| {"id":"prop:GroupoidPathObjectFibrant","label":"GroupoidPathObjectFibrant","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(A\\) be a small groupoid. Recall that \\(\\mathbf{grpd}\\) is Cartesian closed, so we can take the image of the above diagram under the functor \\(A^{-}\\).","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidPathObjectFibrant","lean_link":null,"lean_decl_name":"GroupoidPathObjectFibrant","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"prop:GroupoidFibrationComposition","label":"GroupoidFibrationComposition","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The corresponding fact about fibrations is that the composition of two fibrations is a fibration.","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidFibrationComposition","lean_link":null,"lean_decl_name":"GroupoidFibrationComposition","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"prop:GroupoidFibrationPushforward","label":"GroupoidFibrationPushforward","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Stated in terms of fibrations, we have","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidFibrationPushforward","lean_link":null,"lean_decl_name":"GroupoidFibrationPushforward","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:PCat","label":"PCat","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We will take the category of pointed small categories \\(\\mathbf{cat}_\\bullet \\) to have objects as pairs \\((\\mathbb {C}\\in \\mathbf{cat}, c \\in \\mathbb {C})\\) and morphisms as pairs \\[ (F : \\mathbb {C}_{1} \\to \\mathbb {C}_{0}, \\phi : F c_{1} \\to c_{0}) : (\\mathbb {C}_{1}, c_{1}) \\to (\\mathbb {C}_{0}, c_{0}) \\]","doc_url":null,"latex_link":"sect0004.html#defn:PCat","lean_link":null,"lean_decl_name":"PCat","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/PullbackProofs.lean","lean4_code":"theorem PCat.eqToHom_hom_help {C1 C2 : PCat.{u,v}}(x y: C1)(eq : C1 = C2): (x ⟶ y) = ((eqToHom eq).obj x ⟶ (eqToHom eq).obj y) := by\n cases eq\n simp[CategoryStruct.id]\n\n-- /-- This is the turns the hom part of eqToHom pointed functors into a cast-/\n-- theorem PCat.eqToHom_hom {C1 C2 : PCat.{u,v}}{x y: C1}(f : x ⟶ y)(eq : C1 = C2): (eqToHom eq).map f = (cast (PCat.eqToHom_hom_help x y eq) f) := by\n-- cases eq\n-- simp[CategoryStruct.id]\n\n/-- This shows that two objects are equal in Grothendieck A if they have equal bases and fibers that are equal after being cast-/\n","line":55,"end_line":64,"proof_status":"statement_formalized"} |
| {"id":"defn:CategoryTheory.Grothendieck","label":"CategoryTheory.Grothendieck","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"From \\(\\mathbb {C}\\) a small category and \\(F : \\mathbb {C}\\to \\mathbf{cat}\\) a functor, we construct a small category \\(\\int F\\). For any \\(c\\) in \\(\\mathbb {C}\\) we refer to \\(F c\\) as the fiber over \\(c\\). The objects of \\(\\int F\\) consist of pairs \\((c \\in \\mathbb {C}, x \\in F c)\\), and morphisms between \\((c, x)\\) and \\((d, y)\\) are pairs \\((f : c \\to d, \\phi : F \\, f \\, x \\to y)\\). This makes the following pullback in \\(\\mathbf{Cat}\\)","doc_url":null,"latex_link":"sect0004.html#defn:CategoryTheory.Grothendieck","lean_link":null,"lean_decl_name":"CategoryTheory.Grothendieck","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./HoTTLean/Grothendieck/Groupoidal/Basic.lean","lean4_code":"@[simp] lemma Grothendieck.Groupoidal.eta {Γ : Type*} [Category Γ]\n {A : Γ ⥤ Grpd} (x : ∫(A)) : objMk x.base x.fiber = x :=\n rfl\n\n","line":176,"end_line":179,"proof_status":"statement_formalized"} |
| {"id":"defn:GroupoidInterval","label":"GroupoidInterval","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let the interval groupoid \\(\\mathbb {I}\\) be the small groupoid with two objects and a single non-identity isomorphism. There are two distinct morphisms \\(\\delta _{0}, \\delta _{1} : \\bullet \\to \\mathbb {I}\\) and a natural isomorphism \\(i : \\delta _{0} \\Rightarrow \\delta _{1}\\). Note that \\(\\delta _{0}\\) and \\(\\delta _{1}\\) both form adjoint equivalences with the unique map \\(! : \\mathbb {I}\\to \\bullet \\). Denote by \\(\\bullet +\\bullet \\) the small groupoid with two objects and only identity morphisms. Then let \\(\\partial : \\bullet +\\bullet \\to \\mathbb {I}\\) be the unique map factoring \\(\\delta _{0}\\) and \\(\\delta _{1}\\).","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidInterval","lean_link":null,"lean_decl_name":"GroupoidInterval","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:GroupoidGrothendieck","label":"GroupoidGrothendieck","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\Gamma \\) be a groupoid and \\(A : \\Gamma \\to \\mathbf{grpd}\\) a functor, we can compose \\(F\\) with the inclusion \\(i : \\mathbf{grpd}\\hookrightarrow \\mathbf{Cat}\\) and form the Grothendieck construction which we denote as \\[ \\Gamma \\cdot A := \\int i \\circ A \\quad \\quad \\mathsf{disp}_{A} : \\Gamma \\cdot A \\to \\Gamma \\]","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidGrothendieck","lean_link":null,"lean_decl_name":"GroupoidGrothendieck","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:GroupoidPi","label":"GroupoidPi","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Given \\(A : \\Gamma \\to \\mathbf{grpd}\\) and \\(B : \\Gamma \\cdot A \\to \\mathbf{grpd}\\) we will define \\(\\Pi _{A} B : \\Gamma \\to \\mathbf{grpd}\\) such that for any \\(C : \\Gamma \\to \\mathbf{grpd}\\) we have an isomorphism \\[ [\\Gamma \\cdot A, \\mathbf{grpd}](\\mathsf{disp}_{A}\\circ C, B) \\cong [\\Gamma , \\mathbf{grpd}](C, \\Pi _{A} B) \\]","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidPi","lean_link":null,"lean_decl_name":"GroupoidPi","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/TarskiNaturalModel.lean","lean4_code":"def GroupoidPi (Γ : Grpd) (A : Γ ⥤ Grpd) (B : (Grothendieck.Groupoidal A) ⥤ Grpd) : Γ ⥤ Grpd where\n obj x := by\n have t := g (A.obj x ⥤ (GroupoidSigma Γ A B).obj x) (fun(f) => f ⋙ ((GroupoidSigmaBase Γ A B).app x) = 𝟙 (A.obj x))\n exact Grpd.of {f : A.obj x ⥤ (GroupoidSigma Γ A B).obj x // f ⋙ ((GroupoidSigmaBase Γ A B).app x) = 𝟙 (A.obj x)}\n map f := by\n dsimp[Quiver.Hom]\n rename_i X Y\n sorry\n\n\nend NaturalModelPi\n\nsection NaturalModelEq\n\nend NaturalModelEq\n\n","line":369,"end_line":384,"proof_status":"statement_formalized"} |
| {"id":"defn:GroupoidSigma","label":"GroupoidSigma","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Then given \\(A : \\Gamma \\to \\mathbf{grpd}\\) and \\(B : \\Gamma \\cdot A \\to \\mathbf{grpd}\\) we define \\(\\Sigma _{A}B : \\Gamma \\to \\mathbf{grpd}\\) such that \\(\\Sigma _{A}B\\) acts on objects by forming fiberwise Grothendieck constructions \\[ \\Sigma _{A}B (x) := A(x) \\cdot B \\circ x_{A} \\]","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidSigma","lean_link":null,"lean_decl_name":"GroupoidSigma","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/TarskiNaturalModel.lean","lean4_code":"def GroupoidSigma (Γ : Grpd) (A : Γ ⥤ Grpd) (B : (Grothendieck.Groupoidal A) ⥤ Grpd) : Γ ⥤ Grpd where\n obj x := Grpd.of (Grothendieck.Groupoidal ((PointToFiber A x) ⋙ B))\n map f := by\n rename_i X Y\n have NT' : (PointToFiber A X) ⋙ (B ⋙ Grpd.forgetToCat) ⟶ (A.map f ⋙ PointToFiber A Y) ⋙ (B ⋙ Grpd.forgetToCat) := whiskerRight (PointToFiberNT A X Y f) (B ⋙ Grpd.forgetToCat)\n exact (Grothendieck.map NT') ⋙ (Grothendieck.Groupoidal.functorial (A.map f) (PointToFiber A Y ⋙ B))\n map_id := by\n intro X\n simp[CategoryStruct.id,whiskerRight,Functor.id]\n refine CategoryTheory.Functor.ext ?_ ?_\n . intro X\n simp[PointToFiber,Grpd.forgetToCat]\n sorry\n all_goals sorry\n map_comp := by\n intro X Y Z f g\n simp[Grpd.forgetToCat]\n sorry\n\n","line":198,"end_line":216,"proof_status":"statement_formalized"} |
| {"id":"defn:NaturalModelPi","label":"NaturalModelPi","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:NaturalModelRussell","defn:UvPoly"],"nlp_text":"We will use \\(P_{\\mathsf{tp}_i}\\) to denote the polynomial endofunctor 5.0.1 associated with a natural model \\(\\mathsf{tp}_i\\). Then additional structure of \\(\\Pi \\) types on our \\(N\\) universes consists of, for each \\(i, j \\le N\\), a pullback square","doc_url":null,"latex_link":"sect0002.html#defn:NaturalModelPi","lean_link":null,"lean_decl_name":"NaturalModelPi","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/NaturalModel.lean","lean4_code":"class NaturalModelPi where\n Pi : (P tp).obj Ty ⟶ M.Ty\n lam : (P tp).obj Tm ⟶ M.Tm\n Pi_pullback : IsPullback lam ((P tp).map tp) tp Pi\n\nvariable (Ctx) in\n","line":104,"end_line":109,"proof_status":"statement_formalized"} |
