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of (V) [quiver V] : V ⥤q (free_groupoid V) | { obj := λ X, ⟨X⟩,
map := λ X Y f, quot.mk _ f.to_pos_path } | def | category_theory.groupoid.free.of | category_theory.groupoid | src/category_theory/groupoid/free_groupoid.lean | [
"category_theory.category.basic",
"category_theory.functor.basic",
"category_theory.groupoid",
"tactic.nth_rewrite",
"category_theory.path_category",
"category_theory.quotient",
"combinatorics.quiver.symmetric"
] | [
"quiver"
] | The inclusion of the quiver on `V` to the underlying quiver on `free_groupoid V` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_eq : of V =
(quiver.symmetrify.of ⋙q paths.of).comp (quotient.functor $ @red_step V _).to_prefunctor | begin
apply prefunctor.ext, rotate,
{ rintro X, refl, },
{ rintro X Y f, refl, }
end | lemma | category_theory.groupoid.free.of_eq | category_theory.groupoid | src/category_theory/groupoid/free_groupoid.lean | [
"category_theory.category.basic",
"category_theory.functor.basic",
"category_theory.groupoid",
"tactic.nth_rewrite",
"category_theory.path_category",
"category_theory.quotient",
"combinatorics.quiver.symmetric"
] | [
"of_eq",
"prefunctor.ext",
"quiver.symmetrify.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift (φ : V ⥤q V') : free_groupoid V ⥤ V' | quotient.lift _
(paths.lift $ quiver.symmetrify.lift φ)
(by
{ rintros _ _ _ _ ⟨X,Y,f⟩,
simp only [quiver.symmetrify.lift_reverse, paths.lift_nil, quiver.path.comp_nil,
paths.lift_cons, paths.lift_to_path],
symmetry,
apply groupoid.comp_inv, }) | def | category_theory.groupoid.free.lift | category_theory.groupoid | src/category_theory/groupoid/free_groupoid.lean | [
"category_theory.category.basic",
"category_theory.functor.basic",
"category_theory.groupoid",
"tactic.nth_rewrite",
"category_theory.path_category",
"category_theory.quotient",
"combinatorics.quiver.symmetric"
] | [
"lift",
"quiver.path.comp_nil",
"quiver.symmetrify.lift",
"quiver.symmetrify.lift_reverse"
] | The lift of a prefunctor to a groupoid, to a functor from `free_groupoid V` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_spec (φ : V ⥤q V') : of V ⋙q (lift φ).to_prefunctor = φ | begin
rw [of_eq, prefunctor.comp_assoc, prefunctor.comp_assoc, functor.to_prefunctor_comp],
dsimp [lift],
rw [quotient.lift_spec, paths.lift_spec, quiver.symmetrify.lift_spec],
end | lemma | category_theory.groupoid.free.lift_spec | category_theory.groupoid | src/category_theory/groupoid/free_groupoid.lean | [
"category_theory.category.basic",
"category_theory.functor.basic",
"category_theory.groupoid",
"tactic.nth_rewrite",
"category_theory.path_category",
"category_theory.quotient",
"combinatorics.quiver.symmetric"
] | [
"lift",
"of_eq",
"prefunctor.comp_assoc",
"quiver.symmetrify.lift_spec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_unique (φ : V ⥤q V') (Φ : free_groupoid V ⥤ V')
(hΦ : of V ⋙q Φ.to_prefunctor = φ) : Φ = lift φ | begin
apply quotient.lift_unique,
apply paths.lift_unique,
fapply @quiver.symmetrify.lift_unique _ _ _ _ _ _ _ _ _,
{ rw ←functor.to_prefunctor_comp, exact hΦ, },
{ constructor, rintros X Y f,
simp only [←functor.to_prefunctor_comp,prefunctor.comp_map, paths.of_map, inv_eq_inv],
change Φ.map (inv ((qu... | lemma | category_theory.groupoid.free.lift_unique | category_theory.groupoid | src/category_theory/groupoid/free_groupoid.lean | [
"category_theory.category.basic",
"category_theory.functor.basic",
