imp _ id (h j)
@@ -253,7 +283,8 @@ def primitiveTriangle {n : ℕ} (i : Fin (n+4))
     OrderHom.const_coe_coe, Set.union_singleton, ne_eq, ← Set.univ_subset_iff, Set.subset_def,
     Set.mem_univ, Set.mem_insert_iff, Set.mem_range, Function.const_apply, exists_const,
     forall_true_left, not_forall, not_or, unop_op, not_exists,
-    standardSimplex.triangle, OrderHom.coe_mk, @eq_comm _ _ i]
+    standardSimplex.triangle, OrderHom.coe_mk, @eq_comm _ _ i,
+    standardSimplex.objMk, standardSimplex.objEquiv, Equiv.ulift]
   dsimp
   by_cases hk0 : k = 0
   · subst hk0
@@ -268,9 +299,10 @@ def primitiveTriangle {n : ℕ} (i : Fin (n+4))
 
 /-- The `j`th subface of the `i`-th horn. -/
 @[simps]
-def face {n : ℕ} (i j : Fin (n+2)) (h : j ≠ i) : Λ[n+1, i] _[n] := by
-  refine ⟨SimplexCategory.δ j, ?_⟩
-  simpa [← Set.univ_subset_iff, Set.subset_def, asOrderHom, SimplexCategory.δ, not_or]
+def face {n : ℕ} (i j : Fin (n+2)) (h : j ≠ i) : Λ[n+1, i] _[n] :=
+  ⟨(standardSimplex.objEquiv _ _).symm (SimplexCategory.δ j), by
+    simpa [← Set.univ_subset_iff, Set.subset_def, asOrderHom, SimplexCategory.δ, not_or,
+      standardSimplex.objEquiv, asOrderHom, Equiv.ulift]⟩
 
 /-- Two morphisms from a horn are equal if they are equal on all suitable faces. -/
 protected
@@ -279,13 +311,18 @@ lemma hom_ext {n : ℕ} {i : Fin (n+2)} {S : SSet} (σ₁ σ₂ : Λ[n+1, i] ⟶
     σ₁ = σ₂ := by
   apply NatTrans.ext; apply funext; apply Opposite.rec; apply SimplexCategory.rec
   intro m; ext f
-  obtain ⟨j, hji, hfj⟩ : ∃ j, ¬j = i ∧ ∀ k, f.1.toOrderHom k ≠ j := by
-    simpa [← Set.univ_subset_iff, Set.subset_def, asOrderHom, not_or] using f.2
-  have H : f = (Λ[n+1, i].map (factor_δ f.1 j).op) (face i j hji) := by
+  obtain ⟨f', hf⟩ := (standardSimplex.objEquiv _ _).symm.surjective f.1
+  obtain ⟨j, hji, hfj⟩ : ∃ j, ¬j = i ∧ ∀ k, f'.toOrderHom k ≠ j := by
+    obtain ⟨f, hf'⟩ := f
+    subst hf
+    simpa [← Set.univ_subset_iff, Set.subset_def, asOrderHom, not_or] using hf'
+  have H : f = (Λ[n+1, i].map (factor_δ f' j).op) (face i j hji) := by
     apply Subtype.ext
-    exact (factor_δ_spec f.1 j hfj).symm
-  have H₁ := congrFun (σ₁.naturality (factor_δ f.1 j).op) (face i j hji)
-  have H₂ := congrFun (σ₂.naturality (factor_δ f.1 j).op) (face i j hji)
+    apply (standardSimplex.objEquiv _ _).injective
+    rw [← hf]
+    exact (factor_δ_spec f' j hfj).symm
+  have H₁ := congrFun (σ₁.naturality (factor_δ f' j).op) (face i j hji)
+  have H₂ := congrFun (σ₂.naturality (factor_δ f' j).op) (face i j hji)
   dsimp at H₁ H₂
   erw [H, H₁, H₂, h _ hji]
 
@@ -347,16 +384,12 @@ set_option linter.uppercaseLean3 false in
 
 namespace Augmented
 
--- porting note: added to ease the automation of the proofs in the definition
--- of `standardSimplex`
-attribute [local simp] SSet.standardSimplex
-
 -- porting note: an instance of `Subsingleton (⊤_ (Type u))` was added in
 -- `CategoryTheory.Limits.Types` to ease the automation in this definition
 /-- The functor which sends `[n]` to the simplicial set `Δ[n]` equipped by
 the obvious augmentation towards the terminal object of the category of sets. -/
 @[simps]
-noncomputable def standardSimplex : SimplexCategory ⥤ SSet.Augmented where
+noncomputable def standardSimplex : SimplexCategory ⥤ SSet.Augmented.{u} where
   obj Δ :=
     { left := SSet.standardSimplex.obj Δ
       right := terminal _

