| {"id":"defn:qcat-isofibration","label":"qcat-isofibration","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:coherent-isomorphism"],"nlp_text":"A simplicial map \\(f \\colon A \\to B\\) between quasi-categories is an isofibration if it lifts against the inner horn inclusions, as displayed below-left, and also against the inclusion of either vertex into the free-living isomorphism \\(I\\).","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Isofibration","latex_link":"sec-homotopy-category.html#defn:qcat-isofibration","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Isofibration","lean_decl_name":"SSet.Isofibration","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinity-cosmos","source_path":"InfinityCosmos/ForMathlib/InfinityCosmos/Basic.lean","lean4_code":"/-- Common notation for the hom-spaces in a pre-∞-cosmos.-/\ndef Fun (A B : K) : QCat where\n obj := EnrichedCategory.Hom A B\n property := PreInfinityCosmos.has_qcat_homs (X := A) (Y := B)\n","line":27,"end_line":30,"proof_status":"formalized"} |
| {"id":"prop:qcat-cosmos","label":"qcat-cosmos","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:qcat-isofibration","defn:qcat-trivial-fibration","defn:trivial-fibration"],"nlp_text":"The full subcategory \\({\\mathord {\\mathcal{QCat}}}\\subset {\\mathord {\\mathcal{sSet}}}\\) of quasi-categories defines an \\(\\infty \\)-cosmos in which the isofibrations, equivalences, and trivial fibrations coincide with the classes already bearing these names.","doc_url":null,"latex_link":"sec-examples.html#prop:qcat-cosmos","lean_link":null,"lean_decl_name":"qcat-cosmos","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:one-truncation","label":"one-truncation","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"By 1-truncating, any simplicial set \\(X\\) has an underlying reflexive quiver or reflexive directed graph with the 0-simplices of \\(X\\) defining the objects and the 1-simplices defining the arrows:","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.oneTruncation₂","latex_link":"sec-simplicial-sets.html#defn:one-truncation","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.oneTruncation₂","lean_decl_name":"SSet.oneTruncation₂","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/AlgebraicTopology/SimplicialSet/NerveAdjunction.lean","lean4_code":"theorem oneTruncation₂_toNerve₂Mk' : oneTruncation₂.map (toNerve₂.mk' F hyp) = F := by\n refine ReflPrefunctor.ext (fun _ ↦ ComposableArrows.ext₀ rfl)\n (fun X Y g ↦ eq_of_heq (heq_eqRec_iff_heq.2 <| heq_eqRec_iff_heq.2 ?_))\n simp [oneTruncation₂]\n refine Quiver.heq_of_homOfEq_ext ?_ ?_ (f' := F.map g) ?_\n · exact ComposableArrows.ext₀ rfl\n · exact ComposableArrows.ext₀ rfl\n · apply OneTruncation₂.Hom.ext\n simp only [oneTruncation₂_obj, ReflQuiv.of_val, OneTruncation₂.homOfEq_edge]\n fapply ComposableArrows.ext₁ <;> simp [ReflQuiv.comp_eq_comp]\n · rw [g.src_eq]; exact congr_arg (·.obj 0) (F.map g).src_eq.symm\n · rw [g.tgt_eq]; exact congr_arg (·.obj 1) (F.map g).tgt_eq.symm\n · refine (conj_eqToHom_iff_heq' _ _ _ _).2 ?_\n simp [OneTruncation₂.nerveHomEquiv]\n obtain ⟨g, rfl, rfl⟩ := g\n rfl\n\nend\n\n/-- An equality between maps into the 2-truncated nerve is detected by an equality between their\nunderlying refl prefunctors. -/\n","line":273,"end_line":293,"proof_status":"formalized"} |
| {"id":"defn:homotopy-cat","label":"homotopy-cat","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:one-truncation","prop:free-refl-quiver"],"nlp_text":"The free category on this reflexive directed graph has \\(X_0\\) as its object set, degenerate 1-simplices serving as identity morphisms, and nonidentity morphisms defined to be finite directed paths of nondegenerate 1-simplices. The homotopy category \\({\\mathord {\\mathsf{h}}}{X}\\) of \\(X\\) is the quotient of the free category on its underlying reflexive directed graph by the congruence 4 generated by imposing a composition relation \\(h = g \\circ f\\) witnessed by 2-simplices","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Truncated.HomotopyCategory","latex_link":"sec-simplicial-sets.html#defn:homotopy-cat","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Truncated.HomotopyCategory","lean_decl_name":"SSet.Truncated.HomotopyCategory","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinity-cosmos","source_path":"./InfinityCosmos/ForMathlib/AlgebraicTopology/Quasicategory/TwoTruncated.lean","lean4_code":"def HomotopyCategory₂ (A : Truncated 2) := A _⦋0⦌₂\n\n","line":586,"end_line":587,"proof_status":"formalized"} |
| {"id":"lem:htpy-cat-of-qcat","label":"htpy-cat-of-qcat","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:homotopy-cat","defn:2-truncated-qcat-htpy-cat"],"nlp_text":"If \\(A\\) is a quasi-category then its homotopy category \\({\\mathord {\\mathsf{h}}}{A}\\) is isomorphic to the homotopy category of its underlying 2-truncated quasi-category, as just described.","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Quasicategory₂.isoHomotopyCategories","latex_link":"sec-homotopy-category.html#lem:htpy-cat-of-qcat","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Quasicategory₂.isoHomotopyCategories","lean_decl_name":"SSet.Quasicategory₂.isoHomotopyCategories","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinity-cosmos","source_path":"InfinityCosmos/ForMathlib/AlgebraicTopology/Quasicategory/TwoTruncated.lean","lean4_code":"/--\n Since both `HomotopyCategory A` and `HomotopyCategory₂ A` satisfy the same universal property,\n they are isomorphic.\n-/\nnoncomputable\ndef isoHomotopyCategories : (Cat.of (HomotopyCategory A)) ≅ (Cat.of (HomotopyCategory₂ A)) where\n hom := CategoryTheory.Quotient.lift _ quotientFunctor₂ qFunctor_respects_horel₂\n inv := lift₂ (HomotopyCategory.quotientFunctor A) (by\n intro _ _ _ _ h\n simp only [Cat.of_α, HomotopyCategory.quotientFunctor, Quotient.functor]\n apply Quot.sound\n apply Quotient.CompClosure.of\n exact h)\n hom_inv_id := by\n apply HomotopyCategory.lift_unique'\n dsimp only [Cat.of_α, HomotopyCategory.quotientFunctor, CategoryStruct.comp]\n rw [← Functor.assoc, Quotient.lift_spec, is_lift₂]\n rfl\n inv_hom_id := by\n apply HomotopyCategory₂.lift_unique'\n dsimp only [Cat.of_α, CategoryStruct.comp, HomotopyCategory.quotientFunctor]\n rw [← Functor.assoc, is_lift₂, Quotient.lift_spec]\n rfl\n","line":896,"end_line":918,"proof_status":"formalized"} |
| {"id":"defn:homotopy-2-cat","label":"homotopy-2-cat","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:homotopy-cat","defn:cosmos","prop:2-cat-as-cat-enriched"],"nlp_text":"Let \\({\\mathord {\\mathcal{K}}}\\) be an \\(\\infty \\)-cosmos. Its homotopy 2-category is the 2-category \\(\\mathfrak {h}{\\mathord {\\mathcal{K}}}\\) whose objects are the objects \\(A, B\\) of \\({\\mathord {\\mathcal{K}}}\\), i.e., the \\(\\infty \\)-categories; 1-cells \\(f \\colon A \\to B\\) are the 0-arrows in the functor space \\({\\mathord {\\mathsf{Fun}}}(A,B)\\), i.e., the \\(\\infty \\)-functors; and 2-cells","doc_url":null,"latex_link":"sec-htpy-2-cat.html#defn:homotopy-2-cat","lean_link":null,"lean_decl_name":"homotopy-2-cat","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"prop:ho-nerve-adjunction","label":"ho-nerve-adjunction","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:homotopy-cat","defn:sset-category","prop:nerve-2-coskeletal","defn:nerve-functor"],"nlp_text":"The nerve embedding admits a left adjoint, namely the functor which sends a simplicial set to its homotopy category:","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.nerveAdjunction","latex_link":"sec-simplicial-sets.html#prop:ho-nerve-adjunction","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.nerveAdjunction","lean_decl_name":"CategoryTheory.nerveAdjunction","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/AlgebraicTopology/SimplicialSet/NerveAdjunction.lean","lean4_code":"noncomputable def nerveAdjunction : hoFunctor ⊣ nerveFunctor :=\n Adjunction.ofNatIsoRight ((SSet.coskAdj 2).comp nerve₂Adj) Nerve.cosk₂Iso.symm\n\n/-- Repleteness exists for full and faithful functors but not fully faithful functors, which is\nwhy we do this inefficiently. -/\n","line":486,"end_line":490,"proof_status":"formalized"} |
| {"id":"lem:qcat-htpy-cat-equiv","label":"qcat-htpy-cat-equiv","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:homotopy-cat","defn:qcat-equivalence"],"nlp_text":"If \\(f \\colon A \\to B\\) is an equivalence of quasi-categories, then the functor \\({\\mathord {\\mathsf{h}}}{f} \\colon {\\mathord {\\mathsf{h}}}{A} \\to {\\mathord {\\mathsf{h}}}{B}\\) is an equivalence of categories, where the data displayed above defines an equivalence inverse \\({\\mathord {\\mathsf{h}}}{g} \\colon {\\mathord {\\mathsf{h}}}{B} \\to {\\mathord {\\mathsf{h}}}{A}\\) and natural isomorphisms encoded by the composite 2 functors","doc_url":null,"latex_link":"sec-homotopy-category.html#lem:qcat-htpy-cat-equiv","lean_link":null,"lean_decl_name":"qcat-htpy-cat-equiv","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:htpy-cat-of-infinity-cat","label":"htpy-cat-of-infinity-cat","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:homotopy-cat","defn:cosmos"],"nlp_text":"The homotopy category of an \\(\\infty \\)-category \\(A\\) in an \\(\\infty \\)-cosmos \\({\\mathord {\\mathcal{K}}}\\) is defined to be the homotopy category of its underlying quasi-category, that is: \\[ {\\mathord {\\mathsf{h}}}A :={\\mathord {\\mathsf{hFun}}}(1,A) :={\\mathord {\\mathsf{h}}}({\\mathord {\\mathsf{Fun}}}(1,A)). \\]","doc_url":null,"latex_link":"sec-htpy-2-cat.html#defn:htpy-cat-of-infinity-cat","lean_link":null,"lean_decl_name":"htpy-cat-of-infinity-cat","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:ho-preserves-finite-products","label":"ho-preserves-finite-products","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["lem:htpy-cat-of-qcat","prop:nerve-reflective"],"nlp_text":"The functor \\({\\mathord {\\mathsf{h}}}\\colon {\\mathord {\\mathcal{sSet}}}\\to {\\mathord {\\mathcal{Cat}}}\\) preserves finite products.","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.hoFunctor.preservesFiniteProducts","latex_link":"sec-homotopy-category.html#lem:ho-preserves-finite-products","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.hoFunctor.preservesFiniteProducts","lean_decl_name":"CategoryTheory.hoFunctor.preservesFiniteProducts","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/CategoryTheory/Localization/FiniteProducts.lean","lean4_code":"lemma preservesFiniteProducts :\n PreservesFiniteProducts L where\n preserves _ := preservesProductsOfShape L W _\n\n","line":115,"end_line":118,"proof_status":"formalized"} |