"category_theory.groupoid",
"tactic.nth_rewrite",
"category_theory.path_category",
"category_theory.quotient",
"combinatorics.quiver.symmetric"
] | [
"lift",
"lift_unique",
"quiver.symmetrify.lift_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.category_theory.free_groupoid_functor (φ : V ⥤q V') :
free_groupoid V ⥤ free_groupoid V' | lift (φ ⋙q of V') | def | category_theory.free_groupoid_functor | category_theory.groupoid | src/category_theory/groupoid/free_groupoid.lean | [
"category_theory.category.basic",
"category_theory.functor.basic",
"category_theory.groupoid",
"tactic.nth_rewrite",
"category_theory.path_category",
"category_theory.quotient",
"combinatorics.quiver.symmetric"
] | [
"lift"
] | The functor of free groupoid induced by a prefunctor of quivers | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_groupoid_functor_id :
free_groupoid_functor (prefunctor.id V) = functor.id (free_groupoid V) | begin
dsimp only [free_groupoid_functor], symmetry,
apply lift_unique, refl,
end | lemma | category_theory.groupoid.free.free_groupoid_functor_id | category_theory.groupoid | src/category_theory/groupoid/free_groupoid.lean | [
"category_theory.category.basic",
"category_theory.functor.basic",
"category_theory.groupoid",
"tactic.nth_rewrite",
"category_theory.path_category",
"category_theory.quotient",
"combinatorics.quiver.symmetric"
] | [
"lift_unique",
"prefunctor.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
free_groupoid_functor_comp
(φ : V ⥤q V') (φ' : V' ⥤q V'') :
free_groupoid_functor (φ ⋙q φ') = free_groupoid_functor φ ⋙ free_groupoid_functor φ' | begin
dsimp only [free_groupoid_functor], symmetry,
apply lift_unique, refl,
end | lemma | category_theory.groupoid.free.free_groupoid_functor_comp | category_theory.groupoid | src/category_theory/groupoid/free_groupoid.lean | [
"category_theory.category.basic",
"category_theory.functor.basic",
"category_theory.groupoid",
"tactic.nth_rewrite",
"category_theory.path_category",
"category_theory.quotient",
"combinatorics.quiver.symmetric"
] | [
"lift_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subgroupoid (C : Type u) [groupoid C] | (arrows : ∀ (c d : C), set (c ⟶ d))
(inv : ∀ {c d} {p : c ⟶ d} (hp : p ∈ arrows c d),
inv p ∈ arrows d c)
(mul : ∀ {c d e} {p} (hp : p ∈ arrows c d) {q} (hq : q ∈ arrows d e),
p ≫ q ∈ arrows c e) | structure | category_theory.subgroupoid | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | A sugroupoid of `C` consists of a choice of arrows for each pair of vertices, closed
under composition and inverses. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_mem_iff {c d : C} (f : c ⟶ d) : inv f ∈ S.arrows d c ↔ f ∈ S.arrows c d | begin
split,
{ rintro h,
suffices : inv (inv f) ∈ S.arrows c d,
{ simpa only [inv_eq_inv, is_iso.inv_inv] using this, },
{ apply S.inv h, }, },
{ apply S.inv, },
end | lemma | category_theory.subgroupoid.inv_mem_iff | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"inv_mem_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem_cancel_left {c d e : C} {f : c ⟶ d} {g : d ⟶ e} (hf : f ∈ S.arrows c d) :
f ≫ g ∈ S.arrows c e ↔ g ∈ S.arrows d e | begin
split,
{ rintro h,
suffices : (inv f) ≫ f ≫ g ∈ S.arrows d e,
{ simpa only [inv_eq_inv, is_iso.inv_hom_id_assoc] using this, },
{ apply S.mul (S.inv hf) h, }, },
{ apply S.mul hf, },