@@ -147,7 +147,7 @@ set_option linter.uppercaseLean3 false in
 #align sSet.boundary SSet.boundary
 
 -- mathport name: sSet.boundary
-scoped[Simplicial] notation "∂Δ[" n "]" => SSet.boundary n
+scoped[Simplicial] notation3 "∂Δ[" n "]" => SSet.boundary n
 
 /-- The inclusion of the boundary of the `n`-th standard simplex into that standard simplex. -/
 def boundaryInclusion (n : ℕ) : ∂Δ[n] ⟶ Δ[n] where app m (α : { α : Δ[n].obj m // _ }) := α
@@ -266,6 +266,29 @@ def primitiveTriangle {n : ℕ} (i : Fin (n+4))
     intro j
     fin_cases j <;> simp [Fin.ext_iff, hk0]
 
+/-- The `j`th subface of the `i`-th horn. -/
+@[simps]
+def face {n : ℕ} (i j : Fin (n+2)) (h : j ≠ i) : Λ[n+1, i] _[n] := by
+  refine ⟨SimplexCategory.δ j, ?_⟩
+  simpa [← Set.univ_subset_iff, Set.subset_def, asOrderHom, SimplexCategory.δ, not_or]
+
+/-- Two morphisms from a horn are equal if they are equal on all suitable faces. -/
+protected
+lemma hom_ext {n : ℕ} {i : Fin (n+2)} {S : SSet} (σ₁ σ₂ : Λ[n+1, i] ⟶ S)
+    (h : ∀ (j) (h : j ≠ i), σ₁.app _ (face i j h) = σ₂.app _ (face i j h)) :
+    σ₁ = σ₂ := by
+  apply NatTrans.ext; apply funext; apply Opposite.rec; apply SimplexCategory.rec
+  intro m; ext f
+  obtain ⟨j, hji, hfj⟩ : ∃ j, ¬j = i ∧ ∀ k, f.1.toOrderHom k ≠ j := by
+    simpa [← Set.univ_subset_iff, Set.subset_def, asOrderHom, not_or] using f.2
+  have H : f = (Λ[n+1, i].map (factor_δ f.1 j).op) (face i j hji) := by
+    apply Subtype.ext
+    exact (factor_δ_spec f.1 j hfj).symm
+  have H₁ := congrFun (σ₁.naturality (factor_δ f.1 j).op) (face i j hji)
+  have H₂ := congrFun (σ₂.naturality (factor_δ f.1 j).op) (face i j hji)
+  dsimp at H₁ H₂
+  erw [H, H₁, H₂, h _ hji]
+
 end horn
 
 section Examples

constructors for subfaces of the standard simplex and horn simplices

@@ -7,6 +7,8 @@ import Mathlib.AlgebraicTopology.SimplicialObject
 import Mathlib.CategoryTheory.Limits.Presheaf
 import Mathlib.CategoryTheory.Limits.Shapes.Types
 import Mathlib.CategoryTheory.Yoneda
+import Mathlib.Data.Fin.VecNotation
+import Mathlib.Tactic.FinCases
 
 #align_import algebraic_topology.simplicial_set from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d484ef1"
 
@@ -82,6 +84,44 @@ scoped[Simplicial] notation3 "Δ[" n "]" => SSet.standardSimplex.obj (SimplexCat
 instance : Inhabited SSet :=
   ⟨Δ[0]⟩
 