| {"id":"defn:ProductsExponentials","label":"ProductsExponentials","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:NaturalModelPi","defn:NaturalModelSigma","prop:UvPoly.equiv"],"nlp_text":"In the natural model we can construct these by considering first the map \\[ (\\mathsf{fst}, \\mathsf{snd}) : \\mathsf{Ty}_i \\times \\mathsf{Ty}_j \\to P_{\\mathsf{tp}_i}{\\mathsf{Ty}_j} \\]","doc_url":null,"latex_link":"sect0002.html#defn:ProductsExponentials","lean_link":null,"lean_decl_name":"ProductsExponentials","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:IdComp","label":"IdComp","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:ProductsExponentials"],"nlp_text":"The identity function \\(\\mathsf{id}_{A} : \\Gamma \\to \\mathsf{Tm}\\) of type \\(\\mathsf{Exp}\\circ (A,A) : \\Gamma \\to \\mathsf{Ty}\\) can be defined by the following","doc_url":null,"latex_link":"sect0002.html#defn:IdComp","lean_link":null,"lean_decl_name":"IdComp","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:NaturalModelRussell","label":"NaturalModelRussell","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:NaturalModel"],"nlp_text":"A collection of \\(N+1\\) natural models with \\(N\\) Russell style universes and lifts consists of For each \\(i \\leq N\\) a natural model \\(\\mathsf{tp}_i : \\mathsf{Tm}_i \\to \\mathsf{Ty}_i\\) For each \\(i {\\lt} N\\) a lift \\(\\mathsf{L}_{i}^{i+1} : \\mathsf{Ty}_i \\to \\mathsf{Ty}_{i+1}\\) For each \\(i {\\lt} N\\) a point \\(\\mathsf{U}_i : 1 \\to \\mathsf{Ty}_{i+1}\\) such that","doc_url":null,"latex_link":"sect0002.html#defn:NaturalModelRussell","lean_link":null,"lean_decl_name":"NaturalModelRussell","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:NaturalModelSigma","label":"NaturalModelSigma","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:NaturalModelRussell"],"nlp_text":"We will use the polynomial composition of two maps 5.0.6, \\(\\mathsf{tp}_i \\lhd \\mathsf{tp}_j : Q \\to P_{\\mathsf{tp}_i}(\\mathsf{Ty}_j)\\). Then additional structure of \\(\\Sigma \\) types on our \\(N\\) universes consists of, for each \\(i, j \\le N\\), a pullback square","doc_url":null,"latex_link":"sect0002.html#defn:NaturalModelSigma","lean_link":null,"lean_decl_name":"NaturalModelSigma","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/NaturalModel.lean","lean4_code":"class NaturalModelSigma where\n Sig : (P tp).obj Ty ⟶ M.Ty\n pair : (UvPoly.compDom (uvPoly M.tp) (uvPoly M.tp)) ⟶ M.Tm\n Sig_pullback : IsPullback pair ((uvPoly M.tp).comp (uvPoly M.tp)).p M.tp Sig\n\n","line":110,"end_line":114,"proof_status":"statement_formalized"} |
| {"id":"defn:NaturalModelUnivalentU","label":"NaturalModelUnivalentU","background_color":"white","border_color":"#FFAA33","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:NaturalModelRussell"],"nlp_text":"Univalence for universe \\(\\mathsf{U}\\) states that IdToEquiv itself is an equivalence \\[ \\mathsf{ua} : \\mathsf{IsBigEquiv} (\\mathsf{IdToEquiv} A \\, B) \\]","doc_url":null,"latex_link":"sect0002.html#defn:NaturalModelUnivalentU","lean_link":null,"lean_decl_name":"NaturalModelUnivalentU","kind_description":"theorems and lemmas","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/NaturalModel.lean","lean4_code":"def NaturalModelUnivalentU : Type := sorry\n\n/-\nwe will also want to say that the universe U is closed under Sigma, Pi, and Id,\nso that we can say that U is univalent.\n-/\n/-\nit would probably also be useful to have another universe U1 with U : U1,\nand maybe some type formers for U1 as well .\n-/\n\nend NaturalModel\n\nopen NaturalModel in\nvariable (Ctx) in\n","line":198,"end_line":212,"proof_status":"statement_formalized"} |
| {"id":"defn:NaturalModelIdUIP","label":"NaturalModelIdUIP","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:NaturalModelId"],"nlp_text":"Say an identity type in a natural model satisfies UIP if \\(I \\to \\mathsf{tp}\\times _{\\mathsf{Ty}} \\mathsf{tp}\\) is a strict proposition, meaning for any \\((a,b) : \\Gamma \\to \\mathsf{tp}\\times _{\\mathsf{Ty}} \\mathsf{tp}\\) there it at most one lift","doc_url":null,"latex_link":"sect0002.html#defn:NaturalModelIdUIP","lean_link":null,"lean_decl_name":"NaturalModelIdUIP","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:GroupoidNMId.J","label":"GroupoidNMId.J","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We want to define \\(J : T \\to P_{q}{\\mathsf{Tm}}\\) \\begin{align*} J_{\\Gamma } : \\mathbf{Psh}(\\mathbf{grpd})(\\Gamma , T) \\to \\, & \\mathbf{Psh}(\\mathbf{grpd})(\\Gamma , P_{q}{\\mathsf{Tm}})\\\\ (A, C, \\gamma _{\\mathsf{refl}}) \\mapsto \\, & (A, \\gamma ) \\end{align*}","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidNMId.J","lean_link":null,"lean_decl_name":"GroupoidNMId.J","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./HoTTLean/ForPoly.lean","lean4_code":"instance preservesConnectedLimitsOfShape_of_hasLimitsOfShape {J : Type v₁} [SmallCategory J]\n [IsConnected J] [HasLimitsOfShape J C] (P : UvPoly E B) :\n PreservesLimitsOfShape J (P.functor) := by\n unfold UvPoly.functor\n infer_instance\n\n","line":620,"end_line":625,"proof_status":"statement_formalized"} |