end | lemma | category_theory.subgroupoid.mul_mem_cancel_left | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"mul_mem_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem_cancel_right {c d e : C} {f : c ⟶ d} {g : d ⟶ e} (hg : g ∈ S.arrows d e) :
f ≫ g ∈ S.arrows c e ↔ f ∈ S.arrows c d | begin
split,
{ rintro h,
suffices : (f ≫ g) ≫ (inv g) ∈ S.arrows c d,
{ simpa only [inv_eq_inv, is_iso.hom_inv_id, category.comp_id, category.assoc] using this, },
{ apply S.mul h (S.inv hg), }, },
{ exact λ hf, S.mul hf hg, },
end | lemma | category_theory.subgroupoid.mul_mem_cancel_right | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"mul_mem_cancel_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
objs : set C | {c : C | (S.arrows c c).nonempty} | def | category_theory.subgroupoid.objs | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | The vertices of `C` on which `S` has non-trivial isotropy | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_objs_of_src {c d : C} {f : c ⟶ d} (h : f ∈ S.arrows c d) : c ∈ S.objs | ⟨f ≫ inv f, S.mul h (S.inv h)⟩ | lemma | category_theory.subgroupoid.mem_objs_of_src | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_objs_of_tgt {c d : C} {f : c ⟶ d} (h : f ∈ S.arrows c d) : d ∈ S.objs | ⟨(inv f) ≫ f, S.mul (S.inv h) h⟩ | lemma | category_theory.subgroupoid.mem_objs_of_tgt | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_mem_of_nonempty_isotropy (c : C) :
c ∈ objs S → 𝟙 c ∈ S.arrows c c | begin
rintro ⟨γ,hγ⟩,
convert S.mul hγ (S.inv hγ),
simp only [inv_eq_inv, is_iso.hom_inv_id],
end | lemma | category_theory.subgroupoid.id_mem_of_nonempty_isotropy | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_mem_of_src {c d : C} {f : c ⟶ d} (h : f ∈ S.arrows c d) : (𝟙 c) ∈ S.arrows c c | id_mem_of_nonempty_isotropy S c (mem_objs_of_src S h) | lemma | category_theory.subgroupoid.id_mem_of_src | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_mem_of_tgt {c d : C} {f : c ⟶ d} (h : f ∈ S.arrows c d) : (𝟙 d) ∈ S.arrows d d | id_mem_of_nonempty_isotropy S d (mem_objs_of_tgt S h) | lemma | category_theory.subgroupoid.id_mem_of_tgt | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_wide_quiver : quiver C | ⟨λ c d, subtype $ S.arrows c d⟩ | def | category_theory.subgroupoid.as_wide_quiver | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"quiver"
] | A subgroupoid seen as a quiver on vertex set `C` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe : groupoid S.objs | { hom := λ a b, S.arrows a.val b.val,
id := λ a, ⟨𝟙 a.val, id_mem_of_nonempty_isotropy S a.val a.prop⟩,
comp := λ a b c p q, ⟨p.val ≫ q.val, S.mul p.prop q.prop⟩,
id_comp' := λ a b ⟨p,hp⟩, by simp only [category.id_comp],
comp_id' := λ a b ⟨p,hp⟩, by simp only [category.comp_id],
assoc' := λ a b c d ⟨p,hp⟩ ⟨... | instance | category_theory.subgroupoid.coe | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | The coercion of a subgroupoid as a groupoid | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_inv_coe' {c d : S.objs} (p : c ⟶ d) :
(category_theory.inv p).val = category_theory.inv p.val | by { simp only [subtype.val_eq_coe, ←inv_eq_inv, coe_inv_coe], } | lemma | category_theory.subgroupoid.coe_inv_coe' | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"category_theory.inv",
"subtype.val_eq_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom : S.objs ⥤ C | { obj := λ c, c.val,
map := λ c d f, f.val,
map_id' := λ c, rfl,