+namespace standardSimplex
+
+open Finset Opposite SimplexCategory
+
+/-- The (degenerate) `m`-simplex in the standard simplex concentrated in vertex `k`. -/
+def const (n : ℕ) (k : Fin (n+1)) (m : SimplexCategoryᵒᵖ) : Δ[n].obj m :=
+  Hom.mk <| OrderHom.const _ k
+
+@[simp]
+lemma const_toOrderHom (n : ℕ) (k : Fin (n+1)) (m : SimplexCategoryᵒᵖ) :
+    (const n k m).toOrderHom = OrderHom.const _ k :=
+  rfl
+
+/-- The edge of the standard simplex with endpoints `a` and `b`. -/
+def edge (n : ℕ) (a b : Fin (n+1)) (hab : a ≤ b) : Δ[n] _[1] := by
+  refine Hom.mk ⟨![a, b], ?_⟩
+  rw [Fin.monotone_iff_le_succ]
+  simp only [unop_op, len_mk, Fin.forall_fin_one]
+  apply Fin.mk_le_mk.mpr hab
+
+lemma coe_edge_toOrderHom (n : ℕ) (a b : Fin (n+1)) (hab : a ≤ b) :
+    ↑(edge n a b hab).toOrderHom = ![a, b] :=
+  rfl
+
+/-- The triangle in the standard simplex with vertices `a`, `b`, and `c`. -/
+def triangle {n : ℕ} (a b c : Fin (n+1)) (hab : a ≤ b) (hbc : b ≤ c) : Δ[n] _[2] := by
+  refine Hom.mk ⟨![a, b, c], ?_⟩
+  rw [Fin.monotone_iff_le_succ]
+  simp only [unop_op, len_mk, Fin.forall_fin_two]
+  dsimp
+  simp only [*, Matrix.tail_cons, Matrix.head_cons, true_and]
+
+lemma coe_triangle_toOrderHom {n : ℕ} (a b c : Fin (n+1)) (hab : a ≤ b) (hbc : b ≤ c) :
+    ↑(triangle a b c hab hbc).toOrderHom = ![a, b, c] :=
+  rfl
+
+end standardSimplex
+
 section
 
 /-- The `m`-simplices of the `n`-th standard simplex are
@@ -139,6 +179,95 @@ def hornInclusion (n : ℕ) (i : Fin (n + 1)) : Λ[n, i] ⟶ Δ[n] where
 set_option linter.uppercaseLean3 false in
 #align sSet.horn_inclusion SSet.hornInclusion
 