| {"id":"prop:GroupoidSigmaBeckChevalley","label":"GroupoidSigmaBeckChevalley","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\sigma : \\Delta \\to \\Gamma \\), \\(A : \\Gamma \\to \\mathbf{grpd}\\) and \\(B : \\Gamma \\cdot A \\to \\mathbf{grpd}\\). Then \\[ (\\Sigma _{A}B) \\circ \\sigma = \\Sigma _{A \\circ \\sigma }(B \\circ \\sigma _{A}) \\]","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidSigmaBeckChevalley","lean_link":null,"lean_decl_name":"GroupoidSigmaBeckChevalley","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/TarskiNaturalModel.lean","lean4_code":"theorem GroupoidSigmaBeckChevalley (Δ Γ: Grpd.{v,u}) (σ : Δ ⥤ Γ) (A : Γ ⥤ Grpd.{v,u})\n (B : (Grothendieck.Groupoidal A) ⥤ Grpd.{v,u}) : σ ⋙ GroupoidSigma Γ A B = GroupoidSigma _ (σ ⋙ A)\n (Grothendieck.Groupoidal.Map Δ Γ σ A B) := by\n refine CategoryTheory.Functor.ext ?_ ?_\n . intros X\n sorry\n . intros X Y f\n sorry\n\n","line":234,"end_line":242,"proof_status":"statement_formalized"} |
| {"id":"prop:GroupoidI.epComputation","label":"GroupoidI.epComputation","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We can compute \\(\\varepsilon : P_{q}{\\mathsf{Tm}} \\to T\\) via \\begin{align*} \\varepsilon _{\\Gamma } : \\, & \\mathbf{Psh}(\\mathbf{grpd})(\\Gamma , P_{q}{\\mathsf{Tm}}) \\to \\mathbf{Psh}(\\mathbf{grpd})(\\Gamma , T) \\\\ & (A, \\gamma ) \\mapsto (A, U \\circ \\gamma , \\gamma \\circ A^{*} \\rho ’) \\end{align*}","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidI.epComputation","lean_link":null,"lean_decl_name":"GroupoidI.epComputation","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:PolynomialComposition","label":"PolynomialComposition","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(f : B \\to A\\) and \\(g : D \\to C\\). Define the polynomial composition \\(f \\lhd g : Q \\to P_{f}{C}\\) as the composition of the two vertical maps in the following","doc_url":null,"latex_link":"sect0005.html#defn:PolynomialComposition","lean_link":null,"lean_decl_name":"PolynomialComposition","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"prop:GroupoidNM.Substitution","label":"GroupoidNM.Substitution","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\sigma : \\Delta \\to \\Gamma \\) be a functor between groupoids. Since display maps are pullbacks of the classifier \\(U : \\mathbf{grpd}_\\bullet \\to \\mathbf{grpd}\\) we have the pasting diagram","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidNM.Substitution","lean_link":null,"lean_decl_name":"GroupoidNM.Substitution","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:UvPolySlice","label":"UvPolySlice","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(P_{-} : (\\mathbb {C}/ A)^{\\mathrm{op}} \\to [\\mathbb {C}, \\mathbb {C}]\\) be defined by taking \\(s \\mapsto P_{s}\\) on objects and act on a morphism by","doc_url":null,"latex_link":"sect0005.html#defn:UvPolySlice","lean_link":null,"lean_decl_name":"UvPolySlice","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:NaturalModelId","label":"NaturalModelId","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:UvPolySlice","defn:NaturalModel"],"nlp_text":"Suppose \\(\\mathsf{tp}: \\mathsf{Tm}\\to \\mathsf{Ty}\\) is a natural model and we have a commutative square (this need not be a pullback)","doc_url":null,"latex_link":"sect0002.html#defn:NaturalModelId","lean_link":null,"lean_decl_name":"NaturalModelId","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/NaturalModel.lean","lean4_code":"class NaturalModelId extends NaturalModelIdBase Ctx where\n data : NaturalModelIdData Ctx\n\n","line":150,"end_line":152,"proof_status":"statement_formalized"} |