| {"id":"lem:ho-preserves-small-products","label":"ho-preserves-small-products","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["lem:htpy-cat-of-qcat","prop:nerve-reflective"],"nlp_text":"The functor \\({\\mathord {\\mathsf{h}}}\\colon {\\mathord {\\mathcal{sSet}}}\\to {\\mathord {\\mathcal{Cat}}}\\) preserves finite products.","doc_url":null,"latex_link":"sec-homotopy-category.html#lem:ho-preserves-small-products","lean_link":null,"lean_decl_name":"ho-preserves-small-products","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:underlying-cat-iso","label":"underlying-cat-iso","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:homotopy-2-cat","defn:underlying-cat-of-2cat","defn:underlying-cat"],"nlp_text":"The underlying category of the homotopy 2-category of an \\(\\infty \\)-cosmos is isomorphic to the underlying category of the \\(\\infty \\)-cosmos.","doc_url":null,"latex_link":"sec-htpy-2-cat.html#lem:underlying-cat-iso","lean_link":null,"lean_decl_name":"underlying-cat-iso","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:invertible-2-cell","label":"invertible-2-cell","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:homotopy-2-cat","prop:coherent-iso","lem:qcat-1-simplex-htpy"],"nlp_text":"\\(\\quad \\) Every 2-cell","doc_url":null,"latex_link":"sec-htpy-2-cat.html#lem:invertible-2-cell","lean_link":null,"lean_decl_name":"invertible-2-cell","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":"thm:equiv-are-equiv","label":"equiv-are-equiv","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:homotopy-2-cat","defn:equivalence","defn:2-cat-equivalence"],"nlp_text":"In any \\(\\infty \\)-cosmos \\({\\mathord {\\mathcal{K}}}\\), the following are equivalent and characterize what it means for a functor \\(f \\colon A \\to B\\) between \\(\\infty \\)-categories to define an equivalence. For all \\(X \\in {\\mathord {\\mathcal{K}}}\\), the post-composition map \\(f_* \\colon {\\mathord {\\mathsf{Fun}}}(X,A) \\rightsquigarrow {\\mathord {\\mathsf{Fun}}}(X,B)\\) defines an equivalence of quasi-categories. There exists a functor \\(g \\colon B \\to A\\) and natural isomorphisms \\(\\alpha \\colon \\textup{id}_A \\cong gf\\) and \\(\\beta \\colon fg \\cong \\textup{id}_B\\) in the homotopy 2-category. There exists a functor \\(g \\colon B \\to A\\) and maps","doc_url":null,"latex_link":"sec-htpy-2-cat.html#thm:equiv-are-equiv","lean_link":null,"lean_decl_name":"equiv-are-equiv","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:2-truncated-qcat-htpy-cat","label":"2-truncated-qcat-htpy-cat","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["lem:2-truncated-qcat-htpy","cor:2-truncated-qcat-htpy-comp"],"nlp_text":"If \\(A\\) is a 2-truncated quasi-category then its homotopy category \\({\\mathord {\\mathsf{h}}}{A}\\) has the set of 0-simplices \\(A_0\\) as its objects the set of homotopy classes of 1-simplices \\(A_1\\) as its arrows the identity arrow at \\(a \\in A_0\\) represented by the degenerate 1-simplex \\(a \\cdot \\sigma ^0 \\in A_1\\) a composition relation \\(h = g \\circ f\\) in \\({\\mathord {\\mathsf{h}}}{A}\\) between the homotopy classes of arrows represented by any given 1-simplices \\(f,g,h \\in A_1\\) if and only if there exists a 2-simplex with boundary","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Quasicategory₂.instCategoryHomotopyCategory₂","latex_link":"sec-homotopy-category.html#defn:2-truncated-qcat-htpy-cat","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Quasicategory₂.instCategoryHomotopyCategory₂","lean_decl_name":"SSet.Quasicategory₂.instCategoryHomotopyCategory₂","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/emilyriehl/infinity-cosmos/4879c0d72718cea60c09ee7143aa393e65f2ea91/InfinityCosmos/ForMathlib/AlgebraicTopology/Quasicategory/TwoTruncated.lean","lean4_code":"noncomputable\ninstance instCategoryHomotopyCategory₂ : Category (HomotopyCategory₂ A) where\n id_comp f := by\n rcases f with ⟨f⟩\n apply Quotient.sound\n exact symm (composeEdges_unique (CompStruct.idComp f))\n comp_id f := by\n rcases f with ⟨f⟩\n apply Quotient.sound\n exact symm (composeEdges_unique (CompStruct.compId f))\n assoc f g h := by\n rcases f, g, h with ⟨⟨f⟩, ⟨g⟩, ⟨h⟩⟩\n apply Quotient.sound\n apply composeEdges_unique\n let fg := composeEdges f g\n exact Nonempty.some (Quasicategory₂.fill32\n (composeEdgesIsComposition f g)\n (composeEdgesIsComposition g h)\n (composeEdgesIsComposition fg h))","line":645,"end_line":663,"proof_status":"formalized"} |
| {"id":"prop:htpy-2-cat-closure","label":"htpy-2-cat-closure","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["lem:ho-preserves-finite-products","defn:closed-cosmos"],"nlp_text":"\\(\\quad \\) The homotopy 2-category of any \\(\\infty \\)-cosmos has 2-categorical products. The homotopy 2-category of a cartesian closed \\(\\infty \\)-cosmos is cartesian closed as a 2-category.","doc_url":null,"latex_link":"sec-htpy-2-cat.html#prop:htpy-2-cat-closure","lean_link":null,"lean_decl_name":"htpy-2-cat-closure","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:simplicial-cotensors","label":"simplicial-cotensors","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:simplicial-cotensor"],"nlp_text":"A simplicial category \\({\\mathord {\\mathcal{A}}}\\) has cotensors when all cotensors exist.","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.SimplicialCategory.HasCotensors","latex_link":"sec-enriched-limits.html#defn:simplicial-cotensors","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.SimplicialCategory.HasCotensors","lean_decl_name":"CategoryTheory.SimplicialCategory.HasCotensors","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinity-cosmos","source_path":"InfinityCosmos/ForMathlib/AlgebraicTopology/SimplicialCategory/Cotensors.lean","lean4_code":"/-- `K` has simplicial cotensors when cotensors with any simplicial set exist. -/\nclass HasCotensors : Prop where\n /-- All `U : SSet` and `A : K` have a cotensor. -/\n has_cotensors : ∀ U : SSet, ∀ A : K, HasCotensor U A := by infer_instance\n","line":109,"end_line":112,"proof_status":"formalized"} |
| {"id":"prop:cob-adjunction","label":"cob-adjunction","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:simplicial-cotensors","prop:change-of-base-adjunction"],"nlp_text":"Given an adjunction between cartesian closed categories","doc_url":null,"latex_link":"sec-change-of-base.html#prop:cob-adjunction","lean_link":null,"lean_decl_name":"cob-adjunction","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:cotensor-bifunctor","label":"cotensor-bifunctor","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:simplicial-cotensors"],"nlp_text":"Assuming such objects exist, the simplicial cotensor defines a bifunctor","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.SimplicialCategory.cotensor_bifunctoriality","latex_link":"sec-enriched-limits.html#lem:cotensor-bifunctor","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.SimplicialCategory.cotensor_bifunctoriality","lean_decl_name":"CategoryTheory.SimplicialCategory.cotensor_bifunctoriality","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinity-cosmos","source_path":"InfinityCosmos/ForMathlib/AlgebraicTopology/SimplicialCategory/Cotensors.lean","lean4_code":"theorem cotensor_bifunctoriality {U V : SSet} (i : U ⟶ V) {A B : K} (f : A ⟶ B) :\n (cotensorContraMap i A) ≫ (cotensorCovMap U f) =\n (cotensorCovMap V f) ≫ (cotensorContraMap i B) := cotensor_local_bifunctoriality _ _ _ _ i f\n","line":132,"end_line":134,"proof_status":"formalized"} |
| {"id":"lem:cotensor-associativity","label":"cotensor-associativity","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:simplicial-cotensors"],"nlp_text":"When a simplicial category has cotensors, cotensors are associative: given \\(A \\in {\\mathord {\\mathcal{A}}}\\) and simplicial sets \\(U\\) and \\(V\\) there are canonical isomorphisms \\[ (A^U)^V \\cong A^{U \\times V} \\cong (A^V)^U. \\]","doc_url":null,"latex_link":"sec-enriched-limits.html#lem:cotensor-associativity","lean_link":null,"lean_decl_name":"cotensor-associativity","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:cosmos","label":"cosmos","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["lem:cotensor-bifunctor","defn:simplicial-conical-limit"],"nlp_text":"An \\(\\infty \\)-cosmos \\({\\mathord {\\mathcal{K}}}\\) is a category that is enriched over quasi-categories, 2 meaning in particular that its morphisms \\(f \\colon A \\to B\\) define the vertices of a quasi-category denoted \\({\\mathord {\\mathsf{Fun}}}(A,B)\\) and referred to as a functor space, that is also equipped with a specified collection of maps that we call isofibrations and denote by “\\(\\twoheadrightarrow \\)” satisfying the following two axioms: (completeness) The quasi-categorically enriched category \\({\\mathord {\\mathcal{K}}}\\) possesses a terminal object, small products, pullbacks of isofibrations, limits of countable towers of isofibrations, and cotensors with simplicial sets, each of these limit notions satisfying a universal property that is enriched over simplicial sets. 3 (isofibrations) The isofibrations contain all isomorphisms and any map whose codomain is the terminal object; are closed under composition, product, pullback, forming inverse limits of towers, and Leibniz cotensors with monomorphisms of simplicial sets; and have the property that if \\(f \\colon A \\twoheadrightarrow B\\) is an isofibration and \\(X\\) is any object then \\({\\mathord {\\mathsf{Fun}}}(X,A) \\twoheadrightarrow {\\mathord {\\mathsf{Fun}}}(X,B)\\) is an isofibration of quasi-categories.","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.InfinityCosmos","latex_link":"sec-cosmos.html#defn:cosmos","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.InfinityCosmos","lean_decl_name":"CategoryTheory.InfinityCosmos","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinity-cosmos","source_path":"InfinityCosmos/ForMathlib/InfinityCosmos/Basic.lean","lean4_code":"/-- A `PreInfinityCosmos` is a simplicially enriched category whose hom-spaces are quasi-categories\nand whose morphisms come equipped with a special class of isofibrations.-/\nclass PreInfinityCosmos extends SimplicialCategory K where\n [has_qcat_homs : ∀ {X Y : K}, SSet.Quasicategory (EnrichedCategory.Hom X Y)]\n IsIsofibration : MorphismProperty K\n","line":15,"end_line":19,"proof_status":"formalized"} |
