| {"id":"def:pure_homotopy_k_sheaf","label":"pure_homotopy_k_sheaf","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A homotopy \\(\\mathcal{K}\\)-sheaf \\(\\mathcal{F}^{\\bullet }\\) is said to be pure on \\(X\\) if: For \\(p\\in X\\) and \\(i\\neq 0\\), \\((H^i\\mathcal{F}^{\\bullet })_p=0\\) \\(H^i\\mathcal{F}^{\\bullet }=0\\) for \\(i{\\lt}{\\lt}0\\) locally on \\(X\\): ie for all \\(p\\in X\\) there is an open neighbourhoud \\(U\\) of \\(p\\) and an integer \\(N\\) such that for \\(i\\leq N\\) and \\(K\\subset U\\): \\(H^i\\mathcal{F}^{\\bullet }(K)=0\\)","doc_url":null,"latex_link":"sect0006.html#def:pure_homotopy_k_sheaf","lean_link":null,"lean_decl_name":"pure_homotopy_k_sheaf","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:cech_of_pure_homotopy_k_sheaf","label":"cech_of_pure_homotopy_k_sheaf","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:pure_homotopy_k_sheaf","lem:first_non_zero_homology_of_homotopy_k_sheaf_is_k_sheaf","lem:csq_of_pure_homotopy_k_sheaf","prop:chech_is_determined_by_stalks_3","lem:homotpy_k_sheaves_stable_by_extension","prop:homotopy_k_sheaf_compute_chech"],"nlp_text":"Let \\(\\mathcal{F}^{\\bullet }\\) be a pure-homotopy \\(\\mathcal{K}\\)-sheaf. Then there is a canonical isomorphism: \\[ H^{\\bullet }\\mathcal{F}^{\\bullet }(X)=\\check{H}^{\\bullet }(X;H^0\\mathcal{F}^{\\bullet }) \\]","doc_url":null,"latex_link":"sect0006.html#prop:cech_of_pure_homotopy_k_sheaf","lean_link":null,"lean_decl_name":"cech_of_pure_homotopy_k_sheaf","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:gluing_pure_homotopy_k_sheaves","label":"gluing_pure_homotopy_k_sheaves","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:pure_homotopy_k_sheaf","lem:hocolim_preserves_homotopy_k_sheaves"],"nlp_text":"","doc_url":null,"latex_link":"sect0008.html#lem:gluing_pure_homotopy_k_sheaves","lean_link":null,"lean_decl_name":"gluing_pure_homotopy_k_sheaves","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:poincare-lefschetz_duality","label":"poincare-lefschetz_duality","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["prop:cech_of_pure_homotopy_k_sheaf","def:pushforward_of_K-sheaf","lem:chech_comm_with_exceptional","def:orientation_sheaf_of_a_manifold","lem:Homotopy_k-sheaf_axioms_for_singular_chain","def:chech_of_compact_sup_of_sh"],"nlp_text":"Let \\(M\\) be a topological manifold of dimension \\(n\\) with boundary, \\(i:X\\to M\\) a closed subset, and \\(N\\subset \\partial M\\) a closed subset that locally looks like \\(\\varnothing \\subset \\mathbb {R}^{n-1}, \\mathbb {R}_{{\\gt}0}\\times R^{n-2}\\subset \\mathbb {R}^{n-1}\\) or \\(\\mathbb {R}^{n-1}\\). Let \\(j:\\overset {\\circ }{M}\\cup \\overset {\\circ }{N}\\to M\\) be the canonical inclusion. Then there is a canonical isomorphism: \\[ H^{\\bullet }[C_{n-1-\\bullet }(N,N\\backslash X)\\to C_{n-1-\\bullet }(M,M\\backslash X)]=\\check{H}_c^{\\bullet }(X;i^*j_!j^*\\mathfrak {o}_M) \\]","doc_url":null,"latex_link":"sect0007.html#lem:poincare-lefschetz_duality","lean_link":null,"lean_decl_name":"poincare-lefschetz_duality","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:first_non_zero_homology_of_homotopy_k_sheaf_is_k_sheaf","label":"first_non_zero_homology_of_homotopy_k_sheaf_is_k_sheaf","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["lem:Mayer_Vietoris","def:homotopy_k_sheaf","def:K_sheaves"],"nlp_text":"If \\(\\mathcal{F}^{\\bullet }\\) is a homotopy-\\(\\mathcal{K}\\)-sheaf, and \\(H^{-1}\\mathcal{F}^{\\bullet }=0\\) then \\(H^{0}\\mathcal{F}^{\\bullet }\\) is a \\(\\mathcal{K}\\)-sheaf","doc_url":null,"latex_link":"sect0003.html#lem:first_non_zero_homology_of_homotopy_k_sheaf_is_k_sheaf","lean_link":null,"lean_decl_name":"first_non_zero_homology_of_homotopy_k_sheaf_is_k_sheaf","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:kshv_equiv_shv","label":"kshv_equiv_shv","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["lem:adj_kshv_and_shv","def:K_stalk","lem:equiv_of_adj","def:stalk"],"nlp_text":"The previous adjoint pair give rise to an equivalence of category between shaeves and \\(\\mathcal{K}\\)-sheaves","doc_url":null,"latex_link":"sect0002.html#lem:kshv_equiv_shv","lean_link":null,"lean_decl_name":"kshv_equiv_shv","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:chech_preserve_quasi_iso","label":"chech_preserve_quasi_iso","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["lem:q-iso_iff_cone_acyclic","def:quasi_iso_of_complex_of_k-prsh","lem:chech_of_acyclic_is_zero","lem:chech_long_exact_sequence"],"nlp_text":"If \\(\\mathcal{F}^{\\bullet }\\to \\mathcal{G}^{\\bullet }\\) is a quasi-isomorphism then the induced maps \\(\\check{H}^i\\mathcal{F}^{\\bullet }\\to \\check{H}^i\\mathcal{G}^{\\bullet }\\) are isomorphims.","doc_url":null,"latex_link":"sect0005.html#lem:chech_preserve_quasi_iso","lean_link":null,"lean_decl_name":"chech_preserve_quasi_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:chech_is_determined_by_stalks_2","label":"chech_is_determined_by_stalks_2","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["lem:chech_preserve_quasi_iso","def:homotopy_k_sheaf","lem:chech_is_determined_by_stalks_1"],"nlp_text":"Let \\(\\mathcal{F}^{\\bullet }\\) be a complex of \\(\\mathcal{K}\\)-presheaves that verify 3 and \\(H^i\\mathcal{F}^{\\bullet }=0\\) for \\(i{\\lt}{\\lt}0\\). Then if the stalks of \\(\\mathcal{F}^{\\bullet }\\) are acyclics, \\(\\check{H}^{\\bullet }(X;\\mathcal{F}^{\\bullet })=0\\)","doc_url":null,"latex_link":"sect0005.html#lem:chech_is_determined_by_stalks_2","lean_link":null,"lean_decl_name":"chech_is_determined_by_stalks_2","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:csq_of_pure_homotopy_k_sheaf","label":"csq_of_pure_homotopy_k_sheaf","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["lem:chech_preserve_quasi_iso"],"nlp_text":"Let \\(\\mathcal{F}^{\\bullet }\\) be a pure-homotopy \\(\\mathcal{K}\\)-sheaf. Then: For \\(i{\\lt}0\\) \\(H^i\\mathcal{F}^{\\bullet }=0\\) \\(H^0\\mathcal{F}^{\\bullet }\\) is a \\(\\mathcal{K}\\)-sheaf.","doc_url":null,"latex_link":"sect0006.html#lem:csq_of_pure_homotopy_k_sheaf","lean_link":null,"lean_decl_name":"csq_of_pure_homotopy_k_sheaf","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:chech_is_determined_by_stalks_3","label":"chech_is_determined_by_stalks_3","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["lem:chech_is_determined_by_stalks_2"],"nlp_text":"Let \\(\\mathcal{F}^{\\bullet }\\) and \\(\\mathcal{G}^{\\bullet }\\) be complexes of \\(\\mathcal{K}\\)-presheaves that verify 3 and \\(H^i\\mathcal{F}^{\\bullet }=H^i\\mathcal{G}^{\\bullet }=0\\) for \\(i\\) small enough. Then if a morphism \\(\\mathcal{F}^{\\bullet }\\to \\mathcal{G}^{\\bullet }\\) induces a quasi-isomorphism on stalks, \\(\\check{H}^{\\bullet }(X;\\mathcal{F}^{\\bullet })=\\check{H}^{\\bullet }(X;\\mathcal{G}^{\\bullet })\\)","doc_url":null,"latex_link":"sect0005.html#prop:chech_is_determined_by_stalks_3","lean_link":null,"lean_decl_name":"chech_is_determined_by_stalks_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:q-iso_iff_cone_acyclic","label":"q-iso_iff_cone_acyclic","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A morphism of complex \\(f:A^{\\bullet }\\to B^{\\bullet }\\) is a quasi isomorphism if and only if, its cone is