+namespace horn
+
+open SimplexCategory Finset Opposite
+
+/-- The (degenerate) subsimplex of `Λ[n+2, i]` concentrated in vertex `k`. -/
+@[simps]
+def const (n : ℕ) (i k : Fin (n+3)) (m : SimplexCategoryᵒᵖ) : Λ[n+2, i].obj m := by
+  refine ⟨standardSimplex.const _ k _, ?_⟩
+  suffices ¬ Finset.univ ⊆ {i, k} by
+    simpa [← Set.univ_subset_iff, Set.subset_def, asOrderHom, not_or, Fin.forall_fin_one,
+      subset_iff, mem_univ, @eq_comm _ _ k]
+  intro h
+  have := (card_le_card h).trans card_le_two
+  rw [card_fin] at this
+  omega
+
+/-- The edge of `Λ[n, i]` with endpoints `a` and `b`.
+
+This edge only exists if `{i, a, b}` has cardinality less than `n`. -/
+@[simps]
+def edge (n : ℕ) (i a b : Fin (n+1)) (hab : a ≤ b) (H : Finset.card {i, a, b} ≤ n) :
+    Λ[n, i] _[1] := by
+  refine ⟨standardSimplex.edge n a b hab, ?range⟩
+  case range =>
+    suffices ∃ x, ¬i = x ∧ ¬a = x ∧ ¬b = x by
+      simpa only [unop_op, SimplexCategory.len_mk, asOrderHom, SimplexCategory.Hom.toOrderHom_mk,
+        Set.union_singleton, ne_eq, ← Set.univ_subset_iff, Set.subset_def, Set.mem_univ,
+        Set.mem_insert_iff, @eq_comm _ _ i, Set.mem_range, forall_true_left, not_forall, not_or,
+        not_exists, Fin.forall_fin_two]
+    contrapose! H
+    replace H : univ ⊆ {i, a, b} :=
+      fun x _ ↦ by simpa [or_iff_not_imp_left, eq_comm] using H x
+    replace H := card_le_card H
+    rwa [card_fin] at H
+
+/-- Alternative constructor for the edge of `Λ[n, i]` with endpoints `a` and `b`,
+assuming `3 ≤ n`. -/
+@[simps!]
+def edge₃ (n : ℕ) (i a b : Fin (n+1)) (hab : a ≤ b) (H : 3 ≤ n) :
+    Λ[n, i] _[1] :=
+  horn.edge n i a b hab <| Finset.card_le_three.trans H
+
+/-- The edge of `Λ[n, i]` with endpoints `j` and `j+1`.
+
+This constructor assumes `0 < i < n`,
+which is the type of horn that occurs in the horn-filling condition of quasicategories. -/
+@[simps!]
+def primitiveEdge {n : ℕ} {i : Fin (n+1)}
+    (h₀ : 0 < i) (hₙ : i < Fin.last n) (j : Fin n) :
+    Λ[n, i] _[1] := by
+  refine horn.edge n i j.castSucc j.succ ?_ ?_
+  · simp only [← Fin.val_fin_le, Fin.coe_castSucc, Fin.val_succ, le_add_iff_nonneg_right, zero_le]
+  simp only [← Fin.val_fin_lt, Fin.val_zero, Fin.val_last] at h₀ hₙ
+  obtain rfl|hn : n = 2 ∨ 2 < n := by
+    rw [eq_comm, or_comm, ← le_iff_lt_or_eq]; omega
+  · revert i j; decide
+  · exact Finset.card_le_three.trans hn
+
+/-- The triangle in the standard simplex with vertices `k`, `k+1`, and `k+2`.
+
+This constructor assumes `0 < i < n`,
+which is the type of horn that occurs in the horn-filling condition of quasicategories. -/
+@[simps]
+def primitiveTriangle {n : ℕ} (i : Fin (n+4))
+    (h₀ : 0 < i) (hₙ : i < Fin.last (n+3))
+    (k : ℕ) (h : k < n+2) : Λ[n+3, i] _[2] := by
+  refine ⟨standardSimplex.triangle
+    (n := n+3) ⟨k, by omega⟩ ⟨k+1, by omega⟩ ⟨k+2, by omega⟩ ?_ ?_, ?_⟩
+  · simp only [Fin.mk_le_mk, le_add_iff_nonneg_right, zero_le]
+  · simp only [Fin.mk_le_mk, add_le_add_iff_left, one_le_two]
+  simp only [unop_op, SimplexCategory.len_mk, asOrderHom, SimplexCategory.Hom.toOrderHom_mk,
+    OrderHom.const_coe_coe, Set.union_singleton, ne_eq, ← Set.univ_subset_iff, Set.subset_def,
+    Set.mem_univ, Set.mem_insert_iff, Set.mem_range, Function.const_apply, exists_const,
+    forall_true_left, not_forall, not_or, unop_op, not_exists,
+    standardSimplex.triangle, OrderHom.coe_mk, @eq_comm _ _ i]
+  dsimp
+  by_cases hk0 : k = 0
+  · subst hk0
+    use Fin.last (n+3)
+    simp only [hₙ.ne, not_false_eq_true, Fin.zero_eta, zero_add, true_and]
+    intro j
+    fin_cases j <;> simp [Fin.ext_iff] <;> omega
+  · use 0
+    simp only [h₀.ne', not_false_eq_true, true_and]
+    intro j
+    fin_cases j <;> simp [Fin.ext_iff, hk0]
+
+end horn
+
 section Examples
 
 open Simplicial

@@ -4,15 +4,15 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin, Scott Morrison, Adam Topaz
 -/
 import Mathlib.AlgebraicTopology.SimplicialObject
-import Mathlib.AlgebraicTopology.TopologicalSimplex
 import Mathlib.CategoryTheory.Limits.Presheaf
 import Mathlib.CategoryTheory.Limits.Shapes.Types
 import Mathlib.CategoryTheory.Yoneda
-import Mathlib.Topology.Category.TopCat.Limits.Basic
 
 #align_import algebraic_topology.simplicial_set from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d484ef1"
 
 /-!
+# Simplicial sets
+
 A simplicial set is just a simplicial object in `Type`,
 i.e. a `Type`-valued presheaf on the simplex category.
 