map_comp' := λ c d e f g, rfl } | def | category_theory.subgroupoid.hom | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | The embedding of the coerced subgroupoid to its parent | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom.inj_on_objects : function.injective (hom S).obj | by { rintros ⟨c,hc⟩ ⟨d,hd⟩ hcd, simp only [subtype.mk_eq_mk], exact hcd } | lemma | category_theory.subgroupoid.hom.inj_on_objects | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"subtype.mk_eq_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom.faithful :
∀ c d, function.injective (λ (f : c ⟶ d), (hom S).map f) | by { rintros ⟨c,hc⟩ ⟨d,hd⟩ ⟨f,hf⟩ ⟨g,hg⟩ hfg, simp only [subtype.mk_eq_mk], exact hfg, } | lemma | category_theory.subgroupoid.hom.faithful | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"subtype.mk_eq_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vertex_subgroup {c : C} (hc : c ∈ S.objs) : subgroup (c ⟶ c) | { carrier := S.arrows c c,
mul_mem' := λ f g hf hg, S.mul hf hg,
one_mem' := id_mem_of_nonempty_isotropy _ _ hc,
inv_mem' := λ f hf, S.inv hf } | def | category_theory.subgroupoid.vertex_subgroup | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"subgroup"
] | The subgroup of the vertex group at `c` given by the subgroupoid | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_iff (S : subgroupoid C) (F : Σ c d, c ⟶ d) :
F ∈ S ↔ F.2.2 ∈ S.arrows F.1 F.2.1 | iff.rfl | lemma | category_theory.subgroupoid.mem_iff | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff (S T : subgroupoid C) : (S ≤ T) ↔ (∀ {c d}, (S.arrows c d) ⊆ (T.arrows c d)) | by { rw [set_like.le_def, sigma.forall], exact forall_congr (λ c, sigma.forall) } | lemma | category_theory.subgroupoid.le_iff | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"set_like.le_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_top {c d : C} (f : c ⟶ d) : f ∈ (⊤ : subgroupoid C).arrows c d | trivial | lemma | category_theory.subgroupoid.mem_top | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_top_objs (c : C) : c ∈ (⊤ : subgroupoid C).objs | by { dsimp [has_top.top,objs], simp only [univ_nonempty], } | lemma | category_theory.subgroupoid.mem_top_objs | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_objs {S T : subgroupoid C} (h : S ≤ T) : S.objs ⊆ T.objs | λ s ⟨γ, hγ⟩, ⟨γ, @h ⟨s, s, γ⟩ hγ⟩ | lemma | category_theory.subgroupoid.le_objs | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion {S T : subgroupoid C} (h : S ≤ T) : S.objs ⥤ T.objs | { obj := λ s, ⟨s.val, le_objs h s.prop⟩,
map := λ s t f, ⟨f.val, @h ⟨s, t, f.val⟩ f.prop⟩,
map_id' := λ _, rfl,
map_comp' := λ _ _ _ _ _, rfl } | def | category_theory.subgroupoid.inclusion | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | The functor associated to the embedding of subgroupoids | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inclusion_inj_on_objects {S T : subgroupoid C} (h : S ≤ T) :
function.injective (inclusion h).obj | λ ⟨s,hs⟩ ⟨t,ht⟩, by simpa only [inclusion, subtype.mk_eq_mk] using id | lemma | category_theory.subgroupoid.inclusion_inj_on_objects | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"subtype.mk_eq_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion_faithful {S T : subgroupoid C} (h : S ≤ T) (s t : S.objs) :
function.injective (λ (f : s ⟶ t), (inclusion h).map f) | λ ⟨f,hf⟩ ⟨g,hg⟩, by { dsimp only [inclusion], simpa only [subtype.mk_eq_mk] using id } | lemma | category_theory.subgroupoid.inclusion_faithful | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"subtype.mk_eq_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion_refl {S : subgroupoid C} : inclusion (le_refl S) = 𝟭 S.objs | functor.hext (λ ⟨s,hs⟩, rfl) (λ ⟨s,hs⟩ ⟨t,ht⟩ ⟨f,hf⟩, heq_of_eq rfl) | lemma | category_theory.subgroupoid.inclusion_refl | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion_trans {R S T : subgroupoid C} (k : R ≤ S) (h : S ≤ T) :