| {"id":"prop:GroupoidObjectsFibrant","label":"GroupoidObjectsFibrant","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\bullet \\) denote the terminal groupoid, namely that with a single object and morphism. Then the unique map \\(\\Gamma \\to \\bullet \\) is a fibration.","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidObjectsFibrant","lean_link":null,"lean_decl_name":"GroupoidObjectsFibrant","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"prop:GroupoidPiBeckChevalley","label":"GroupoidPiBeckChevalley","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\sigma : \\Delta \\to \\Gamma \\), \\(A : \\Gamma \\to \\mathbf{grpd}\\) and \\(B : \\Gamma \\cdot A \\to \\mathbf{grpd}\\). Then \\[ (\\Pi _{A}B) \\circ \\sigma = \\Pi _{A \\circ \\sigma }(B \\circ \\sigma _{A}) \\]","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidPiBeckChevalley","lean_link":null,"lean_decl_name":"GroupoidPiBeckChevalley","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:GroupoidQ","label":"GroupoidQ","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Recall the following definition of composition of polynomial endofunctors, specialized to our situation","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidQ","lean_link":null,"lean_decl_name":"GroupoidQ","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"prop:GroupoidNMId.JSection","label":"GroupoidNMId.JSection","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(J : T \\to P_{q}{\\mathsf{Tm}}\\), as defined above is a section of \\(\\varepsilon \\).","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidNMId.JSection","lean_link":null,"lean_decl_name":"GroupoidNMId.JSection","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:UvPolyEvEq","label":"UvPolyEvEq","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:UvPolyEv"],"nlp_text":"Suppose \\((\\beta ,y) : \\Gamma \\to R\\), as in 5.0.3 \\[ \\beta : \\Gamma \\to B \\quad \\text{ and } \\quad y : \\Gamma \\cdot t \\circ \\beta \\to Y \\]","doc_url":null,"latex_link":"sect0005.html#lem:UvPolyEvEq","lean_link":null,"lean_decl_name":"UvPolyEvEq","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:GroupoidI.q","label":"GroupoidI.q","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Specialized to \\(q : I \\to \\mathsf{Ty}\\) in \\(\\mathbf{Psh}(\\mathbf{grpd})\\), the characterizing property of polynomial endofunctors 5.0.2 says that a map from a representable \\(\\varepsilon : \\Gamma \\to P_{q}X\\) corresponds to the data of \\[ A : \\Gamma \\to \\mathsf{Ty}\\quad \\quad \\text{ and } \\quad \\quad C : \\Gamma \\cdot A \\cdot A \\cdot \\mathsf{Id}\\to X \\]","doc_url":null,"latex_link":"sect0004.html#lem:GroupoidI.q","lean_link":null,"lean_decl_name":"GroupoidI.q","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/NaturalModel.lean","lean4_code":"def q : I ⟶ M.Ty := pullback.fst .. ≫ pullback.fst .. ≫ tp\n","line":133,"end_line":133,"proof_status":"statement_formalized"} |
| {"id":"lem:GroupoidT","label":"GroupoidT","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"","doc_url":null,"latex_link":"sect0004.html#lem:GroupoidT","lean_link":null,"lean_decl_name":"GroupoidT","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"prop:GroupoidFibrationPullbackStable","label":"GroupoidFibrationPullbackStable","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidFibrationPullbackStable","lean_link":null,"lean_decl_name":"GroupoidFibrationPullbackStable","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:GroupoidNMDiscreteU","label":"GroupoidNMDiscreteU","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\mathsf{U}\\) be the (large) groupoid of small sets, i.e. let \\(\\mathsf{U}\\) have \\(\\mathbf{set}\\) as its objects and morphisms between two small sets as all the bijections between them. This gives us \\(\\ulcorner \\mathsf{U}\\urcorner : \\bullet \\to \\mathsf{Ty}\\). Then we define \\(\\mathsf{El}: \\mathsf{y}\\mathsf{U}\\to \\mathsf{Ty}\\) by defining \\(\\mathsf{El}: \\mathsf{U}\\to \\mathbf{Grpd}\\) as the inclusion - any small set can be regarded as a large discrete groupoid.","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidNMDiscreteU","lean_link":null,"lean_decl_name":"GroupoidNMDiscreteU","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:GroupoidNM","label":"GroupoidNM","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We would like to define a natural transformation in \\(\\mathbf{Psh}(\\mathbf{grpd})\\) \\[ \\mathsf{tp}: \\mathsf{Tm}\\to \\mathsf{Ty} \\]","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidNM","lean_link":null,"lean_decl_name":"GroupoidNM","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:GroupoidNMPi","label":"GroupoidNMPi","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We define the natural transformation \\(\\Pi : P_{\\mathsf{tp}} \\mathsf{Ty}\\to \\mathsf{Ty}\\) as that which is induced (4.4.1) by the \\(\\Pi \\)-former