@@ -218,29 +218,3 @@ set_option linter.uppercaseLean3 false in
 end Augmented
 
 end SSet
-
-/-- The functor associating the singular simplicial set to a topological space. -/
-noncomputable def TopCat.toSSet : TopCat ⥤ SSet :=
-  ColimitAdj.restrictedYoneda SimplexCategory.toTop
-set_option linter.uppercaseLean3 false in
-#align Top.to_sSet TopCat.toSSet
-
-/-- The geometric realization functor. -/
-noncomputable def SSet.toTop : SSet ⥤ TopCat :=
-  ColimitAdj.extendAlongYoneda SimplexCategory.toTop
-set_option linter.uppercaseLean3 false in
-#align sSet.to_Top SSet.toTop
-
-/-- Geometric realization is left adjoint to the singular simplicial set construction. -/
-noncomputable def sSetTopAdj : SSet.toTop ⊣ TopCat.toSSet :=
-  ColimitAdj.yonedaAdjunction _
-set_option linter.uppercaseLean3 false in
-#align sSet_Top_adj sSetTopAdj
-
-/-- The geometric realization of the representable simplicial sets agree
-  with the usual topological simplices. -/
-noncomputable def SSet.toTopSimplex :
-    (yoneda : SimplexCategory ⥤ _) ⋙ SSet.toTop ≅ SimplexCategory.toTop :=
-  ColimitAdj.isExtensionAlongYoneda _
-set_option linter.uppercaseLean3 false in
-#align sSet.to_Top_simplex SSet.toTopSimplex

add delaborator for standard simplex and horn notation

@@ -77,7 +77,7 @@ set_option linter.uppercaseLean3 false in
 #align sSet.standard_simplex SSet.standardSimplex
 
 -- mathport name: standard_simplex
-scoped[Simplicial] notation "Δ[" n "]" => SSet.standardSimplex.obj (SimplexCategory.mk n)
+scoped[Simplicial] notation3 "Δ[" n "]" => SSet.standardSimplex.obj (SimplexCategory.mk n)
 
 instance : Inhabited SSet :=
   ⟨Δ[0]⟩
@@ -131,7 +131,7 @@ set_option linter.uppercaseLean3 false in
 #align sSet.horn SSet.horn
 
 -- mathport name: sSet.horn
-scoped[Simplicial] notation "Λ[" n ", " i "]" => SSet.horn (n : ℕ) i
+scoped[Simplicial] notation3 "Λ[" n ", " i "]" => SSet.horn (n : ℕ) i
 
 /-- The inclusion of the `i`-th horn of the `n`-th standard simplex into that standard simplex. -/
 def hornInclusion (n : ℕ) (i : Fin (n + 1)) : Λ[n, i] ⟶ Δ[n] where

Autoimplicits are highly controversial and also defeat the performance-improving work in #6474.

The intent of this PR is to make autoImplicit opt-in on a per-file basis, by disabling it in the lakefile and enabling it again with set_option autoImplicit true in the few files that rely on it.

That also keeps this PR small, as opposed to attempting to "fix" files to not need it any more.

I claim that many of the uses of autoImplicit in these files are accidental; situations such as:

• Assuming variables are in scope, but pasting the lemma in the wrong section
• Pasting in a lemma from a scratch file without checking to see if the variable names are consistent with the rest of the file
• Making a copy-paste error between lemmas and forgetting to add an explicit arguments.

Having set_option autoImplicit false as the default prevents these types of mistake being made in the 90% of files where autoImplicits are not used at all, and causes them to be caught by CI during review.

I think there were various points during the port where we encouraged porters to delete the universes u v lines; I think having autoparams for universe variables only would cover a lot of the cases we actually use them, while avoiding any real shortcomings.

A Zulip poll (after combining overlapping votes accordingly) was in favor of this change with 5:5:18 as the no:dontcare:yes vote ratio.

While this PR was being reviewed, a handful of files gained some more likely-accidental autoImplicits. In these places, set_option autoImplicit true has been placed locally within a section, rather than at the top of the file.

@@ -32,6 +32,8 @@ from a non-surjective order preserving function `Fin n → Fin n`
 a morphism `Δ[n] ⟶ ∂Δ[n]`.
 -/
 
+set_option autoImplicit true
+
 
 universe v u
 

@@ -6,7 +6,7 @@ Authors: Johan Commelin, Scott Morrison, Adam Topaz
 import Mathlib.AlgebraicTopology.SimplicialObject
 import Mathlib.AlgebraicTopology.TopologicalSimplex
 import Mathlib.CategoryTheory.Limits.Presheaf
-import Mathlib.CategoryTheory.Limits.Types
+import Mathlib.CategoryTheory.Limits.Shapes.Types
 import Mathlib.CategoryTheory.Yoneda
 import Mathlib.Topology.Category.TopCat.Limits.Basic
 

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,11 +2,6 @@
 Copyright (c) 2021 Scott Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin, Scott Morrison, Adam Topaz
-
-! This file was ported from Lean 3 source module algebraic_topology.simplicial_set
-! leanprover-community/mathlib commit 178a32653e369dce2da68dc6b2694e385d484ef1
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.AlgebraicTopology.SimplicialObject
 import Mathlib.AlgebraicTopology.TopologicalSimplex
@@ -15,6 +10,8 @@ import Mathlib.CategoryTheory.Limits.Types
 import Mathlib.CategoryTheory.Yoneda
 import Mathlib.Topology.Category.TopCat.Limits.Basic
 
+#align_import algebraic_topology.simplicial_set from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d484ef1"
+
 /-!
 A simplicial set is just a simplicial object in `Type`,
 i.e. a `Type`-valued presheaf on the simplex category.

