Documentation

InfinityCosmos.ForMathlib.AlgebraicTopology.SimplicialSet.HomotopyCat

theorem CategoryTheory.Fin.le_succ {n : } (i : Fin n) :
i.castSucc i.succ
def CategoryTheory.Fin.hom_succ {n : } (i : Fin n) :
i.castSucc i.succ
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        def CategoryTheory.SimplexCategory.mkOfLeComp {n : } (i : Fin (n + 1)) (j : Fin (n + 1)) (k : Fin (n + 1)) (h₁ : i j) (h₂ : j k) :
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          @[reducible, inline]

          The fully faithful inclusion of the truncated simplex category into the usual simplex category.

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            The ulift functor SSet.Truncated.{u} ⥤ SSet.Truncated.{max u v} on truncated simplicial sets.

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              This is called "sk" in SimplicialSet and SimplicialObject, but this is a better name.

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                The identity natural transformation exhibits nerve C as a right extension of its restriction to (Δ 2).op along (Δ.ι 2).op.

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                  The natural transformation in nerveRightExtension C defines a cone with summit nerve C _[n] over the diagram (StructuredArrow.proj (op ([n] : SimplexCategory)) (Δ.ι 2).op ⋙ nerveFunctor₂.obj C) indexed by the category StructuredArrow (op [n]) (Δ.ι 2).op.

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                        It follows that we have a natural isomorphism between nerveFunctor and nerveFunctor ⋙ cosk₂ whose components are the isomorphisms just established.