operation (4.2.8). Then we define the natural transformation \\(\\lambda : P_{\\mathsf{tp}} \\mathsf{Ty}\\to \\mathsf{Ty}\\) as the natural transformation induced by the following operation: given \\(A : \\Gamma \\to \\mathbf{grpd}\\) and \\(\\beta : \\Gamma \\cdot A \\to \\mathbf{grpd}_\\bullet \\), \\(\\lambda _{A}\\beta : \\Gamma \\to \\mathbf{grpd}_\\bullet \\) will be the functor such that on objects \\(x \\in \\Gamma \\) \\[ \\lambda _{A}\\beta \\, (x) := (\\Pi _{A} B \\, (x) , s_x : a \\mapsto (a, b (x , a))) \\]","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidNMPi","lean_link":null,"lean_decl_name":"GroupoidNMPi","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:GroupoidFibration","label":"GroupoidFibration","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(p : \\mathbb {C}_{1} \\to \\mathbb {C}_{0}\\) be a functor. We say \\(p\\) is a split Grothendieck fibration if we have a dependent function \\(\\mathsf{lift} \\, a \\, f\\) satisfying the following: for any object \\(a\\) in \\(\\mathbb {C}_{1}\\) and morphism \\(f : p \\, a \\to y\\) in the base \\(\\mathbb {C}_{0}\\) we have \\(\\mathsf{lift} \\, a \\, f : a \\to b\\) in \\(\\mathbb {C}_{1}\\) such that \\(p (\\mathsf{lift} \\, a \\, f) = f\\) and moreover \\(\\mathsf{lift} \\, a \\, g \\circ f = \\mathsf{lift} \\, b \\, g \\circ \\mathsf{lift} \\, a \\, f\\)","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidFibration","lean_link":null,"lean_decl_name":"GroupoidFibration","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:NaturalModel","label":"NaturalModel","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Following Awodey [ Awo17 ] , we say that a map \\(\\mathsf{tp}: \\mathsf{Tm}\\to \\mathsf{Ty}\\) in \\(\\mathbf{Psh}(\\mathbb {C})\\) is fiberwise representable or a natural model when every fiber is representable. In other words, given any \\(\\Gamma \\in \\mathbb {C}\\) and any map \\(A : \\mathsf{y}(\\Gamma ) \\to \\mathsf{Ty}\\), there is some representable \\(\\Gamma \\cdot A \\in \\mathbb {C}\\) and maps \\(\\mathsf{disp}_{A} : \\Gamma \\cdot A \\to \\Gamma \\) and \\(\\mathsf{var}_{A} : \\mathsf{y}(\\Gamma \\cdot A) \\to \\mathsf{Tm}\\) forming a pullback","doc_url":null,"latex_link":"sect0002.html#defn:NaturalModel","lean_link":null,"lean_decl_name":"NaturalModel","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/TarskiNaturalModel.lean","lean4_code":"instance GroupoidNM : NaturalModel.NaturalModelBase sGrpd.{u} where\n Ty := Ty\n Tm := Tm\n tp := tp\n ext _ A := sGrpd.of (ext A)\n disp _ A := disp A\n var _ A := var A\n disp_pullback A := disp_pullback A\n\n","line":115,"end_line":123,"proof_status":"statement_formalized"} |
| {"id":"defn:NaturalModelEq","label":"NaturalModelEq","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:NaturalModel"],"nlp_text":"The first option is fully extensional identity types, i.e. those satisfying equality reflection and uniqueness of identity proofs (UIP). Equality reflection says that if one can construct a term satisfying \\(\\mathsf{Id}(a,b)\\) then we have that definitionally \\(a \\equiv b\\), i.e. they are equal morphisms in the natural model. This amounts to just requiring that 3 is a pullback, i.e. \\(\\rho \\) is an isomorphism","doc_url":null,"latex_link":"sect0002.html#defn:NaturalModelEq","lean_link":null,"lean_decl_name":"NaturalModelEq","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/NaturalModel.lean","lean4_code":"class NaturalModelEq where\n Eq : pullback tp tp ⟶ M.Ty\n refl : Tm ⟶ M.Tm\n Eq_pullback : IsPullback refl δ tp Eq\n\nvariable (Ctx) in\n","line":117,"end_line":122,"proof_status":"statement_formalized"} |
| {"id":"defn:GroupoidEv","label":"GroupoidEv","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Define the operation of evaluation \\(\\mathsf{ev}_{\\alpha } \\, B\\) to take \\(\\alpha : \\Gamma \\to \\mathbf{grpd}_\\bullet \\) and \\(B : \\Gamma \\cdot U \\circ \\alpha \\to \\mathbf{grpd}\\) and return \\(\\mathsf{ev}_{\\alpha } \\, B : \\Gamma \\to \\mathbf{grpd}\\), described below.","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidEv","lean_link":null,"lean_decl_name":"GroupoidEv","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:GroupoidNMSigma","label":"GroupoidNMSigma","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We define the natural transformation \\[ \\Sigma : P_{\\mathsf{tp}} \\mathsf{Ty}\\to \\mathsf{Ty} \\]","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidNMSigma","lean_link":null,"lean_decl_name":"GroupoidNMSigma","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/TarskiNaturalModel.lean","lean4_code":"def GroupoidNMSigma : (P.obj.