| {"id":"prop:isofib-define-isofib","label":"isofib-define-isofib","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:cosmos"],"nlp_text":"An isofibration \\(p \\colon E \\twoheadrightarrow B\\) in an \\(\\infty \\)-cosmos \\({\\mathord {\\mathcal{K}}}\\) also defines an isofibration in the homotopy 2-category \\(\\mathfrak {h}{\\mathord {\\mathcal{K}}}\\): given any invertible 2-cell as displayed below-left abutting to \\(B\\) with a specified lift of one of its boundary 1-cells through \\(p\\), there exists an invertible 2-cell abutting to \\(E\\) with this boundary 1-cell as displayed below-right that whiskers with \\(p\\) to the original 2-cell.","doc_url":null,"latex_link":"sec-htpy-2-cat.html#prop:isofib-define-isofib","lean_link":null,"lean_decl_name":"isofib-define-isofib","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:closed-cosmos","label":"closed-cosmos","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:cosmos"],"nlp_text":"An \\(\\infty \\)-cosmos \\({\\mathord {\\mathcal{K}}}\\) is cartesian closed if the product bifunctor \\(- \\times -\\colon {\\mathord {\\mathcal{K}}}\\times {\\mathord {\\mathcal{K}}}\\to {\\mathord {\\mathcal{K}}}\\) extends to a simplicially enriched two-variable adjunction \\[ {\\mathord {\\mathsf{Fun}}}(A \\times B,C) \\cong {\\mathord {\\mathsf{Fun}}}(A,C^B) \\cong {\\mathord {\\mathsf{Fun}}}(B,C^A) \\]","doc_url":null,"latex_link":"sec-examples.html#defn:closed-cosmos","lean_link":null,"lean_decl_name":"closed-cosmos","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:underlying-cat-of-cosmos","label":"underlying-cat-of-cosmos","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:cosmos"],"nlp_text":"The underlying category \\({\\mathord {\\mathcal{K}}}_0\\) of an \\(\\infty \\)-cosmos \\({\\mathord {\\mathcal{K}}}\\) is the category whose objects are the \\(\\infty \\)-categories in \\({\\mathord {\\mathcal{K}}}\\) and whose morphisms are the \\(0\\)-arrows, i.e., the vertices in the functor spaces.","doc_url":null,"latex_link":"sec-cosmos.html#defn:underlying-cat-of-cosmos","lean_link":null,"lean_decl_name":"underlying-cat-of-cosmos","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:equivalence","label":"equivalence","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:cosmos","defn:qcat-equivalence"],"nlp_text":"In an \\(\\infty \\)-cosmos \\({\\mathord {\\mathcal{K}}}\\), a morphism \\(f\\colon A \\to B\\) is an equivalence just when the induced map \\(f_* \\colon {\\mathord {\\mathsf{Fun}}}(X,A) \\rightsquigarrow {\\mathord {\\mathsf{Fun}}}(X,B)\\) on functor spaces is an equivalence of quasi-categories for all \\(X \\in {\\mathord {\\mathcal{K}}}\\).","doc_url":null,"latex_link":"sec-cosmos.html#defn:equivalence","lean_link":null,"lean_decl_name":"equivalence","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Topology/Homotopy/Basic.lean","lean4_code":"theorem equivalence : Equivalence (@Homotopic X Y _ _) :=\n ⟨refl, by apply symm, by apply trans⟩\n\nnonrec theorem prodMk {f₀ f₁ : C(X, Y)} {g₀ g₁ : C(X, Z)} :\n Homotopic f₀ f₁ → Homotopic g₀ g₁ → Homotopic (f₀.prodMk g₀) (f₁.prodMk g₁)\n | ⟨F⟩, ⟨G⟩ => ⟨F.prodMk G⟩\n\nnonrec theorem prodMap {f₀ f₁ : C(X, Y)} {g₀ g₁ : C(Z, Z')} :\n Homotopic f₀ f₁ → Homotopic g₀ g₁ → Homotopic (f₀.prodMap g₀) (f₁.prodMap g₁)\n | ⟨F⟩, ⟨G⟩ => ⟨F.prodMap G⟩\n\n/-- If each `f₀ i : C(X, Y i)` is homotopic to `f₁ i : C(X, Y i)`, then `ContinuousMap.pi f₀` is\nhomotopic to `ContinuousMap.pi f₁`. -/\n","line":349,"end_line":361,"proof_status":"statement_formalized"} |
| {"id":"lem:equiv-htpy-equiv","label":"equiv-htpy-equiv","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:equivalence"],"nlp_text":"A map \\(f \\colon A \\to B\\) between \\(\\infty \\)-categories in an \\(\\infty \\)-cosmos \\({\\mathord {\\mathcal{K}}}\\) is an equivalence if and only if it extends to the data of a “homotopy equivalence” with the free-living isomorphism \\(I\\) serving as the interval: that is, if there exist maps \\(g \\colon B \\to A\\)","doc_url":null,"latex_link":"sec-cosmos.html#lem:equiv-htpy-equiv","lean_link":null,"lean_decl_name":"equiv-htpy-equiv","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:trivial-fibration","label":"trivial-fibration","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:equivalence"],"nlp_text":"In an \\(\\infty \\)-cosmos \\({\\mathord {\\mathcal{K}}}\\), a morphism \\(f\\colon A \\to B\\) is a trivial fibration just when \\(f\\) is both an isofibration and an equivalence.","doc_url":null,"latex_link":"sec-cosmos.html#defn:trivial-fibration","lean_link":null,"lean_decl_name":"trivial-fibration","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:simplicial-conical-limit","label":"simplicial-conical-limit","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:simplicial-category"],"nlp_text":"Consider a limit cone \\((\\lim _{j \\in J}A_j \\to A_j)_{j \\in J}\\) in the underlying category \\({\\mathord {\\mathcal{A}}}_0\\) of a simplicially-enriched category \\({\\mathord {\\mathcal{A}}}\\). By applying the covariant representable functor \\({\\mathord {\\mathcal{A}}}(X,-) \\colon {\\mathord {\\mathcal{A}}}_0 \\to {\\mathord {\\mathcal{sSet}}}\\) to a limit cone \\((\\lim _{j \\in J}A_j \\to A_j)_{j \\in J}\\) in \\({\\mathord {\\mathcal{A}}}_0\\), we obtain a natural comparison map \\begin{equation} \\label{eq:simplicial-limit-map} {\\mathord {\\mathcal{A}}}(X,\\lim _{j \\in J}A_j) \\to \\lim _{j \\in J}{\\mathord {\\mathcal{A}}}(X,A_j). \\end{equation}","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.Enriched.IsConicalLimit","latex_link":"sec-enriched-limits.html#defn:simplicial-conical-limit","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.Enriched.IsConicalLimit","lean_decl_name":"CategoryTheory.Enriched.IsConicalLimit","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinity-cosmos","source_path":"./InfinityCosmos/ForMathlib/CategoryTheory/Enriched/Limits/IsConicalLimit.lean","lean4_code":"structure IsConicalLimit {J : Type u₁} [Category.{v₁} J]\n (V : outParam <| Type u') [Category.{v'} V] [MonoidalCategory V]\n {C : Type u} [Category.{v} C] [EnrichedOrdinaryCategory V C]\n {F : J ⥤ C} (c : Cone F) where\n /-- A conical limit cone is a limit cone. -/\n isLimit : IsLimit c\n /--\n The cone obtained by applying the coyoneda functor `(X ⟶[V] -)` to `c` is a limit cone in `V`.\n -/\n isConicalLimit (X : C) : IsLimit <| (eCoyoneda V X).mapCone c\n\nvariable {J : Type u₁} [Category.{v₁} J]\nvariable (V : Type u') [Category.{v'} V] [MonoidalCategory V]\nvariable {C : Type u} [Category.{v} C] [EnrichedOrdinaryCategory V C]\nvariable {F : J ⥤ C} (c : Cone F)\n\nnamespace IsConicalLimit\n\nvariable {V} {c} {d : Cone F}\n\n","line":43,"end_line":62,"proof_status":"formalized"} |
| {"id":"defn:underlying-cat-of-2cat","label":"underlying-cat-of-2cat","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The underlying category of a 2-category is defined by simply forgetting its 2-cells. Note that an \\(\\infty \\)-cosmos \\({\\mathord {\\mathcal{K}}}\\) and its homotopy 2-category \\(\\mathfrak {h}{\\mathord {\\mathcal{K}}}\\) share the same underlying category \\({\\mathord {\\mathcal{K}}}_0\\) of \\(\\infty \\)-categories and \\(\\infty \\)-functors in \\({\\mathord {\\mathcal{K}}}\\).","doc_url":null,"latex_link":"sec-htpy-2-cat.html#defn:underlying-cat-of-2cat","lean_link":null,"lean_decl_name":"underlying-cat-of-2cat","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:qcat-trivial-fibration","label":"qcat-trivial-fibration","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A map \\(f \\colon X \\to Y\\) between simplicial sets is a trivial fibration if it admits lifts against the boundary inclusions for all simplices","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.TrivialFibration","latex_link":"sec-homotopy-category.html#defn:qcat-trivial-fibration","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.TrivialFibration","lean_decl_name":"SSet.TrivialFibration","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinity-cosmos","source_path":"InfinityCosmos/ForMathlib/AlgebraicTopology/SimplicialSet/MorphismProperty.lean","lean4_code":"/-- a morphism of simplicial sets is a trivial fibration if it has the right lifting property wrt\n every boundary inclusion `∂Δ[n] ⟶ Δ[n]`. -/\ndef TrivialFibration : MorphismProperty SSet := fun _ _ p ↦ BoundaryInclusions.rlp p\n","line":26,"end_line":28,"proof_status":"formalized"} |
| {"id":"lem:trivial-fib-representability","label":"trivial-fib-representability","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:qcat-trivial-fibration","defn:trivial-fibration"],"nlp_text":"If \\(E \\twoheadrightarrow B\\) is a trivial fibration in an \\(\\infty \\)-cosmos, then is \\({\\mathord {\\mathsf{Fun}}}(X,E)\\twoheadrightarrow {\\mathord {\\mathsf{Fun}}}(X,B)\\) is a trivial fibration of quasi-categories.","doc_url":null,"latex_link":"sec-cosmos.html#lem:trivial-fib-representability","lean_link":null,"lean_decl_name":"trivial-fib-representability","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:face-map","label":"face-map","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:simplex-category"],"nlp_text":"The elementary face operators are the maps","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SimplexCategory.δ","latex_link":"sec-simplicial-sets.html#defn:face-map","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SimplexCategory.δ","lean_decl_name":"SimplexCategory.δ","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinity-cosmos","source_path":"./InfinityCosmos/ForMathlib/AlgebraicTopology/SimplexCategory.lean","lean4_code":"def δ_zero_mkOfLe {n : ℕ} (i j : Fin (n + 1)) (h : i ≤ j) : SimplexCategory.δ 0 ≫ mkOfLe i j h =\n (SimplexCategory.mk 0).const (SimplexCategory.mk n) j := by\n ext x\n fin_cases x\n aesop\n\n","line":7,"end_line":12,"proof_status":"formalized"} |