acyclic.","doc_url":null,"latex_link":"sect0001.html#lem:q-iso_iff_cone_acyclic","lean_link":null,"lean_decl_name":"q-iso_iff_cone_acyclic","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:complex_total_of_three_is_acyclic","label":"complex_total_of_three_is_acyclic","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["lem:q-iso_iff_cone_acyclic"],"nlp_text":"If a complex \\([A^{\\bullet }\\to B^{\\bullet -1}\\to C^{\\bullet -2}]\\) is acyclic then there is a long exact sequence \\[ \\ldots \\to H^k A\\to H^k B\\to H^k C\\to \\ldots \\]","doc_url":null,"latex_link":"sect0001.html#lem:complex_total_of_three_is_acyclic","lean_link":null,"lean_decl_name":"complex_total_of_three_is_acyclic","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:Mayer_Vietoris","label":"Mayer_Vietoris","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["lem:complex_total_of_three_is_acyclic"],"nlp_text":"By using 1.4, 3 give rise to a \"Mayer-Vietoris\" long exact sequence: \\[ \\ldots \\to H^k \\mathcal{F}^{\\bullet }(K_1\\cup K_2)\\to H^k \\mathcal{F}^{\\bullet }(K_1)\\bigoplus H^k\\mathcal{F}^{\\bullet }(K_2)\\to H^k \\mathcal{F}^{\\bullet }(K_1\\cap K_2)\\to \\ldots \\]","doc_url":null,"latex_link":"sect0003.html#lem:Mayer_Vietoris","lean_link":null,"lean_decl_name":"Mayer_Vietoris","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":"def:pushforward_of_K-sheaf","label":"pushforward_of_K-sheaf","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:pushforward_of_sheaf","def:K_sheaves","lem:basis_of_open_of_fm1K"],"nlp_text":"If \\(\\mathcal{F}\\) is a pre-\\(\\mathcal{K}\\)-sheaf over \\(X\\), then the rule \\(K\\mapsto \\mathcal{F}(f^{-1}(K))\\) defines a pre-\\(\\mathcal{K}\\)-sheaf over \\(Y\\). The functor obtained is denoted \\(f_*\\) and named the pushforward by \\(f\\). \\(f_*\\) send \\(\\mathcal{K}\\)-sheaves over \\(X\\) into \\(\\mathcal{K}\\)-sheaves over \\(Y\\).","doc_url":null,"latex_link":"sect0004.html#def:pushforward_of_K-sheaf","lean_link":null,"lean_decl_name":"pushforward_of_K-sheaf","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":"def:total_complex","label":"total_complex","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(A_0^{\\bullet }\\to \\ldots \\to A_n^{\\bullet }\\) is a sequence of maps (with \\(f_i:A_i\\to A_{i+1}\\) )of complex such that the composition of two consecutive maps is \\(0\\), then let’s denote \\([A_0^{\\bullet }\\to \\ldots \\to A_n^{\\bullet -n}]\\) the total complex of this double complex, defined by the following data: The object in degré \\(k\\) is \\(\\bigoplus \\limits _{i=0}^n A_i[-i]^k\\) The diferential is given by the matrix \\(\\begin{pmatrix} d_{A_0} & 0 & \\ldots & \\ldots & 0 \\\\ f_0 & -d_{A_1} & \\ldots & \\ldots & 0 \\\\ 0 & f_1 & \\ldots & \\ldots & 0 \\\\ \\vdots & \\vdots & \\ddots & & \\vdots \\\\ \\vdots & \\vdots & & (-1)^{n-1}d_{A_{n-1}} & 0 \\\\ 0 & 0 & \\ldots & f_{n-1} & (-1)^nd_{A_n} \\\\ \\end{pmatrix}\\)","doc_url":null,"latex_link":"sect0001.html#def:total_complex","lean_link":null,"lean_decl_name":"total_complex","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":"def:homotopy_k_sheaf","label":"homotopy_k_sheaf","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:total_complex","def:K_pre_sheaves"],"nlp_text":"A complex of \\(\\mathcal{K}\\)-presheaves \\(\\mathcal{F}^{\\bullet }\\) is said to be a Homotopy-\\(\\mathcal{K}\\)-sheave if the folowing conditions are satisfied: \\begin{equation} \\label{axiom:hKsh1} \\mathcal{F}^{\\bullet }(\\varnothing ) \\text{ is an acyclic complex} \\end{equation}","doc_url":null,"latex_link":"sect0003.html#def:homotopy_k_sheaf","lean_link":null,"lean_decl_name":"homotopy_k_sheaf","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:homotpy_k_sheaves_stable_by_extension","label":"homotpy_k_sheaves_stable_by_extension","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:homotopy_k_sheaf"],"nlp_text":"Let \\(\\mathcal{F}^{\\bullet }\\) be a complex of \\(\\mathcal{K}\\)-presheaves. If \\(\\mathcal{F}^{\\bullet }\\) has a finite filtration whose associated graded is a Homotopy-\\(\\mathcal{K}\\)-sheaf, then \\(\\mathcal{F}^{\\bullet }\\) is a Homotopy-\\(\\mathcal{K}\\)-sheaf.","doc_url":null,"latex_link":"sect0003.html#lem:homotpy_k_sheaves_stable_by_extension","lean_link":null,"lean_decl_name":"homotpy_k_sheaves_stable_by_extension","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:homotopy_k_sheaf_compute_chech","label":"homotopy_k_sheaf_compute_chech","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:homotopy_k_sheaf","def:chech_of_compelx_of_k_prsh"],"nlp_text":"Let \\(\\mathcal{F}^{\\bullet }\\) be a complex of \\(\\mathcal{K}\\)-presheaves that verify 1 and 2 then the canonical map \\(H^{\\bullet }\\mathcal{F}^{\\bullet }\\to \\check{H}^{\\bullet }(X;\\mathcal{F}^{\\bullet })\\) is an isomorphism.","doc_url":null,"latex_link":"sect0005.html#prop:homotopy_k_sheaf_compute_chech","lean_link":null,"lean_decl_name":"homotopy_k_sheaf_compute_chech","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:hocolim_preserves_homotopy_k_sheaves","label":"hocolim_preserves_homotopy_k_sheaves","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["lem:homotpy_k_sheaves_stable_by_extension","def:homotopy_colimit"],"nlp_text":"","doc_url":null,"latex_link":"sect0008.html#lem:hocolim_preserves_homotopy_k_sheaves","lean_link":null,"lean_decl_name":"hocolim_preserves_homotopy_k_sheaves","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":"def:sheaves","label":"sheaves","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:pre_sheaves"],"nlp_text":"If \\(\\mathcal{F}\\) is a presheaf on \\(X\\), it is said to be a sheaf if for any \\(U\\subset X\\) open and any covering family of \\(U\\) \\((U_a)_{a\\in A}\\) one has the exact sequence: \\begin{equation} \\label{axiom:Sh} 0\\to \\mathcal{F}(U)\\to \\prod \\limits _{a\\in A}\\mathcal{F}(U_a)\\to \\prod \\limits _{a,b\\in A}\\mathcal{F}(U_a\\cap U_b) \\end{equation}","doc_url":null,"latex_link":"sect0002.html#def:sheaves","lean_link":null,"lean_decl_name":"sheaves","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/AlgebraicGeometry/StructureSheaf.lean","lean4_code":"a morphism between the structure sheaves of `R` and `S`.\n-/\n","line":949,"end_line":950,"proof_status":"formalized"} |
| {"id":"def:pullback_of_sheaf","label":"pullback_of_sheaf","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:sheaves"],"nlp_text":"If \\(\\mathcal{F}\\) is a pre-sheaf over \\(Y\\), then the rule \\(U\\mapsto \\varinjlim \\limits _{f(U)\\subset V}\\mathcal{F}(V)\\) defines a pre-sheaf over \\(Y\\). If \\(\\mathcal{F}\\) is a sheaf, the sheafification of the previous pre-sheaf is a denoted \\(f^*\\mathcal{F}\\) and called the pullback by \\(f\\).","doc_url":null,"latex_link":"sect0004.html#def:pullback_of_sheaf","lean_link":null,"lean_decl_name":"pullback_of_sheaf","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:adj_kshv_and_shv","label":"adj_kshv_and_shv","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:sheaves","def:K_sheaves","def:adj_kprshv_and_prshv","lem:cofinal_syst_of_inter_compact"],"nlp_text":"\\(\\alpha ^*\\) send sheaves to \\(\\mathcal{K}\\)-sheaves \\(\\alpha _*\\) send \\(\\mathcal{K}\\)-sheaves to sheaves The reistrictions obtained still form an adjoint pair between shaeves and \\(\\mathcal{K}\\)-sheaves.","doc_url":null,"latex_link":"sect0002.html#lem:adj_kshv_and_shv","lean_link":null,"lean_decl_name":"adj_kshv_and_shv","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":"def:pushforward_of_sheaf","label":"pushforward_of_sheaf","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:sheaves"],"nlp_text":"If \\(\\mathcal{F}\\) is a pre-sheaf over \\(X\\), then the rule \\(U\\mapsto \\mathcal{F}(f^{-1}(U))\\) defines a pre-shaef over \\(Y\\). The functor obtained is denoted \\(f_*\\) and named the pushforward by \\(f\\). \\(f_*\\) send sheaves over \\(X\\) into sheaves over \\(Y\\).","doc_url":null,"latex_link":"sect0004.html#def:pushforward_of_sheaf","lean_link":null,"lean_decl_name":"pushforward_of_sheaf","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":"def:exceptional_pushforward_of_sheaf","label":"exceptional_pushforward_of_sheaf","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:pushforward_of_sheaf"],"nlp_text":"If \\(f:X\\to Y\\) is the inclusion of an open subset, the exceptional pushforward by \\(f\\): \\(f_!\\) is defined by \\(f_!\\mathcal{F}(U)\\) being the subset of \\(f_*\\mathcal{F}(U)\\) of sections that vanish over a neighborhood of \\(Y-X\\). It send the sheaves over \\(X\\) into the sheaves over \\(Y\\)","doc_url":null,"latex_link":"sect0004.html#def:exceptional_pushforward_of_sheaf","lean_link":null,"lean_decl_name":"exceptional_pushforward_of_sheaf","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:chech_comm_with_exceptional","label":"chech_comm_with_exceptional","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:exceptional_pushforward_of_sheaf"],"nlp_text":"Let \\(f:X\\to Y\\) be an inclusion of open subset, then there is a natural isomorphism \\(f_!:\\check{H}_c^{\\bullet }(X;\\mathcal{F})\\to \\check{H}^{\\bullet }(Y;f_!\\mathcal{F})\\)","doc_url":null,"latex_link":"sect0005.html#lem:chech_comm_with_exceptional","lean_link":null,"lean_decl_name":"chech_comm_with_exceptional","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":"def:K_pre_sheaves","label":"K_pre_sheaves","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A \\(\\mathcal{K}\\)-presheave on \\(X\\) is a contravariant functor from the category of compact sets of \\(X\\) to abélian groups.","doc_url":null,"latex_link":"sect0002.html#def:K_pre_sheaves","lean_link":null,"lean_decl_name":"K_pre_sheaves","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"def:cohomology_of_k-prsh","label":"cohomology_of_k-prsh","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:K_pre_sheaves"],"nlp_text":"Let \\(\\mathcal{F}^{\\bullet }\\) be complex of \\(\\mathcal{K}\\)-presheaves then taking the cohomology defines a \\(\\mathcal{K}\\)-presheaf denoted \\(H^{\\bullet }\\mathcal{F}^{\\bullet }\\).","doc_url":null,"latex_link":"sect0003.html#def:cohomology_of_k-prsh","lean_link":null,"lean_decl_name":"cohomology_of_k-prsh","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":"def:K_stalk","label":"K_stalk","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:K_pre_sheaves"],"nlp_text":"If \\(\\mathcal{F}\\) is a \\(\\mathcal{K}\\)-presheaf on \\(X\\) and \\(p\\in X\\) then the stalk of \\(\\mathcal{F}\\) at \\(p\\) is the abelian group \\(\\mathcal{F}_p:=\\varinjlim \\limits _{p\\in K\\text{ compact}}\\mathcal{F}(K)=\\mathcal{F}(\\{ p\\} )\\).","doc_url":null,"latex_link":"sect0002.html#def:K_stalk","lean_link":null,"lean_decl_name":"K_stalk","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"def:K_sheaves","label":"K_sheaves","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:K_pre_sheaves"],"nlp_text":"If \\(\\mathcal{F}\\) is a \\(\\mathcal{K}\\)-presheaf on \\(X\\), it is said to be a \\(\\mathcal{K}\\)-sheaf if the folowing conditions are satisfied: \\begin{equation} \\label{axiom:Ksh1} \\mathcal{F}(\\varnothing )=0 \\end{equation}","doc_url":null,"latex_link":"sect0002.html#def:K_sheaves","lean_link":null,"lean_decl_name":"K_sheaves","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"def:adj_kprshv_and_prshv","label":"adj_kprshv_and_prshv","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:K_pre_sheaves","def:pre_sheaves"],"nlp_text":"If \\(\\mathcal{F}\\) is a presheaf then let \\(\\alpha ^*\\mathcal{F}\\) be the \\(\\mathcal{K}\\)-presheaf: \\[ K\\mapsto \\varinjlim \\limits _{K\\subset U \\text{ open}}\\mathcal{F}(U) \\]","doc_url":null,"latex_link":"sect0002.html#def:adj_kprshv_and_prshv","lean_link":null,"lean_decl_name":"adj_kprshv_and_prshv","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:chech_is_determined_by_stalks_1","label":"chech_is_determined_by_stalks_1","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:cohomology_of_k-prsh","def:K_stalk","def:chech_of_compelx_of_k_prsh"],"nlp_text":"Let \\(\\mathcal{F}^{\\bullet }\\) be a complex of \\(\\mathcal{K}\\)-presheaves that verify 4 and such that all the stalks are \\(0\\) then \\(\\check{H}^{\\bullet }(X;\\mathcal{F})=0\\)","doc_url":null,"latex_link":"sect0005.html#lem:chech_is_determined_by_stalks_1","lean_link":null,"lean_decl_name":"chech_is_determined_by_stalks_1","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":"def:quasi_iso_of_complex_of_k-prsh","label":"quasi_iso_of_complex_of_k-prsh","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:cohomology_of_k-prsh"],"nlp_text":"A morphisme of complex of \\(\\mathcal{K}\\)-presheave \\(\\mathcal{F}^{\\bullet }\\to \\mathcal{G}^{\\bullet }\\) is called quasi-isomorphism if it induces isomorphisms \\(H^i\\mathcal{F}^{\\bullet }\\to H^i\\mathcal{G}^{\\bullet }\\) for all \\(i\\).","doc_url":null,"latex_link":"sect0003.html#def:quasi_iso_of_complex_of_k-prsh","lean_link":null,"lean_decl_name":"quasi_iso_of_complex_of_k-prsh","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"pro:adj_kprshv_and_prshv","label":"adj_kprshv_and_prshv","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:adj_kprshv_and_prshv"],"nlp_text":"The pair \\((\\alpha ^*,\\alpha _*)\\) is an adjonit pair.","doc_url":null,"latex_link":"sect0002.html#pro:adj_kprshv_and_prshv","lean_link":null,"lean_decl_name":"adj_kprshv_and_prshv","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":"def:chech_of_sh","label":"chech_of_sh","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(X\\) is a topological space, and \\(\\mathcal{F}\\) a sheaf over \\(X\\), then let \\(\\check{H}^{\\bullet }(X;\\mathcal{F}))\\) be the Čech cohomology of \\(X\\) with coeficient in \\(\\mathcal{F}\\)","doc_url":null,"latex_link":"sect0005.html#def:chech_of_sh","lean_link":null,"lean_decl_name":"chech_of_sh","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":"def:pullback_for_chech","label":"pullback_for_chech","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:chech_of_sh","def:chech_of_compact_sup_of_sh"],"nlp_text":"Let \\(f:X\\to Y\\) be a continuous map between topological spaces, and \\(\\mathcal{F}\\) a sheaf over \\(X\\). \\(f\\) induces a natural map \\(\\check{H}^{\\bullet }(Y;f_*\\mathcal{F})\\to \\check{H}^{\\bullet }(X;\\mathcal{F})\\). Moreover if \\(f\\) is proper, one gets a natural map \\(\\check{H}_c^{\\bullet }(Y;f_*\\mathcal{F})\\to \\check{H}_c^{\\bullet }(X;\\mathcal{F})\\)","doc_url":null,"latex_link":"sect0005.html#def:pullback_for_chech","lean_link":null,"lean_decl_name":"pullback_for_chech","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"def:orientation_sheaf_of_a_manifold","label":"orientation_sheaf_of_a_manifold","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(M\\) be a topological manifold, the rule \\(\\mathfrak {o}_M:K\\mapsto H_{\\dim M}(M,M\\backslash K)\\) defines a \\(\\mathcal{K}-\\)sheaf, called the orientation \\(\\mathcal{K}-\\)sheaf of \\(M\\). If \\(M\\) is a manifold with boundary, let \\(j:M\\backslash \\partial M\\to M\\) denote the canonical inclusion. The the orientation shaeves of \\(M\\) are defined as follows: \\(\\mathfrak {o}_M:= j_*\\mathfrak {o}_{M\\backslash \\partial M}\\) \\(\\mathfrak {o}_{M\\quad rel \\partial }:= j_!