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                            theorem CategoryTheory.opstuff {C : Type u} [CategoryTheory.Category.{v, u} C] (V : CategoryTheory.Functor Cᵒᵖ (Type w)) {X : C} {Y : C} {Z : C} {α : X Y} {β : Y Z} {γ : X Z} {φ : V.obj (Opposite.op Z)} :
                            CategoryTheory.CategoryStruct.comp α β = γV.map α.op (V.map β.op φ) = V.map γ.op φ
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                                    theorem CategoryTheory.SSet.HoRel.ext_triangle {V : SSet} (X : CategoryTheory.SSet.OneTruncation V) (X' : CategoryTheory.SSet.OneTruncation V) (Y : CategoryTheory.SSet.OneTruncation V) (Y' : CategoryTheory.SSet.OneTruncation V) (Z : CategoryTheory.SSet.OneTruncation V) (Z' : CategoryTheory.SSet.OneTruncation V) (hX : X = X') (hY : Y = Y') (hZ : Z = Z') (f : X Z) (f' : X' Z') (hf : f = f') (g : X Y) (g' : X' Y') (hg : g = g') (h : Y Z) (h' : Y' Z') (hh : h = h') :
                                    CategoryTheory.SSet.HoRel ((CategoryTheory.Quotient.functor CategoryTheory.Cat.FreeReflRel).obj X) ((CategoryTheory.Quotient.functor CategoryTheory.Cat.FreeReflRel).obj Z) ((CategoryTheory.Quotient.functor CategoryTheory.Cat.FreeReflRel).map (Quiver.Path.nil.cons f)) ((CategoryTheory.Quotient.functor CategoryTheory.Cat.FreeReflRel).map ((Quiver.Path.nil.cons g).cons h)) CategoryTheory.SSet.HoRel ((CategoryTheory.Quotient.functor CategoryTheory.Cat.FreeReflRel).obj X') ((CategoryTheory.Quotient.functor CategoryTheory.Cat.FreeReflRel).obj Z') ((CategoryTheory.Quotient.functor CategoryTheory.Cat.FreeReflRel).map (Quiver.Path.nil.cons f')) ((CategoryTheory.Quotient.functor CategoryTheory.Cat.FreeReflRel).map ((Quiver.Path.nil.cons g').cons h'))
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                                          @[reducible, inline]
                                          abbrev CategoryTheory.SSet.δ₂ {n : } (i : Fin (n + 2)) (hn : autoParam ((SimplexCategory.mk n).len 2) _auto✝) (hn' : autoParam ((SimplexCategory.mk (n + 1)).len 2) _auto✝) :
                                          { obj := SimplexCategory.mk n, property := hn } { obj := SimplexCategory.mk (n + 1), property := hn' }
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                                            @[reducible, inline]
                                            abbrev CategoryTheory.SSet.σ₂ {n : } (i : Fin (n + 1)) (hn : autoParam ((SimplexCategory.mk (n + 1)).len 2) _auto✝) (hn' : autoParam ((SimplexCategory.mk n).len 2) _auto✝) :
                                            { obj := SimplexCategory.mk (n + 1), property := hn } { obj := SimplexCategory.mk n, property := hn' }
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                                                    theorem CategoryTheory.SSet.HoRel₂.ext_triangle {V : SSet.Truncated 2} (X : CategoryTheory.SSet.OneTruncation₂ V) (X' : CategoryTheory.SSet.OneTruncation₂ V) (Y : CategoryTheory.SSet.OneTruncation₂ V) (Y' : CategoryTheory.SSet.OneTruncation₂ V) (Z : CategoryTheory.SSet.OneTruncation₂ V) (Z' : CategoryTheory.SSet.OneTruncation₂ V) (hX : X = X') (hY : Y = Y') (hZ : Z = Z') (f : X Z) (f' : X' Z') (hf : f = f') (g : X Y) (g' : X' Y') (hg : g = g') (h : Y Z) (h' : Y' Z') (hh : h = h') :
                                                    CategoryTheory.SSet.HoRel₂ ((CategoryTheory.Quotient.functor CategoryTheory.Cat.FreeReflRel).obj X) ((CategoryTheory.Quotient.functor CategoryTheory.Cat.FreeReflRel).obj Z) ((CategoryTheory.Quotient.functor CategoryTheory.Cat.FreeReflRel).map (Quiver.Path.nil.cons f)) ((CategoryTheory.Quotient.functor CategoryTheory.Cat.FreeReflRel).map ((Quiver.Path.nil.cons g).cons h)) CategoryTheory.SSet.HoRel₂ ((CategoryTheory.Quotient.functor CategoryTheory.Cat.FreeReflRel).obj X') ((CategoryTheory.Quotient.functor CategoryTheory.Cat.FreeReflRel).obj Z') ((CategoryTheory.Quotient.functor CategoryTheory.Cat.FreeReflRel).map (Quiver.Path.nil.cons f')) ((CategoryTheory.Quotient.functor CategoryTheory.Cat.FreeReflRel).map ((Quiver.Path.nil.cons g').cons h'))
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                                                          @[simp]
                                                          theorem CategoryTheory.nerve₂Adj.counit.component_map (C : CategoryTheory.Cat) (a : CategoryTheory.Quotient CategoryTheory.SSet.HoRel₂) (b : CategoryTheory.Quotient CategoryTheory.SSet.HoRel₂) (hf : a b) :
                                                          (CategoryTheory.nerve₂Adj.counit.component C).map hf = Quot.liftOn hf (fun (f : a.as b.as) => (CategoryTheory.ReflQuiv.adj.counit.app C).map (Quot.liftOn f (fun (f : a.as.as b.as.as) => (CategoryTheory.Cat.FreeRefl.quotientFunctor C).map ((CategoryTheory.forgetToReflQuiv.natIso.hom.app C).mapPath f)) ))
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                                                            def CategoryTheory.nerve₂.seagull (C : CategoryTheory.Cat) :
                                                            (CategoryTheory.nerveFunctor₂.obj C).obj (Opposite.op { obj := SimplexCategory.mk 2, property := }) (CategoryTheory.nerveFunctor₂.obj C).obj (Opposite.op { obj := SimplexCategory.mk 1, property := }) (CategoryTheory.nerveFunctor₂.obj C).obj (Opposite.op { obj := SimplexCategory.mk 1, property := })

                                                            This is similiar to one of the famous Segal maps, except valued in a product rather than a pullback.

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                                                              Now do a case split. For n = 0 and n = 1 this is covered by the hypothesis. For n = 2 this is covered by the new lemma above.

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                                                                The adjunction between forming the free category on a quiver, and forgetting a category to a quiver.

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                                                                  Repleteness exists for full and faithful functors but not fully faithful functors, which is why we do this inefficiently.

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