| {"id":"prop:simplex-cat-factorization","label":"simplex-cat-factorization","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:face-map","defn:degeneracy-map"],"nlp_text":"Every morphism in \\(\\Delta \\) factors uniquely as an epimorphism followed by a monomorphism; these epimorphisms, the degeneracy operators, decompose as composites of elementary degeneracy operators, while the monomorphisms, the face operators, decompose as composites of elementary face operators.","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SimplexCategory.instHasStrongEpiImages","latex_link":"sec-simplicial-sets.html#prop:simplex-cat-factorization","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SimplexCategory.instHasStrongEpiImages","lean_decl_name":"SimplexCategory.instHasStrongEpiImages","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/ef921c9a5139539a505bc3291f1a076a1e250d92/Mathlib/AlgebraicTopology/SimplexCategory/Basic.lean","lean4_code":"instance : HasStrongEpiImages SimplexCategory :=\n Limits.hasStrongEpiImages_of_hasStrongEpiMonoFactorisations","line":839,"end_line":840,"proof_status":"formalized"} |
| {"id":"defn:simplicial-cotensor","label":"simplicial-cotensor","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:simplicial-category"],"nlp_text":"Let \\({\\mathord {\\mathcal{A}}}\\) be a simplicial category. The cotensor of an object \\(A \\in {\\mathord {\\mathcal{A}}}\\) by a simplicial set \\(U\\) is given by the data of an object \\(A^U \\in {\\mathord {\\mathcal{A}}}\\) together with a cone \\(U \\to {\\mathord {\\mathcal{A}}}(A^U,A)\\) so that the induced map defines an isomorphism of simplicial sets: \\begin{equation} \\label{eq:cotensor-defn} {\\mathord {\\mathcal{A}}}(X,A^U) \\cong {\\mathord {\\mathcal{A}}}(X,A)^U \\end{equation}","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.SimplicialCategory.HasCotensor","latex_link":"sec-enriched-limits.html#defn:simplicial-cotensor","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.SimplicialCategory.HasCotensor","lean_decl_name":"CategoryTheory.SimplicialCategory.HasCotensor","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinity-cosmos","source_path":"InfinityCosmos/ForMathlib/AlgebraicTopology/SimplicialCategory/Cotensors.lean","lean4_code":"/-- `HasCotensor U A` represents the mere existence of a simplicial cotensor. -/\nclass HasCotensor (U : SSet) (A : K) : Prop where mk' ::\n /-- There is some cotensor. -/\n exists_cotensor : Nonempty (Cotensor U A)\n","line":63,"end_line":66,"proof_status":"formalized"} |
| {"id":"lem:brown-fact","label":"brown-fact","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:trivial-fibration"],"nlp_text":"Any functor \\(f\\colon A \\to B\\) in an \\(\\infty \\)-cosmos may be factored as an equivalence followed by an isofibration, where this equivalence is constructed as a section of a trivial fibration.","doc_url":null,"latex_link":"sec-cosmos.html#lem:brown-fact","lean_link":null,"lean_decl_name":"brown-fact","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":"prop:sliced-cosmoi","label":"sliced-cosmoi","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["lem:brown-fact","lem:trivial-fib-conical"],"nlp_text":"For any \\(\\infty \\)-cosmos \\({\\mathord {\\mathcal{K}}}\\) and any \\(\\infty \\)-category \\(B \\in {\\mathord {\\mathcal{K}}}\\) there is an \\(\\infty \\)-cosmos \\({\\mathord {\\mathcal{K}}}_{/B}\\) of isofibrations over \\(B\\) whose objects are isofibrations \\(p \\colon E \\twoheadrightarrow B\\) with codomain \\(B\\) functor spaces, say from \\(p \\colon E \\twoheadrightarrow B\\) to \\(q \\colon F \\twoheadrightarrow B\\), are defined by pullback","doc_url":null,"latex_link":"sec-examples.html#prop:sliced-cosmoi","lean_link":null,"lean_decl_name":"sliced-cosmoi","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:coherent-isomorphism","label":"coherent-isomorphism","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:nerve"],"nlp_text":"The homotopy coherent isomorphism \\(I\\), is the nerve of the free-living isomorphism.","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.coherentIso","latex_link":"sec-homotopy-category.html#defn:coherent-isomorphism","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.coherentIso","lean_decl_name":"SSet.coherentIso","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinity-cosmos","source_path":"InfinityCosmos/ForMathlib/AlgebraicTopology/SimplicialSet/CoherentIso.lean","lean4_code":"def coherentIso : SSet.{u} := nerve WalkingIso\n","line":79,"end_line":79,"proof_status":"formalized"} |
| {"id":"defn:qcat-equivalence","label":"qcat-equivalence","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:coherent-isomorphism","eq:qcat-defn"],"nlp_text":"w= A map \\(f \\colon A \\to B\\) between quasi-categories is an equivalence if it extends to the data of a “homotopy equivalence” with the free-living isomorphism \\(I\\) serving as the interval: that is, if there exist maps \\(g \\colon B \\to A\\),","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/QCat.Equiv","latex_link":"sec-homotopy-category.html#defn:qcat-equivalence","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/QCat.Equiv","lean_decl_name":"QCat.Equiv","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinity-cosmos","source_path":"InfinityCosmos/ForMathlib/AlgebraicTopology/SimplicialSet/Homotopy.lean","lean4_code":"/-- For the correct interval, this defines a good notion of equivalences for both Kan complexes and quasi-categories.-/\nstructure Equiv (A B : SSet.{u}) : Type u where\n toFun : A ⟶ B\n invFun : B ⟶ A\n left_inv : Homotopy (I := I) (toFun ≫ invFun) (𝟙 A)\n right_inv : Homotopy (I := I) (invFun ≫ toFun) (𝟙 B)\n","line":78,"end_line":83,"proof_status":"formalized"} |
| {"id":"prop:coherent-iso","label":"coherent-iso","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:coherent-isomorphism","eq:qcat-defn","defn:isomorphism"],"nlp_text":"An arrow \\(f\\) in a quasi-category \\(A\\) is an isomorphism if and only if it extends to a homotopy coherent isomorphism","doc_url":null,"latex_link":"sec-homotopy-category.html#prop:coherent-iso","lean_link":null,"lean_decl_name":"coherent-iso","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:equivalence-2-of-3","label":"equivalence-2-of-3","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["lem:equiv-htpy-equiv"],"nlp_text":"The equivalences in an \\(\\infty \\)-cosmos are closed under retracts and satisfy the 2-of-3 property: given a composable pair of functors and their composite, if any two of these are equivalences so is the third.","doc_url":null,"latex_link":"sec-cosmos.html#lem:equivalence-2-of-3","lean_link":null,"lean_decl_name":"equivalence-2-of-3","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:qcat-1-simplex-htpy","label":"qcat-1-simplex-htpy","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:1-simplex-htpy","eq:qcat-defn"],"nlp_text":"Parallel 1-simplices \\(f\\) and \\(g\\) in a quasi-category are homotopic if and only if there exists a 2-simplex of any or equivalently all of the forms displayed in Definition 1.2.19.","doc_url":null,"latex_link":"sec-simplicial-sets.html#lem:qcat-1-simplex-htpy","lean_link":null,"lean_decl_name":"qcat-1-simplex-htpy","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:trivial-fib-leibniz","label":"trivial-fib-leibniz","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:trivial-fibration"],"nlp_text":"In an \\(\\infty \\)-cosmos, the Leibniz cotensors of any trivial fibration with a monomorphism of simplicial sets is a trivial fibration as is the Leibniz cotensor of an isofibration with a map in the class cellularly generated by the inner horn inclusions and the map \\([0]\\hookrightarrow I\\).","doc_url":null,"latex_link":"sec-cosmos.html#lem:trivial-fib-leibniz","lean_link":null,"lean_decl_name":"trivial-fib-leibniz","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":"prop:nerve-reflective","label":"nerve-reflective","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The homotopy category of the nerve of a 1-category is isomorphic to the original category, as the 2-simplices in the nerve witness all of the composition relations satisfied by the arrows in the underlying reflexive directed graph.","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.nerveFunctorCompHoFunctorIso","latex_link":"sec-simplicial-sets.html#prop:nerve-reflective","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.nerveFunctorCompHoFunctorIso","lean_decl_name":"CategoryTheory.nerveFunctorCompHoFunctorIso","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/AlgebraicTopology/SimplicialSet/NerveAdjunction.lean","lean4_code":"noncomputable def nerveFunctorCompHoFunctorIso : nerveFunctor.{u, u} ⋙ hoFunctor ≅ 𝟭 Cat :=\n asIso (nerveAdjunction.counit)\n\nnoncomputable instance : Reflective nerveFunctor where\n L := hoFunctor\n adj := nerveAdjunction\n\n","line":510,"end_line":516,"proof_status":"formalized"} |
| {"id":"cor:sset-cat-colimits","label":"sset-cat-colimits","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:sset-category"],"nlp_text":"The category of simplicial sets is cocomplete.","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.hasColimits","latex_link":"sec-simplicial-sets.html#cor:sset-cat-colimits","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.hasColimits","lean_decl_name":"SSet.hasColimits","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/ef921c9a5139539a505bc3291f1a076a1e250d92/Mathlib/AlgebraicTopology/SimplicialSet/Basic.lean","lean4_code":"instance hasColimits : HasColimits SSet := by\n dsimp only [SSet]\n infer_instance","line":49,"end_line":51,"proof_status":"formalized"} |
| {"id":"defn:simplex-boundary","label":"simplex-boundary","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:standard-simplex"],"nlp_text":"We write \\(\\partial \\Delta [n] \\subset \\Delta [n]\\) for the boundary sphere of the \\(n\\)-simplex. The sphere \\(\\partial \\Delta [n]\\) is the simplicial subset generated by the codimension-one faces of the \\(n\\)-simplex.","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.boundary","latex_link":"sec-simplicial-sets.html#defn:simplex-boundary","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.boundary","lean_decl_name":"SSet.boundary","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Order/Heyting/Boundary.lean","lean4_code":"def boundary (a : α) : α :=\n a ⊓ ￢a\n\n/-- The boundary of an element of a co-Heyting algebra. -/\nscoped[Heyting] prefix:120 \"∂ \" => Coheyting.boundary\n\nopen Heyting\n\n-- TODO: Should hnot be named hNot?\n","line":35,"end_line":43,"proof_status":"formalized"} |