Changes are of the form

• some_tactic at h⊢ -> some_tactic at h ⊢
• some_tactic at h -> some_tactic at h

@@ -124,7 +124,7 @@ def horn (n : ℕ) (i : Fin (n + 1)) : SSet where
   map {m₁ m₂} f α :=
     ⟨f.unop ≫ (α : Δ[n].obj m₁), by
       intro h; apply α.property
-      rw [Set.eq_univ_iff_forall] at h⊢; intro j
+      rw [Set.eq_univ_iff_forall] at h ⊢; intro j
       apply Or.imp _ id (h j)
       intro hj
       exact Set.range_comp_subset_range _ _ hj⟩

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au>

@@ -64,6 +64,12 @@ instance hasColimits : HasColimits SSet := by
   dsimp only [SSet]
   infer_instance
 
+-- Porting note: added an `ext` lemma.
+-- See https://github.com/leanprover-community/mathlib4/issues/5229
+@[ext]
+lemma hom_ext {X Y : SSet} {f g : X ⟶ Y} (w : ∀ n, f.app n = g.app n) : f = g :=
+  SimplicialObject.hom_ext _ _ w
+
 /-- The `n`-th standard simplex `Δ[n]` associated with a nonempty finite linear order `n`
 is the Yoneda embedding of `n`. -/
 def standardSimplex : SimplexCategory ⥤ SSet :=
@@ -129,8 +135,8 @@ set_option linter.uppercaseLean3 false in
 scoped[Simplicial] notation "Λ[" n ", " i "]" => SSet.horn (n : ℕ) i
 
 /-- The inclusion of the `i`-th horn of the `n`-th standard simplex into that standard simplex. -/
-def hornInclusion (n : ℕ) (i : Fin (n + 1)) : Λ[n, i] ⟶ Δ[n]
-    where app m (α : { α : Δ[n].obj m // _ }) := α
+def hornInclusion (n : ℕ) (i : Fin (n + 1)) : Λ[n, i] ⟶ Δ[n] where
+  app m (α : { α : Δ[n].obj m // _ }) := α
 set_option linter.uppercaseLean3 false in
 #align sSet.horn_inclusion SSet.hornInclusion
 
@@ -166,6 +172,12 @@ instance Truncated.hasColimits : HasColimits (Truncated n) := by
   dsimp only [Truncated]
   infer_instance
 
+-- Porting note: added an `ext` lemma.
+-- See https://github.com/leanprover-community/mathlib4/issues/5229
+@[ext]
+lemma Truncated.hom_ext {X Y : Truncated n} {f g : X ⟶ Y} (w : ∀ n, f.app n = g.app n) : f = g :=
+  NatTrans.ext _ _ (funext w)
+
 /-- The skeleton functor on simplicial sets. -/
 def sk (n : ℕ) : SSet ⥤ SSet.Truncated n :=
   SimplicialObject.sk n

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

@@ -154,7 +154,7 @@ def Truncated (n : ℕ) :=
 set_option linter.uppercaseLean3 false in
 #align sSet.truncated SSet.Truncated
 
-instance Truncated.largeCategory (n : ℕ) : LargeCategory (Truncated n):= by
+instance Truncated.largeCategory (n : ℕ) : LargeCategory (Truncated n) := by
   dsimp only [Truncated]
   infer_instance
 

@@ -13,7 +13,7 @@ import Mathlib.AlgebraicTopology.TopologicalSimplex
 import Mathlib.CategoryTheory.Limits.Presheaf
 import Mathlib.CategoryTheory.Limits.Types
 import Mathlib.CategoryTheory.Yoneda
-import Mathlib.Topology.Category.Top.Limits.Basic
+import Mathlib.Topology.Category.TopCat.Limits.Basic
 
 /-!
 A simplicial set is just a simplicial object in `Type`,