{u} Ty.{u}) ⟶ Ty.{u} := by\n fconstructor\n . dsimp [Quiver.Hom]\n intros sObj poly\n let poly' := yonedaEquiv.invFun poly\n let poly_as_pair := (UvPoly.equiv uv_tp (yoneda.obj (Opposite.unop sObj)) Ty).toFun poly'\n rcases poly_as_pair with ⟨A, B⟩\n have B := yonedaEquiv.toFun ((Limits_Sub A).inv ≫ B)\n exact downFunctor ⋙ (GroupoidSigma (sGrpd.forget.obj (Opposite.unop sObj)) (upFunctor ⋙ (yonedaEquiv.toFun A)) (upFunctor ⋙ B))\n . intros X Y f\n funext a\n refine CategoryTheory.Functor.ext ?_ ?_\n . intro b\n sorry\n . intros b₁ b₂ g\n sorry\n\n","line":255,"end_line":271,"proof_status":"statement_formalized"} |
| {"id":"defn:UvPoly","label":"UvPoly","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\mathbb {C}\\) be a locally Cartesian closed category (in our case, presheaves on the category of contexts). This means for each morphism \\(t : B \\to A\\) we have an adjoint triple","doc_url":null,"latex_link":"sect0005.html#defn:UvPoly","lean_link":null,"lean_decl_name":"UvPoly","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./HoTTLean/ForPoly.lean","lean4_code":"instance preservesPullbacks (P : UvPoly E B)\n {Pb X Y Z : C} (fst : Pb ⟶ X) (snd : Pb ⟶ Y)\n (f : X ⟶ Z) (g : Y ⟶ Z)\n (h: IsPullback fst snd f g) :\n IsPullback (P.functor.map fst) (P.functor.map snd) (P.functor.map f) (P.functor.map g) :=\n P.functor.map_isPullback h\n\n\n","line":626,"end_line":633,"proof_status":"statement_formalized"} |
| {"id":"prop:UvPoly.equiv","label":"UvPoly.equiv","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:UvPoly"],"nlp_text":"The data of a map into the polynomial applied to an object in \\(\\mathbb {C}\\)","doc_url":null,"latex_link":"sect0005.html#prop:UvPoly.equiv","lean_link":null,"lean_decl_name":"UvPoly.equiv","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/FibrationForMathlib/FibredCats/Basic.lean","lean4_code":"def equiv (c : C) : (P.op ⁻¹ (op c)) ≃ (P⁻¹ c) where\n toFun := fun x => (⟨unop x.1, by rw [obj_over] ⟩)\n invFun := fun x => ⟨op x.1 , by simp only [Functor.op_obj, unop_op, Fiber.over]⟩\n left_inv := fun _ ↦ rfl\n right_inv := fun _ ↦ rfl\n\n/-- The fibres of the opposite functor `P.op` are isomorphic\nto the the fibres of `P`. -/\n","line":90,"end_line":97,"proof_status":"statement_formalized"} |
| {"id":"defn:UvPolyEv","label":"UvPolyEv","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:UvPoly"],"nlp_text":"Let \\(\\, \\mathsf{counit} \\, : \\rho _B \\to B \\to B^* Y\\) denote the counit of the adjunction \\(f^* \\dashv f_*\\) at the object \\(B^* Y\\), recalling that \\(\\rho _B = t^* t_* B^* Y\\). Then viewing the object \\(B^* Y\\) in the slice as the object \\(Y \\times B\\) in the ambient category, we define \\(\\mathsf{ev}_{} \\, : R \\to Y\\) as the composition","doc_url":null,"latex_link":"sect0005.html#defn:UvPolyEv","lean_link":null,"lean_decl_name":"UvPolyEv","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:R","label":"R","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["prop:UvPoly.equiv"],"nlp_text":"Use \\(R\\) to denote the fiber product","doc_url":null,"latex_link":"sect0005.html#lem:R","lean_link":null,"lean_decl_name":"R","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Groupoids-NaturalModel/Id.lean","lean4_code":"def R : PGrpd ⥤ ∫ id :=\n (isPullback id).lift refl diag refl_forgetToGrpd\n\n/- This is the composition\n\nPGrpd\n |----> ∫Id\n | R |\n | |\n Diag | forget\n | |\n | V\n ---> BPGrpd\n |\n |\n | BPGrpd.forgetToGrpd\n |\n V\n Grpd\n-/\n-- TODO : consider removal?\n-- def K : Grothendieck.Groupoidal Id ⥤ Grpd := Grothendieck.Groupoidal.forget ⋙ BPGrpd.forgetToGrpd\n\n/- This is the universal lift\n Refl\nPGrpd ------------>\n |----> ∫Id -----> PGrpd\n | R | |\n | | |\n Diag | | forget\n | | |\n | V V\n ---> BPGrpd -----> Grpd\n Id\n-/\n","line":223,"end_line":257,"proof_status":"statement_formalized"} |