| {"id":"defn:2-truncated-qcat","label":"2-truncated-qcat","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A 2-truncated simplicial set \\(A\\) is a 2-truncated quasi-category if it admits the following three operations: (2,1)-filling: any path \\(f_\\bullet \\) of length 2 in \\(A\\) may be filled to a \\(2\\)-simplex whose spine equals the given path. (3,1)-filling: given any path \\(f_\\bullet \\) of length 3 in \\(A\\), 2-simplices \\(\\sigma _3\\) and \\(\\sigma _0\\) filling the restricted paths \\(f_{012}\\) and \\(f_{123}\\) respectively, and 2-simplex \\(\\sigma _2\\) filling the path formed by \\(f_{01}\\) and the diagonal of \\(\\sigma _0\\), there is a 2-simplex \\(\\sigma _1\\) filling the path formed by the diagonal of \\(\\sigma _3\\) and \\(f_{23}\\) and whose diagonal is the diagonal of \\(\\sigma _2\\). (3,2)-filling: given any path \\(f_\\bullet \\) of length 3 in \\(A\\), 2-simplices \\(\\sigma _3\\) and \\(\\sigma _0\\) filling the restricted paths \\(f_{012}\\) and \\(f_{123}\\) respectively, and 2-simplex \\(\\sigma _1\\) filling the path formed by the diagonal of \\(\\sigma _3\\) and \\(f_{23}\\), there is a 2-simplex \\(\\sigma _2\\) filling the path formed by \\(f_{01}\\) and the diagonal of \\(\\sigma _0\\) and whose diagonal is the diagonal of \\(\\sigma _1\\).","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Truncated.Quasicategory₂","latex_link":"sec-homotopy-category.html#defn:2-truncated-qcat","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Truncated.Quasicategory₂","lean_decl_name":"SSet.Truncated.Quasicategory₂","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/emilyriehl/infinity-cosmos/4879c0d72718cea60c09ee7143aa393e65f2ea91/InfinityCosmos/ForMathlib/AlgebraicTopology/Quasicategory/TwoTruncated.lean","lean4_code":"/--\nA 2-truncated quasicategory is a 2-truncated simplicial set with 3 properties:\n (2, 1)-filling: any path of length 2 in may be filled to a 2-simplex whose\n spine equals the given path.\n (3, 1)-filling: given any path f of length 3, 2-simplices σ₃ and σ₀ filling the restricted paths\n f₀₁₂ and f₁₂₃ respectively, and a 2-simplex σ₂ filling the path formed by f₀₁ and the diagonal\n of σ₀, there is a 2-simplex σ₁ filling the path formed by the diagonal of σ₃ and f₂₃ and whose\n diagonal is the diagonal of σ₂.\n (3, 2)-filling: given any path f of length 3, 2-simplices σ₃ and σ₀ filling the restricted paths\n f₀₁₂ and f₁₂₃ respectively, and a 2-simplex σ₁ filling the path formed by f₂₃ and the diagonal\n of σ₃, there is a 2-simplex σ₂ filling the path formed by f₀₁ and the diagonal of σ₀ and whose\n diagonal is the diagonal of σ₁.\n-/\nclass Quasicategory₂ (X : Truncated 2) where\n fill21 {x₀ x₁ x₂ : X _⦋0⦌₂}\n (e₀₁ : Edge x₀ x₁) (e₁₂ : Edge x₁ x₂) :\n Nonempty (Σ e₀₂ : Edge x₀ x₂, CompStruct e₀₁ e₁₂ e₀₂)\n fill31 {x₀ x₁ x₂ x₃ : X _⦋0⦌₂}\n {e₀₁ : Edge x₀ x₁} {e₁₂ : Edge x₁ x₂} {e₂₃ : Edge x₂ x₃}\n {e₀₂ : Edge x₀ x₂} {e₁₃ : Edge x₁ x₃} {e₀₃ : Edge x₀ x₃}\n (f₃ : CompStruct e₀₁ e₁₂ e₀₂)\n (f₀ : CompStruct e₁₂ e₂₃ e₁₃)\n (f₂ : CompStruct e₀₁ e₁₃ e₀₃) :\n Nonempty (CompStruct e₀₂ e₂₃ e₀₃)\n fill32 {x₀ x₁ x₂ x₃ : X _⦋0⦌₂}\n {e₀₁ : Edge x₀ x₁} {e₁₂ : Edge x₁ x₂} {e₂₃ : Edge x₂ x₃}\n {e₀₂ : Edge x₀ x₂} {e₁₃ : Edge x₁ x₃} {e₀₃ : Edge x₀ x₃}\n (f₃ : CompStruct e₀₁ e₁₂ e₀₂)\n (f₀ : CompStruct e₁₂ e₂₃ e₁₃)\n (f₁ : CompStruct e₀₂ e₂₃ e₀₃) :\n Nonempty (CompStruct e₀₁ e₁₃ e₀₃)","line":46,"end_line":76,"proof_status":"formalized"} |
| {"id":"lem:2-truncated-qcat","label":"2-truncated-qcat","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:2-truncated-qcat","eq:qcat-defn"],"nlp_text":"The 2-truncation of a quasi-category is a 2-truncated quasi-category.","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Truncated.two_truncatation_of_qc_is_2_trunc_qc","latex_link":"sec-homotopy-category.html#lem:2-truncated-qcat","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Truncated.two_truncatation_of_qc_is_2_trunc_qc","lean_decl_name":"SSet.Truncated.two_truncatation_of_qc_is_2_trunc_qc","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/emilyriehl/infinity-cosmos/4879c0d72718cea60c09ee7143aa393e65f2ea91/InfinityCosmos/ForMathlib/AlgebraicTopology/Quasicategory/TwoTruncated.lean","lean4_code":"/--\nThe 2-truncation of a quasi-category is a 2-truncated quasi-category.\n-/\ninstance two_truncatation_of_qc_is_2_trunc_qc {X : SSet} [Quasicategory X] :\n Quasicategory₂ ((truncation 2).obj X) where\n fill21 e₀₁ e₁₂ := by\n obtain ⟨g, h⟩ := Quasicategory.hornFilling Fin.zero_lt_one (by simp)\n (horn₂₁.fromEdges e₀₁ e₁₂)\n apply Nonempty.intro\n exact (horn₂₁.fromHornExtension e₀₁ e₁₂ g h)\n fill31 f₃ f₀ f₂ := by\n obtain ⟨g, h⟩ := Quasicategory.hornFilling Fin.zero_lt_one (by simp)\n (horn₃₁.fromFaces f₃ f₀ f₂)\n apply Nonempty.intro\n exact (horn₃₁.fromHornExtension f₃ f₀ f₂ g h)\n fill32 f₃ f₀ f₁ := by\n obtain ⟨g, h⟩ := Quasicategory.hornFilling (by simp) (by simp)\n (horn₃₂.fromFaces f₃ f₀ f₁)\n apply Nonempty.intro\n exact (horn₃₂.fromHornExtension f₃ f₀ f₁ g h)","line":380,"end_line":399,"proof_status":"formalized"} |
| {"id":"lem:2-truncated-qcat-htpy","label":"2-truncated-qcat-htpy","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:2-truncated-qcat","defn:1-simplex-left-right-htpy"],"nlp_text":"If \\(A\\) is a 2-truncated quasi-category then: The left and right homotopy relations are reflexive. The left and right homotopy relations are symmetric. The left and right homotopy relations are transitive. The left homotopy relation coincides with the right homotopy relation.","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Quasicategory₂.HomotopicL_iff_HomotopicR","latex_link":"sec-homotopy-category.html#lem:2-truncated-qcat-htpy","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Quasicategory₂.HomotopicL_iff_HomotopicR","lean_decl_name":"SSet.Quasicategory₂.HomotopicL_iff_HomotopicR","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinity-cosmos","source_path":"InfinityCosmos/ForMathlib/AlgebraicTopology/Quasicategory/TwoTruncated.lean","lean4_code":"/--\nThe right and left homotopy relations coincide\n-/\ntheorem HomotopicL_iff_HomotopicR {x y : A _⦋0⦌₂} {f g : Edge x y} :\n HomotopicL f g ↔ HomotopicR f g := by\n constructor\n . rintro ⟨lhfg⟩\n exact Quasicategory₂.fill32 (idComp f) (compId f) lhfg\n . rintro ⟨rhfg⟩\n exact Quasicategory₂.fill31 (idComp f) (compId f) rhfg\n","line":523,"end_line":532,"proof_status":"formalized"} |
| {"id":"lem:2-truncated-qcat-htpy-comp","label":"2-truncated-qcat-htpy-comp","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:2-truncated-qcat","defn:1-simplex-left-right-htpy"],"nlp_text":"\\(\\quad \\) If \\(\\sigma \\) and \\(\\tau \\) are 2-simplices in a 2-truncated quasi-category filling the same path, their diagonal edges are homotopic. If \\(h\\) is the diagonal edge of a 2-simplex filling the path formed by \\(f\\) and \\(g\\) and \\(g\\) is homotopic to \\(g'\\), then \\(h\\) is the diagonal edge of a 2-simplex filling the path formed by \\(f\\) and \\(g'\\). If \\(h\\) is the diagonal edge of a 2-simplex filling the path formed by \\(f\\) and \\(g\\) and \\(f\\) is homotopic to \\(f'\\), then \\(h\\) is the diagonal edge of a 2-simplex filling the path formed by \\(f'\\) and \\(g\\).","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Quasicategory₂.transport_edge₀","latex_link":"sec-homotopy-category.html#lem:2-truncated-qcat-htpy-comp","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Quasicategory₂.transport_edge₀","lean_decl_name":"SSet.Quasicategory₂.transport_edge₀","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/emilyriehl/infinity-cosmos/4879c0d72718cea60c09ee7143aa393e65f2ea91/InfinityCosmos/ForMathlib/AlgebraicTopology/Quasicategory/TwoTruncated.lean","lean4_code":"lemma transport_edge₀ {f : Edge x y} {g g' : Edge y z} {h : Edge x z}\n (s : CompStruct f g h) (htpy : HomotopicL g g') : Nonempty (CompStruct f g' h) := by\n rcases htpy with ⟨htpy⟩\n exact Quasicategory₂.fill32 s htpy (compId h)","line":551,"end_line":554,"proof_status":"formalized"} |
| {"id":"cor:2-truncated-qcat-htpy-comp","label":"2-truncated-qcat-htpy-comp","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["lem:2-truncated-qcat-htpy-comp"],"nlp_text":"Suppose there is a 2-simplex in a 2-truncated quasi-category with spine formed by the paths \\(f\\) and \\(g\\) and diagonal \\(h\\). Then if \\(f \\sim f'\\), \\(g \\sim g'\\), and \\(h \\sim h'\\), there is a 2-simplex with spine formed by \\(f'\\) and \\(g'\\) and diagonal \\(h'\\).","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Quasicategory₂.transport_all_edges","latex_link":"sec-homotopy-category.html#cor:2-truncated-qcat-htpy-comp","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Quasicategory₂.transport_all_edges","lean_decl_name":"SSet.Quasicategory₂.transport_all_edges","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinity-cosmos","source_path":"InfinityCosmos/ForMathlib/AlgebraicTopology/Quasicategory/TwoTruncated.lean","lean4_code":"lemma transport_all_edges {f f' : Edge x y} {g g' : Edge y z}\n {h h' : Edge x z} (hf : HomotopicL f f') (hg : HomotopicL g g') (hh : HomotopicL h h')\n (s : CompStruct f g h) :\n Nonempty (CompStruct f' g' h') := by\n have a : Nonempty (CompStruct f' g h) := transport_edge₂ s hf\n have b : Nonempty (CompStruct f' g' h) := by\n rcases a with ⟨a⟩\n exact transport_edge₀ a hg\n rcases b with ⟨b⟩\n exact transport_edge₁ b hh\n","line":566,"end_line":575,"proof_status":"formalized"} |
| {"id":"defn:1-simplex-left-right-htpy","label":"1-simplex-left-right-htpy","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A parallel pair of 1-simplices \\(f,g\\) in a 2-truncated simplicial set \\(X\\) are left homotopic if there exists a 2-simplex whose boundary takes the form below-left and right homotopic if there exists a 2-simplex whose boundary takes the form below-right:","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Truncated.HomotopicL","latex_link":"sec-homotopy-category.html#defn:1-simplex-left-right-htpy","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Truncated.HomotopicL","lean_decl_name":"SSet.Truncated.HomotopicL","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinity-cosmos","source_path":"InfinityCosmos/ForMathlib/AlgebraicTopology/SimplicialSet/Homotopy.lean","lean4_code":"def HomotopicL : Prop :=\n Nonempty (HomotopyL f g)\n","line":131,"end_line":132,"proof_status":"formalized"} |