\\mathfrak {o}_{M\\backslash \\partial M}\\)","doc_url":null,"latex_link":"sect0007.html#def:orientation_sheaf_of_a_manifold","lean_link":null,"lean_decl_name":"orientation_sheaf_of_a_manifold","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:equiv_of_adj","label":"equiv_of_adj","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(\\mathcal{C}\\) and \\(\\mathcal{D}\\) are two categories, \\(F:\\mathcal{C}\\to \\mathcal{D}\\) and \\(G:\\mathcal{D}\\to \\mathcal{C}\\) two functors such that \\((F,G)\\) is an adjoint pair. Then for \\((F,G)\\) to be an equivalence of category, it’s enough to have that thes canonical naturals transformations \\(\\text{id}_{\\mathcal{D}}\\Rightarrow F\\circ G\\) and \\(G\\circ F\\Rightarrow \\text{id}_{\\mathcal{D}}\\) are isomorphisms.","doc_url":null,"latex_link":"sect0002.html#lem:equiv_of_adj","lean_link":null,"lean_decl_name":"equiv_of_adj","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:basis_of_open_of_fm1K","label":"basis_of_open_of_fm1K","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(K\\) is a compact of \\(Y\\), then \\(\\{ f^{-1}(U)\\} _{K\\subset U}\\) is a basis of open neighborhoods of \\(f^{-1}(K)\\).","doc_url":null,"latex_link":"sect0004.html#lem:basis_of_open_of_fm1K","lean_link":null,"lean_decl_name":"basis_of_open_of_fm1K","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:Homotopy_k-sheaf_axioms_for_singular_chain","label":"Homotopy_k-sheaf_axioms_for_singular_chain","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:singular_chain"],"nlp_text":"Let \\(X\\) be a topological manifold, then we have: \\(C_{\\bullet }(X,X)\\) is an acyclic complex For \\(A\\) and \\(B\\) two closed subsets of \\(X\\) the folowing complex is acyclic: \\[ [C_{\\bullet }(X,X\\backslash (A\\cup B))\\to C_{\\bullet +1}(X,X\\backslash A)\\bigoplus C_{\\bullet +1}(X,X\\backslash B)\\to C_{\\bullet +2}(X,X\\backslash (A\\cap B)) ] \\]","doc_url":null,"latex_link":"sect0007.html#lem:Homotopy_k-sheaf_axioms_for_singular_chain","lean_link":null,"lean_decl_name":"Homotopy_k-sheaf_axioms_for_singular_chain","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:chech_of_acyclic_is_zero","label":"chech_of_acyclic_is_zero","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(mathcal{F}^{\\bullet }\\) be an acyclic complex of \\(\\mathcal{K}\\)-presheaves, then \\(\\check{H}^k(X;mathcal{F}^{\\bullet })=0\\)","doc_url":null,"latex_link":"sect0005.html#lem:chech_of_acyclic_is_zero","lean_link":null,"lean_decl_name":"chech_of_acyclic_is_zero","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:chech_long_exact_sequence","label":"chech_long_exact_sequence","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:chech_of_compelx_of_k_prsh"],"nlp_text":"Let \\(0\\to \\mathcal{F}^{\\bullet }\\to \\mathcal{G}^{\\bullet }\\to \\mathcal{H}^{\\bullet }\\to 0\\) be a short exact sequence of complex of \\(\\mathcal{K}\\)-presheaves. Then there is a long exact seuquence in čech cohomology: \\[ \\ldots \\to \\check{H}^k(X;\\mathcal{F}^{\\bullet })\\to \\check{H}^k(X;\\mathcal{G}^{\\bullet })\\to \\check{H}^k(X;\\mathcal{H}^{\\bullet })\\to \\ldots \\]","doc_url":null,"latex_link":"sect0005.html#lem:chech_long_exact_sequence","lean_link":null,"lean_decl_name":"chech_long_exact_sequence","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":"def:chech_of_compelx_of_k_prsh","label":"chech_of_compelx_of_k_prsh","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\mathcal{F}^{\\bullet }\\) be a complex of \\(\\mathcal{K}\\)-presheaves on a compact space \\(X\\) then we define the Čech cohomology \\(\\check{H}(X;\\mathcal{F}^{\\bullet })\\) by TODO","doc_url":null,"latex_link":"sect0005.html#def:chech_of_compelx_of_k_prsh","lean_link":null,"lean_decl_name":"chech_of_compelx_of_k_prsh","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":"def:chech_of_compact_sup_of_sh","label":"chech_of_compact_sup_of_sh","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(X\\) is a topological space, \\(K\\) a comapct subset of \\(X\\) and \\(\\mathcal{F}\\) a sheaf over \\(X\\), then let \\(\\check{H}_K^{\\bullet }(X;\\mathcal{F}))\\) be the Čech cohomology of \\(X\\) with support in \\(K\\) with coeficient in \\(\\mathcal{F}\\)","doc_url":null,"latex_link":"sect0005.html#def:chech_of_compact_sup_of_sh","lean_link":null,"lean_decl_name":"chech_of_compact_sup_of_sh","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":"def:pre_sheaves","label":"pre_sheaves","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A presheave on \\(X\\) is a contravariant functor from the category of open sets of \\(X\\) to abélian groups.","doc_url":null,"latex_link":"sect0002.html#def:pre_sheaves","lean_link":null,"lean_decl_name":"pre_sheaves","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"def:stalk","label":"stalk","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:pre_sheaves"],"nlp_text":"If \\(\\mathcal{F}\\) is a presheaf on \\(X\\) and \\(p\\in X\\) then the stalk of \\(\\mathcal{F}\\) at \\(p\\) is the abelian group \\(\\mathcal{F}_p:=\\varinjlim \\limits _{p\\in U\\text{ open}}\\mathcal{F}(U)\\).","doc_url":null,"latex_link":"sect0002.html#def:stalk","lean_link":null,"lean_decl_name":"stalk","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"M2","source_path":"./M2/alphaup_and_stalk.lean","lean4_code":"def FUtoStalkι : FU (pC p) F (trueCond ) ⟶ (Functor.const _ ).obj (F.stalk p) where\n app U := germ _ U.unop.obj p (U.unop.property.1 (by rfl)) ≫ F.stalkSpecializes (by rfl)\n\n/-- The cocone induced by FUtoStalkι -/\n","line":29,"end_line":32,"proof_status":"formalized"} |
| {"id":"def:homotopy_diagram","label":"homotopy_diagram","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"","doc_url":null,"latex_link":"sect0008.html#def:homotopy_diagram","lean_link":null,"lean_decl_name":"homotopy_diagram","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":"def:homotopy_colimit","label":"homotopy_colimit","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:homotopy_diagram"],"nlp_text":"","doc_url":null,"latex_link":"sect0008.html#def:homotopy_colimit","lean_link":null,"lean_decl_name":"homotopy_colimit","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:cofinal_syst_of_inter_compact","label":"cofinal_syst_of_inter_compact","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(K_1,\\ldots K_n\\) are comapcts of \\(X\\) then \\(\\{ U_1\\cap \\ldots \\cap U_n\\} _{U_i\\supset K_i\\text{ open in }X}\\) is a cofinal system of neighborhoods of \\(K_1\\cap \\ldots \\cap K_n\\).","doc_url":null,"latex_link":"sect0002.html#lem:cofinal_syst_of_inter_compact","lean_link":null,"lean_decl_name":"cofinal_syst_of_inter_compact","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":"def:singular_chain","label":"singular_chain","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Singular chains","doc_url":null,"latex_link":"sect0007.html#def:singular_chain","lean_link":null,"lean_decl_name":"singular_chain","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"} |
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