| {"id":"prop:GroupoidPtpUniversalProperty","label":"GroupoidPtpUniversalProperty","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Specialized to \\(\\mathsf{tp}: \\mathsf{Tm}\\to \\mathsf{Ty}\\) in \\(\\mathbf{Psh}(\\mathbf{grpd})\\), the characterizing property of polynomial endofunctors 5.0.2 says that a map from a representable \\(\\Gamma \\to P_{\\mathsf{tp}}X\\) corresponds to the data of \\[ A : \\Gamma \\to \\mathsf{Ty}\\quad \\quad \\text{ and } \\quad \\quad B : \\Gamma \\cdot A \\to X \\]","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidPtpUniversalProperty","lean_link":null,"lean_decl_name":"GroupoidPtpUniversalProperty","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"prop:GroupoidEvComputation","label":"GroupoidEvComputation","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The functor \\(\\mathsf{ev}_{\\alpha } \\, B : \\Gamma \\to \\mathbf{grpd}\\) can be computed as \\[ \\mathsf{ev}_{\\alpha } \\, B = B \\circ a \\]","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidEvComputation","lean_link":null,"lean_decl_name":"GroupoidEvComputation","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:GroupoidI","label":"GroupoidI","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We can then construct the pullback \\(I'\\)","doc_url":null,"latex_link":"sect0004.html#lem:GroupoidI","lean_link":null,"lean_decl_name":"GroupoidI","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:GroupoidBeckChevalleyIffNatural","label":"GroupoidBeckChevalleyIffNatural","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(X \\in \\mathbf{Psh}(\\mathbf{grpd})\\) be a presheaf. Let \\(F\\) be an operation that takes a groupoid \\(\\Gamma \\), a functor \\(A : \\Gamma \\to \\mathbf{grpd}\\) and \\(B : \\Gamma \\cdot A \\to X\\) and returns a natural transformation \\(F_{A} B : \\Gamma \\to X\\). Then using Yoneda to define \\(\\tilde{F} : P_{\\mathsf{tp}} X \\to X\\) pointwise as \\begin{align*} \\tilde{F}_{\\Gamma } : \\mathbf{Psh}(\\mathbf{grpd})(\\Gamma , P_{\\mathsf{tp}} X) & \\to \\mathbf{Psh}(\\mathbf{grpd})(\\Gamma , X) \\\\ (A, B) & \\mapsto F_{A} B \\end{align*}","doc_url":null,"latex_link":"sect0004.html#lem:GroupoidBeckChevalleyIffNatural","lean_link":null,"lean_decl_name":"GroupoidBeckChevalleyIffNatural","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"prop:ContextExtension","label":"ContextExtension","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For any small groupoid \\(\\Gamma \\) and \\(A : \\mathsf{y}\\Gamma \\to \\mathsf{Ty}\\), the pullback of \\(\\mathsf{tp}\\) along \\(A\\) can be given by the representable map \\(\\mathsf{y}\\mathsf{disp}_{A}\\).","doc_url":null,"latex_link":"sect0004.html#prop:ContextExtension","lean_link":null,"lean_decl_name":"ContextExtension","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"mate","label":"mate","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Suppose","doc_url":null,"latex_link":"sect0005.html#mate","lean_link":null,"lean_decl_name":"mate","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"prop:GroupoidFibrationClassified","label":"GroupoidFibrationClassified","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"There is an adjoint equivalence","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidFibrationClassified","lean_link":null,"lean_decl_name":"GroupoidFibrationClassified","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:UvPolyCartesianAction","label":"UvPolyCartesianAction","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We can also view taking polynomial endofunctors as a covariant functor on the category of arrows with cartesian squares as morphisms \\[ P_{-} : \\mathsf{CartArr}(\\mathbb {C}) \\to [\\mathbb {C}, \\mathbb {C}] \\]","doc_url":null,"latex_link":"sect0005.html#defn:UvPolyCartesianAction","lean_link":null,"lean_decl_name":"UvPolyCartesianAction","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:GroupoidR","label":"GroupoidR","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"This is a specialization of 5.0.3. Use \\(R\\) to denote the fiber product","doc_url":null,"latex_link":"sect0004.html#lem:GroupoidR","lean_link":null,"lean_decl_name":"GroupoidR","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"prop:UvPolySliceCartesian","label":"UvPolySliceCartesian","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If we have","doc_url":null,"latex_link":"sect0005.html#prop:UvPolySliceCartesian","lean_link":null,"lean_decl_name":"UvPolySliceCartesian","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:GroupoidNMId","label":"GroupoidNMId","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"To define the commutative square in \\(\\mathbf{Psh}(\\mathbf{grpd})\\)","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidNMId","lean_link":null,"lean_decl_name":"GroupoidNMId","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
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