| {"id":"defn:underlying-cat","label":"underlying-cat","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:simplicial-category"],"nlp_text":"The category \\({\\mathord {\\mathcal{A}}}_0\\) of 0-arrows is the underlying category of the simplicial category \\({\\mathord {\\mathcal{A}}}\\), which forgets the higher dimensional simplicial structure.","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.categoryForgetEnrichment","latex_link":"sec-enriched-limits.html#defn:underlying-cat","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.categoryForgetEnrichment","lean_decl_name":"CategoryTheory.categoryForgetEnrichment","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/ef921c9a5139539a505bc3291f1a076a1e250d92/Mathlib/CategoryTheory/Enriched/Basic.lean","lean4_code":"instance categoryForgetEnrichment : Category (ForgetEnrichment W C) :=\n enrichedCategoryTypeEquivCategory C (inferInstanceAs (EnrichedCategory (Type w)\n (TransportEnrichment (coyoneda.obj (op (𝟙_ W))) C)))","line":229,"end_line":231,"proof_status":"formalized"} |
| {"id":"prop:free-refl-quiver","label":"free-refl-quiver","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The functor that carries a category to its underlying reflexive quiver has a left adjoint, defining the free category on a reflexive quiver:","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.ReflQuiv.adj","latex_link":"sec-simplicial-sets.html#prop:free-refl-quiver","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.ReflQuiv.adj","lean_decl_name":"CategoryTheory.ReflQuiv.adj","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Topology/Category/TopCat/Adjunctions.lean","lean4_code":"def adj₁ : discrete ⊣ forget TopCat.{u} where\n unit := { app := fun _ => id }\n counit := { app := fun X => TopCat.ofHom (X := discrete.obj X) ⟨id, continuous_bot⟩ }\n\n/-- Equipping a type with the trivial topology is right adjoint to the forgetful functor\n`Top ⥤ Type`. -/\n","line":29,"end_line":34,"proof_status":"formalized"} |
| {"id":"cor:equiv-invar-fun","label":"equiv-invar-fun","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["thm:equiv-are-equiv"],"nlp_text":"Equivalences of \\(\\infty \\)-categories \\(A' \\rightsquigarrow A\\) and \\(B \\rightsquigarrow B'\\) induce an equivalence of functor spaces \\({\\mathord {\\mathsf{Fun}}}(A,B) \\rightsquigarrow {\\mathord {\\mathsf{Fun}}}(A',B')\\).","doc_url":null,"latex_link":"sec-htpy-2-cat.html#cor:equiv-invar-fun","lean_link":null,"lean_decl_name":"equiv-invar-fun","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":"cor:free-underlying-2-adj","label":"free-underlying-2-adj","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["prop:change-of-base-adjunction"],"nlp_text":"For any cartesian closed category \\({\\mathord {\\mathcal{V}}}\\) with coproducts, the underlying category construction and free category construction define adjoint 2-functors","doc_url":null,"latex_link":"sec-change-of-base.html#cor:free-underlying-2-adj","lean_link":null,"lean_decl_name":"free-underlying-2-adj","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:simplex-category","label":"simplex-category","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\Delta \\) denote the simplex category of finite nonempty ordinals \\([n] = \\{ 0 {\\lt}1 {\\lt}\\cdots {\\lt} n\\} \\) and order-preserving maps.","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SimplexCategory.smallCategory","latex_link":"sec-simplicial-sets.html#defn:simplex-category","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SimplexCategory.smallCategory","lean_decl_name":"SimplexCategory.smallCategory","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/CategoryTheory/Category/Preorder.lean","lean4_code":"instance (priority := 100) smallCategory (α : Type u) [Preorder α] : SmallCategory α where\n Hom U V := ULift (PLift (U ≤ V))\n id X := ⟨⟨le_refl X⟩⟩\n comp f g := ⟨⟨le_trans _ _ _ f.down.down g.down.down⟩⟩\n\n","line":44,"end_line":48,"proof_status":"formalized"} |
| {"id":"defn:sset-category","label":"sset-category","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:simplex-category"],"nlp_text":"The category of simplicial sets is the category \\({\\mathord {\\mathcal{sSet}}}:={\\mathord {\\mathcal{Set}}}^{\\Delta ^{\\mathord {\\textup{op}}}}\\) of presheaves on the simplex category.","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.largeCategory","latex_link":"sec-simplicial-sets.html#defn:sset-category","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.largeCategory","lean_decl_name":"SSet.largeCategory","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/ef921c9a5139539a505bc3291f1a076a1e250d92/Mathlib/AlgebraicTopology/SimplicialSet/Basic.lean","lean4_code":"instance largeCategory : LargeCategory SSet := by\n dsimp only [SSet]\n infer_instance","line":41,"end_line":43,"proof_status":"formalized"} |
| {"id":"defn:degeneracy-map","label":"degeneracy-map","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:simplex-category"],"nlp_text":"The elementary degeneracy operators are the maps","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SimplexCategory.σ","latex_link":"sec-simplicial-sets.html#defn:degeneracy-map","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SimplexCategory.σ","lean_decl_name":"SimplexCategory.σ","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Topology/Category/CompHausLike/SigmaComparison.lean","lean4_code":"instance isIsoSigmaComparison : IsIso <| sigmaComparison X σ := by\n rw [sigmaComparison_eq_comp_isos]\n infer_instance\n\n","line":62,"end_line":65,"proof_status":"formalized"} |
| {"id":"defn:simplicial-category","label":"simplicial-category","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:sset-category"],"nlp_text":"The data of a simplicial category is a simplicially enriched category with a set of objects and a simplicial set \\({\\mathord {\\mathcal{A}}}(x,y)\\) of morphisms between each ordered pair of objects. Each endo-hom space contains a distinguished 0-simplex \\(\\textup{id}_x \\in {\\mathord {\\mathcal{A}}}(x,y)_0\\), and composition is required to define a simplicial map","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.SimplicialCategory","latex_link":"sec-enriched-limits.html#defn:simplicial-category","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.SimplicialCategory","lean_decl_name":"CategoryTheory.SimplicialCategory","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinity-cosmos","source_path":"./InfinityCosmos/ForMathlib/AlgebraicTopology/SimplicialCategory/Basic.lean","lean4_code":"noncomputable instance : SimplicialCategory SSet where\n toEnrichedCategory := inferInstanceAs (EnrichedCategory (_ ⥤ Type _) (_ ⥤ Type _))\n homEquiv {K} {L} :=\n letI e : (K ⟶ L) ≃ (K ⊗ 𝟙_ SSet ⟶ L) :=\n ⟨fun f => (ρ_ _).hom ≫ f, fun f => (ρ_ _).inv ≫ f, by aesop_cat, by aesop_cat⟩\n e.trans (Functor.homObjEquiv _ _ _).symm |>.trans (Functor.functorHomEquiv K L (𝟙_ SSet)).symm\n homEquiv_id := by aesop_cat\n homEquiv_comp := by aesop_cat\n\nnoncomputable instance : MonoidalClosed SSet where\n closed A := {\n rightAdj := (sHomFunctor _).obj ⟨A⟩\n adj := FunctorToTypes.adj _\n }\n\n/-- Required apparently due to some refactoring. -/\n","line":75,"end_line":90,"proof_status":"formalized"} |
| {"id":"cor:sset-cat-limits","label":"sset-cat-limits","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:sset-category"],"nlp_text":"The category of simplicial sets is complete.","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.hasLimits","latex_link":"sec-simplicial-sets.html#cor:sset-cat-limits","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.hasLimits","lean_decl_name":"SSet.hasLimits","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Algebra/Category/ModuleCat/Limits.lean","lean4_code":"instance (priority := high) hasLimits' : HasLimits (ModuleCat.{u} R) :=\n ModuleCat.hasLimitsOfSize.{u, u, u}\n\n/-- An auxiliary declaration to speed up typechecking.\n-/\n","line":144,"end_line":148,"proof_status":"formalized"} |
| {"id":"defn:n-arrow","label":"n-arrow","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:simplicial-category"],"nlp_text":"For each \\(n \\geq 0\\), an \\(n\\)-simplex in \\({\\mathord {\\mathcal{A}}}(x,y)\\) is referred to as an \\(n\\)-arrow from \\(x\\) to \\(y\\).","doc_url":null,"latex_link":"sec-enriched-limits.html#defn:n-arrow","lean_link":null,"lean_decl_name":"n-arrow","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn:kan-complex","label":"kan-complex","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A Kan complex is a simplicial set admitting extensions as in 1.2.13 along all horn inclusions \\(n \\geq 1, 0 \\leq k \\leq n\\).","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.KanComplex","latex_link":"sec-simplicial-sets.html#defn:kan-complex","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.KanComplex","lean_decl_name":"SSet.KanComplex","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/AlgebraicTopology/SimplicialSet/KanComplex.lean","lean4_code":"lemma KanComplex.hornFilling {S : SSet.{u}} [KanComplex S]\n {n : ℕ} {i : Fin (n + 2)} (σ₀ : (Λ[n + 1, i] : SSet) ⟶ S) :\n ∃ σ : Δ[n + 1] ⟶ S, σ₀ = Λ[n + 1, i].ι ≫ σ := by\n have sq' : CommSq σ₀ Λ[n + 1, i].ι (terminal.from S) (terminal.from _) := ⟨by simp⟩\n exact ⟨sq'.lift, by simp⟩\n\n","line":40,"end_line":45,"proof_status":"formalized"} |
| {"id":"prop:kan-cosmos","label":"kan-cosmos","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:kan-complex","defn:trivial-fibration"],"nlp_text":"The category \\({\\mathord {\\mathcal{Kan}}}\\) of Kan complexes defines an \\(\\infty \\)-cosmos whose isofibrations are the Kan fibrations: maps that lift against all horn inclusions \\(\\Lambda ^k[n] \\hookrightarrow \\Delta [n]\\) for \\(n \\geq 1\\) and \\(0 \\leq k \\leq n\\).","doc_url":null,"latex_link":"sec-examples.html#prop:kan-cosmos","lean_link":null,"lean_decl_name":"kan-cosmos","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":"prop:cat-cosmos","label":"cat-cosmos","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:trivial-fibration","defn:cat-isofibration"],"nlp_text":"The category \\({\\mathord {\\mathcal{Cat}}}\\) of 1-categories defines an \\(\\infty \\)-cosmos whose isofibrations are the isofibrations. The equivalences are the equivalences of categories and the trivial fibrations are surjective equivalences: equivalences of categories that are also surjective on objects.","doc_url":null,"latex_link":"sec-examples.html#prop:cat-cosmos","lean_link":null,"lean_decl_name":"cat-cosmos","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:simplex-yoneda","label":"simplex-yoneda","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:standard-simplex"],"nlp_text":"Each \\(n\\)-simplex \\(x \\in X_n\\) corresponds to a map of simplicial sets \\(x \\colon \\Delta [n] \\to X\\). Accordingly, we write \\(x \\cdot \\delta ^i\\) for the \\(i\\)th face of the \\(n\\)-simplex, an \\((n-1)\\)-simplex classified by the composite map","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.yonedaEquiv","latex_link":"sec-simplicial-sets.html#lem:simplex-yoneda","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.yonedaEquiv","lean_decl_name":"SSet.yonedaEquiv","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/CategoryTheory/Yoneda.lean","lean4_code":"def yonedaEquiv {X : C} {F : Cᵒᵖ ⥤ Type v₁} : (yoneda.obj X ⟶ F) ≃ F.obj (op X) where\n toFun η := η.app (op X) (𝟙 X)\n invFun ξ := { app := fun _ f ↦ F.map f.op ξ }\n left_inv := by\n intro η\n ext Y f\n dsimp\n rw [← FunctorToTypes.naturality]\n simp\n right_inv := by intro ξ; simp\n\n","line":442,"end_line":452,"proof_status":"formalized"} |
| {"id":"lem:trivial-fib-conical","label":"trivial-fib-conical","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:trivial-fibration"],"nlp_text":"The trivial fibrations in an \\(\\infty \\)-cosmos define a subcategory containing the isomorphisms and are stable under product, pullback, and forming inverse limits of towers.","doc_url":null,"latex_link":"sec-cosmos.html#lem:trivial-fib-conical","lean_link":null,"lean_decl_name":"trivial-fib-conical","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":"prop:nerve-qcat","label":"nerve-qcat","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["prop:nerve-2-coskeletal","eq:qcat-defn"],"nlp_text":"Nerves of categories are quasi-categories.","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.Nerve.quasicategory","latex_link":"sec-simplicial-sets.html#prop:nerve-qcat","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.Nerve.quasicategory","lean_decl_name":"CategoryTheory.Nerve.quasicategory","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/AlgebraicTopology/Quasicategory/StrictSegal.lean","lean4_code":"theorem quasicategory {X : SSet.{u}} (sx : StrictSegal X) : Quasicategory X := by\n apply quasicategory_of_filler X\n intro n i σ₀ h₀ hₙ\n use sx.spineToSimplex <| Path.map (horn.spineId i h₀ hₙ) σ₀\n intro j hj\n apply sx.spineInjective\n ext k\n dsimp only [spineEquiv, spine_arrow, Function.comp_apply, Equiv.coe_fn_mk]\n rw [← types_comp_apply (σ₀.app _) (X.map _), ← σ₀.naturality]\n let ksucc := k.succ.castSucc\n obtain hlt | hgt | heq : ksucc < j ∨ j < ksucc ∨ j = ksucc := by omega\n · rw [← spine_arrow, spine_δ_arrow_lt sx _ hlt]\n dsimp only [Path.map_arrow, spine_arrow, Fin.coe_eq_castSucc]\n apply congr_arg\n apply Subtype.ext\n dsimp [horn.face, CosimplicialObject.δ]\n rw [Subcomplex.yonedaEquiv_coe, Subpresheaf.lift_ι, stdSimplex.map_apply,\n Quiver.Hom.unop_op, stdSimplex.yonedaEquiv_map, Equiv.apply_symm_apply,\n mkOfSucc_δ_lt hlt]\n rfl\n · rw [← spine_arrow, spine_δ_arrow_gt sx _ hgt]\n dsimp only [Path.map_arrow, spine_arrow, Fin.coe_eq_castSucc]\n apply congr_arg\n apply Subtype.ext\n dsimp [horn.face, CosimplicialObject.δ]\n rw [Subcomplex.yonedaEquiv_coe, Subpresheaf.lift_ι, stdSimplex.map_apply,\n Quiver.Hom.unop_op, stdSimplex.yonedaEquiv_map, Equiv.apply_symm_apply,\n mkOfSucc_δ_gt hgt]\n rfl\n · obtain _ | n := n\n · /- The only inner horn of `Δ[2]` does not contain the diagonal edge. -/\n obtain rfl : k = 0 := by omega\n fin_cases i <;> contradiction\n · /- We construct the triangle in the standard simplex as a 2-simplex in\n the horn. While the triangle is not contained in the inner horn `Λ[2, 1]`,\n it suffices to inhabit `Λ[n + 3, i] _⦋2⦌`. -/\n let triangle : (Λ[n + 3, i] : SSet.{u}) _⦋2⦌ :=\n horn.primitiveTriangle i h₀ hₙ k (by omega)\n /- The interval spanning from `k` to `k + 2` is equivalently the spine\n of the triangle with vertices `k`, `k + 1`, and `k + 2`. -/\n have hi : ((horn.spineId i h₀ hₙ).map σ₀).interval k 2 (by omega) =\n X.spine 2 (σ₀.app _ triangle) := by\n ext m\n dsimp [spine_arrow, Path.map_interval, Path.map_arrow]\n rw [← types_comp_apply (σ₀.app _) (X.map _), ← σ₀.naturality]\n apply congr_arg\n apply Subtype.ext\n ext a : 1\n fin_cases a <;> fin_cases m <;> rfl\n rw [← spine_arrow, spine_δ_arrow_eq sx _ heq, hi]\n","line":28,"end_line":77,"proof_status":"formalized"} |
| {"id":"defn:lax-monoidal-functor","label":"lax-monoidal-functor","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A (lax) monoidal functor between cartesian closed categories \\({\\mathord {\\mathcal{V}}}\\) and \\({\\mathord {\\mathcal{W}}}\\) is a functor \\(T \\colon {\\mathord {\\mathcal{V}}}\\to {\\mathord {\\mathcal{W}}}\\) equipped with natural transformations","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.LaxMonoidalFunctor","latex_link":"sec-change-of-base.html#defn:lax-monoidal-functor","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.LaxMonoidalFunctor","lean_decl_name":"CategoryTheory.LaxMonoidalFunctor","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/CategoryTheory/Monoidal/Functor.lean","lean4_code":"structure LaxMonoidalFunctor extends C ⥤ D where\n laxMonoidal : toFunctor.LaxMonoidal := by infer_instance\n\nnamespace LaxMonoidalFunctor\n\nattribute [instance] laxMonoidal\n\nvariable {C D}\n\n/-- Constructor for `LaxMonoidalFunctor C D`. -/\n","line":1152,"end_line":1161,"proof_status":"formalized"} |
| {"id":"prop:change-of-base","label":"change-of-base","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:lax-monoidal-functor"],"nlp_text":"A finite-product-preserving functor \\(T \\colon {\\mathord {\\mathcal{V}}}\\to {\\mathord {\\mathcal{W}}}\\) between cartesian closed categories induces a change-of-base 2-functor","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.instEnrichedCategoryTransportEnrichment","latex_link":"sec-change-of-base.html#prop:change-of-base","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.instEnrichedCategoryTransportEnrichment","lean_decl_name":"CategoryTheory.instEnrichedCategoryTransportEnrichment","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/ef921c9a5139539a505bc3291f1a076a1e250d92/Mathlib/CategoryTheory/Enriched/Basic.lean","lean4_code":"instance : EnrichedCategory W (TransportEnrichment F C) where\n Hom := fun X Y : C => F.obj (X ⟶[V] Y)\n id := fun X : C => ε F ≫ F.map (eId V X)\n comp := fun X Y Z : C => μ F _ _ ≫ F.map (eComp V X Y Z)\n id_comp X Y := by\n simp only [comp_whiskerRight, Category.assoc, Functor.LaxMonoidal.μ_natural_left_assoc,\n Functor.LaxMonoidal.left_unitality_inv_assoc]\n simp_rw [← F.map_comp]\n convert F.map_id _\n simp\n comp_id X Y := by\n simp only [MonoidalCategory.whiskerLeft_comp, Category.assoc,\n Functor.LaxMonoidal.μ_natural_right_assoc,\n Functor.LaxMonoidal.right_unitality_inv_assoc]\n simp_rw [← F.map_comp]\n convert F.map_id _\n simp\n assoc P Q R S := by\n rw [comp_whiskerRight, Category.assoc, μ_natural_left_assoc,\n ← associativity_inv_assoc, ← F.map_comp, ← F.map_comp, e_assoc,\n F.map_comp, MonoidalCategory.whiskerLeft_comp, Category.assoc,\n Functor.LaxMonoidal.μ_natural_right_assoc]","line":122,"end_line":143,"proof_status":"statement_formalized"} |
| {"id":"prop:change-of-base-adjunction","label":"change-of-base-adjunction","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["prop:change-of-base"],"nlp_text":"Any adjunction between cartesian closed categories whose left adjoint preserves finite products induces a change-of-base 2-adjunction","doc_url":null,"latex_link":"sec-change-of-base.html#prop:change-of-base-adjunction","lean_link":null,"lean_decl_name":"change-of-base-adjunction","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:right-of-monoidal-adj-ucats","label":"right-of-monoidal-adj-ucats","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["prop:change-of-base-adjunction"],"nlp_text":"Consider a finite-product-preserving adjunction between cartesian closed categories:","doc_url":null,"latex_link":"sec-change-of-base.html#lem:right-of-monoidal-adj-ucats","lean_link":null,"lean_decl_name":"right-of-monoidal-adj-ucats","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:enriched-cob-adjunction","label":"enriched-cob-adjunction","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["prop:change-of-base-adjunction"],"nlp_text":"Any adjunction comprised of finite-product-preserving functors between cartesian closed categories","doc_url":null,"latex_link":"sec-change-of-base.html#lem:enriched-cob-adjunction","lean_link":null,"lean_decl_name":"enriched-cob-adjunction","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:cat-of-n-arrows","label":"cat-of-n-arrows","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:n-arrow"],"nlp_text":"For any simplicial category \\({\\mathord {\\mathcal{A}}}\\) and \\(n \\geq 0\\), the \\(n\\)-arrows assemble into the arrows of an ordinary category \\({\\mathord {\\mathcal{A}}}_n\\) with the same set of objects as \\({\\mathord {\\mathcal{A}}}\\).","doc_url":null,"latex_link":"sec-enriched-limits.html#lem:cat-of-n-arrows","lean_link":null,"lean_decl_name":"cat-of-n-arrows","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:nerve-2-coskeletal","label":"nerve-2-coskeletal","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:nerve"],"nlp_text":"The nerve of a category \\(C\\) is 2-coskeletal as a simplicial set, meaning that every sphere \\(\\partial \\Delta [n] \\to C\\) with \\(n \\geq 3\\) is filled uniquely by an \\(n\\)-simplex in \\(C\\), or equivalently that the nerve is canonically isomorphic to the right Kan extension of its restriction to 2-truncated simplicial sets. 2","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.Nerve.cosk₂Iso","latex_link":"sec-simplicial-sets.html#prop:nerve-2-coskeletal","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.Nerve.cosk₂Iso","lean_decl_name":"CategoryTheory.Nerve.cosk₂Iso","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/AlgebraicTopology/SimplicialSet/Coskeletal.lean","lean4_code":"noncomputable def cosk₂Iso : nerveFunctor.{v, u} ≅ nerveFunctor₂.{v, u} ⋙ Truncated.cosk 2 :=\n NatIso.ofComponents (fun C ↦ (nerve C).isoCoskOfIsCoskeletal 2)\n (fun _ ↦ (coskAdj 2).unit.naturality _)\n\nend Nerve\n\n","line":249,"end_line":254,"proof_status":"formalized"} |
| {"id":"prop:nerve-fully-faithful","label":"nerve-fully-faithful","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:nerve"],"nlp_text":"The nerve functor is fully faithful.","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.nerveFunctor.fullyfaithful","latex_link":"sec-simplicial-sets.html#prop:nerve-fully-faithful","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.nerveFunctor.fullyfaithful","lean_decl_name":"CategoryTheory.nerveFunctor.fullyfaithful","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/AlgebraicTopology/SimplicialSet/NerveAdjunction.lean","lean4_code":"noncomputable def nerveFunctor₂.fullyfaithful : nerveFunctor₂.FullyFaithful :=\n FullyFaithful.ofFullyFaithful nerveFunctor₂\n\n","line":476,"end_line":478,"proof_status":"formalized"} |
| {"id":"defn:1-simplex-htpy","label":"1-simplex-htpy","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A parallel pair of 1-simplices \\(f,g\\) in a simplicial set \\(X\\) are homotopic if there exists a 2-simplex whose boundary takes either of the following forms 3","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Truncated.HomotopicL","latex_link":"sec-simplicial-sets.html#defn:1-simplex-htpy","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Truncated.HomotopicL","lean_decl_name":"SSet.Truncated.HomotopicL","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinity-cosmos","source_path":"InfinityCosmos/ForMathlib/AlgebraicTopology/SimplicialSet/Homotopy.lean","lean4_code":"def HomotopicL : Prop :=\n Nonempty (HomotopyL f g)\n","line":131,"end_line":132,"proof_status":"formalized"} |
| {"id":"defn:cat-isofibration","label":"cat-isofibration","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"An isofibration between categories is a functor \\(f \\colon A \\twoheadrightarrow B\\) satisfying the displayed right lifting property for the inclusion of both endpoints of the free-living isomorphism:","doc_url":null,"latex_link":"sec-examples.html#defn:cat-isofibration","lean_link":null,"lean_decl_name":"cat-isofibration","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:pro-normal-monoidal","label":"pro-normal-monoidal","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The change-of-base 2-functor induced by a finite-product-preserving functor \\(T \\colon {\\mathord {\\mathcal{V}}}\\to {\\mathord {\\mathcal{W}}}\\) between cartesian closed categories preserves underlying categories, if and only if, for each \\(v \\in {\\mathord {\\mathcal{V}}}\\) the composite function on hom-sets","doc_url":null,"latex_link":"sec-change-of-base.html#lem:pro-normal-monoidal","lean_link":null,"lean_decl_name":"pro-normal-monoidal","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:standard-simplex","label":"standard-simplex","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We write \\(\\Delta [n]\\) for the standard \\(n\\)-simplex the simplicial set represented by \\([n] \\in \\Delta \\).","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.stdSimplex","latex_link":"sec-simplicial-sets.html#defn:standard-simplex","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.stdSimplex","lean_decl_name":"SSet.stdSimplex","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Analysis/Convex/Basic.lean","lean4_code":"def stdSimplex : Set (ι → 𝕜) :=\n { f | (∀ x, 0 ≤ f x) ∧ ∑ x, f x = 1 }\n\n","line":599,"end_line":601,"proof_status":"formalized"} |
| {"id":"defn:simplicial-horn","label":"simplicial-horn","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:standard-simplex"],"nlp_text":"We write \\(\\Lambda ^k[n] \\subset \\Delta [n]\\) for the \\(k\\)-horn in the \\(n\\)-simplex. The horn \\(\\Lambda ^k[n]\\) is the further simplicial subset of \\(\\partial \\Delta [n]\\) that omits the face opposite the vertex \\(k\\), but it is defined as a subset of \\(\\Delta [n]\\).","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.horn","latex_link":"sec-simplicial-sets.html#defn:simplicial-horn","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.horn","lean_decl_name":"SSet.horn","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/AlgebraicTopology/SimplicialSet/Horn.lean","lean4_code":"def horn (n : ℕ) (i : Fin (n + 1)) : (Δ[n] : SSet.{u}).Subcomplex where\n obj _ := setOf (fun s ↦ Set.range (stdSimplex.asOrderHom s) ∪ {i} ≠ Set.univ)\n map φ s hs h := hs (by\n rw [Set.eq_univ_iff_forall] at h ⊢; intro j\n apply Or.imp _ id (h j)\n intro hj\n exact Set.range_comp_subset_range _ _ hj)\n\n/-- The `i`-th horn `Λ[n, i]` of the standard `n`-simplex -/\nscoped[Simplicial] notation3 \"Λ[\" n \", \" i \"]\" => SSet.horn (n : ℕ) i\n\n","line":27,"end_line":37,"proof_status":"formalized"} |
| {"id":"eq:qcat-defn","label":"qcat-defn","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A quasi-category is a simplicial set \\(A\\) in which any inner horn can be extended to a simplex, solving the displayed lifting problem:","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Quasicategory","latex_link":"sec-simplicial-sets.html#eq:qcat-defn","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/SSet.Quasicategory","lean_decl_name":"SSet.Quasicategory","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinity-cosmos","source_path":"./InfinityCosmos/ForMathlib/AlgebraicTopology/Quasicategory/TwoTruncated.lean","lean4_code":"class Quasicategory₂ (X : Truncated 2) where\n fill21 {x₀ x₁ x₂ : X _⦋0⦌₂}\n (e₀₁ : Edge x₀ x₁) (e₁₂ : Edge x₁ x₂) :\n Nonempty (Σ e₀₂ : Edge x₀ x₂, CompStruct e₀₁ e₁₂ e₀₂)\n fill31 {x₀ x₁ x₂ x₃ : X _⦋0⦌₂}\n {e₀₁ : Edge x₀ x₁} {e₁₂ : Edge x₁ x₂} {e₂₃ : Edge x₂ x₃}\n {e₀₂ : Edge x₀ x₂} {e₁₃ : Edge x₁ x₃} {e₀₃ : Edge x₀ x₃}\n (f₃ : CompStruct e₀₁ e₁₂ e₀₂)\n (f₀ : CompStruct e₁₂ e₂₃ e₁₃)\n (f₂ : CompStruct e₀₁ e₁₃ e₀₃) :\n Nonempty (CompStruct e₀₂ e₂₃ e₀₃)\n fill32 {x₀ x₁ x₂ x₃ : X _⦋0⦌₂}\n {e₀₁ : Edge x₀ x₁} {e₁₂ : Edge x₁ x₂} {e₂₃ : Edge x₂ x₃}\n {e₀₂ : Edge x₀ x₂} {e₁₃ : Edge x₁ x₃} {e₀₃ : Edge x₀ x₃}\n (f₃ : CompStruct e₀₁ e₁₂ e₀₂)\n (f₀ : CompStruct e₁₂ e₂₃ e₁₃)\n (f₁ : CompStruct e₀₂ e₂₃ e₀₃) :\n Nonempty (CompStruct e₀₁ e₁₃ e₀₃)\n\nend Truncated\n\nnamespace horn₂₁\nopen Truncated (Edge edgeMap CompStruct truncEquiv trunc_map trunc_map')\n\nvariable {S : SSet} {x₀ x₁ x₂ : ((truncation 2).obj S) _⦋0⦌₂}\n (e₀₁ : Edge x₀ x₁) (e₁₂ : Edge x₁ x₂)\n\n","line":59,"end_line":85,"proof_status":"formalized"} |
| {"id":"defn:nerve-functor","label":"nerve-functor","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":["defn:nerve"],"nlp_text":"The map \\([n] \\mapsto [n]\\) defines a fully faithful embedding \\(\\Delta \\hookrightarrow {\\mathord {\\mathcal{Cat}}}\\). From this point of view, the nerve functor can be described as a “restricted Yoneda embedding” which carries a category \\(C\\) to the restriction of the representable functor \\(\\hom (-,C)\\) to the image of this inclusion.","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.nerveFunctor","latex_link":"sec-simplicial-sets.html#defn:nerve-functor","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.nerveFunctor","lean_decl_name":"CategoryTheory.nerveFunctor","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/AlgebraicTopology/SimplicialSet/NerveAdjunction.lean","lean4_code":"instance nerveFunctor₂.faithful : nerveFunctor₂.{u, u}.Faithful :=\n Functor.Faithful.of_comp_iso\n (G := oneTruncation₂) (H := ReflQuiv.forget) OneTruncation₂.ofNerve₂.natIso\n\n","line":408,"end_line":411,"proof_status":"formalized"} |
| {"id":"defn:2-cat-equivalence","label":"2-cat-equivalence","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"An equivalence in a 2-category is given by a pair of objects \\(A\\) and \\(B\\); a pair of 1-cells \\(f \\colon A \\to B\\) and \\(g \\colon B \\to A\\); and a pair of invertible 2-cells","doc_url":null,"latex_link":"sec-htpy-2-cat.html#defn:2-cat-equivalence","lean_link":null,"lean_decl_name":"2-cat-equivalence","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"prop:2-cat-as-cat-enriched","label":"2-cat-as-cat-enriched","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"There is an equivalence between categories enriched in categories and strict bicategories. In particular, each can be converted into the other.","doc_url":null,"latex_link":"sec-htpy-2-cat.html#prop:2-cat-as-cat-enriched","lean_link":null,"lean_decl_name":"2-cat-as-cat-enriched","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:nerve","label":"nerve","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The category \\({\\mathord {\\mathcal{Cat}}}\\) of 1-categories embeds fully faithfully into the category of simplicial sets via the nerve functor. An \\(n\\)-simplex in the nerve of a 1-category \\(C\\) is a sequence of \\(n\\) composable arrows in \\(C\\), or equally a functor \\([n]\\to C\\) from the ordinal category \\([n]\\) with objects \\(0,\\ldots , n\\) and a unique arrow \\(i \\to j\\) just when \\(i \\leq j\\).","doc_url":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.nerve","latex_link":"sec-simplicial-sets.html#defn:nerve","lean_link":"https://emilyriehl.github.io/infinity-cosmos/docs/find/#doc/CategoryTheory.nerve","lean_decl_name":"CategoryTheory.nerve","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/AlgebraicTopology/SimplicialSet/NerveAdjunction.lean","lean4_code":"def nerve₂Adj.counit.app (C : Type u) [SmallCategory C] :\n (nerveFunctor₂.obj (Cat.of C)).HomotopyCategory ⥤ C := by\n fapply Quotient.lift\n · exact\n (whiskerRight (OneTruncation₂.ofNerve₂.natIso).hom _ ≫ ReflQuiv.adj.{u}.counit).app (Cat.of C)\n · intro x y f g rel\n obtain ⟨φ⟩ := rel\n simpa [ReflQuiv.adj, Quot.liftOn, Cat.FreeRefl.quotientFunctor, Quotient.functor,\n pathComposition, Quiv.adj, OneTruncation₂.nerveHomEquiv] using\n φ.map_comp (X := 0) (Y := 1) (Z := 2) (homOfLE (by decide)) (homOfLE (by decide))\n\n","line":42,"end_line":52,"proof_status":"formalized"} |
| {"id":"defn:isomorphism","label":"isomorphism","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://emilyriehl.github.io/infinity-cosmos/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A 1-simplex in a quasi-category is an isomorphism 1 just when it represents an isomorphism in the homotopy category. By Lemma 1.3.8 this means that \\(f \\colon a \\to b\\) is an isomorphism if and only if there exists a 1-simplex \\(f^{-1} \\colon b \\to a\\) together with a pair of 2-simplices","doc_url":null,"latex_link":"sec-homotopy-category.html#defn:isomorphism","lean_link":null,"lean_decl_name":"isomorphism","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/CategoryTheory/Limits/Shapes/Pullback/Categorical/Basic.lean","lean4_code":"/-- The structure isomorphism of a `CatCommSqOver` as a natural transformation. -/\n","line":231,"end_line":231,"proof_status":"statement_formalized"} |
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