inclusion (k.trans h) = (inclusion k) ⋙ (inclusion h) | rfl | lemma | category_theory.subgroupoid.inclusion_trans | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion_comp_embedding {S T : subgroupoid C} (h : S ≤ T) :
(inclusion h) ⋙ T.hom = S.hom | rfl | lemma | category_theory.subgroupoid.inclusion_comp_embedding | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete.arrows : Π (c d : C), (c ⟶ d) → Prop
| id (c : C) : discrete.arrows c c (𝟙 c) | inductive | category_theory.subgroupoid.discrete.arrows | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | The family of arrows of the discrete groupoid | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete : subgroupoid C | { arrows := discrete.arrows,
inv := by { rintros _ _ _ ⟨⟩, simp only [inv_eq_inv, is_iso.inv_id], split, },
mul := by { rintros _ _ _ _ ⟨⟩ _ ⟨⟩, rw category.comp_id, split, } } | def | category_theory.subgroupoid.discrete | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | The only arrows of the discrete groupoid are the identity arrows. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_discrete_iff {c d : C} (f : c ⟶ d) :
(f ∈ (discrete).arrows c d) ↔ (∃ (h : c = d), f = eq_to_hom h) | ⟨by { rintro ⟨⟩, exact ⟨rfl, rfl⟩ }, by { rintro ⟨rfl, rfl⟩, split }⟩ | lemma | category_theory.subgroupoid.mem_discrete_iff | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_wide : Prop | (wide : ∀ c, (𝟙 c) ∈ (S.arrows c c)) | structure | category_theory.subgroupoid.is_wide | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | A subgroupoid is wide if its carrier set is all of `C` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_wide_iff_objs_eq_univ : S.is_wide ↔ S.objs = set.univ | begin
split,
{ rintro h,
ext, split; simp only [top_eq_univ, mem_univ, implies_true_iff, forall_true_left],
apply mem_objs_of_src S (h.wide x), },
{ rintro h,
refine ⟨λ c, _⟩,
obtain ⟨γ,γS⟩ := (le_of_eq h.symm : ⊤ ⊆ S.objs) (set.mem_univ c),
exact id_mem_of_src S γS, },
end | lemma | category_theory.subgroupoid.is_wide_iff_objs_eq_univ | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"forall_true_left",
"set.mem_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_wide.id_mem {S : subgroupoid C} (Sw : S.is_wide) (c : C) :
(𝟙 c) ∈ S.arrows c c | Sw.wide c | lemma | category_theory.subgroupoid.is_wide.id_mem | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_wide.eq_to_hom_mem {S : subgroupoid C} (Sw : S.is_wide) {c d : C} (h : c = d) :
(eq_to_hom h) ∈ S.arrows c d | by
{ cases h, simp only [eq_to_hom_refl], apply Sw.id_mem c, } | lemma | category_theory.subgroupoid.is_wide.eq_to_hom_mem | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal extends (is_wide S) : Prop | (conj : ∀ {c d} (p : c ⟶ d) {γ : c ⟶ c} (hs : γ ∈ S.arrows c c),
((inv p) ≫ γ ≫ p) ∈ (S.arrows d d)) | structure | category_theory.subgroupoid.is_normal | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | A subgroupoid is normal if it is wide and satisfies the expected stability under conjugacy. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_normal.conj' {S : subgroupoid C} (Sn : is_normal S) :
∀ {c d} (p : d ⟶ c) {γ : c ⟶ c} (hs : γ ∈ S.arrows c c), (p ≫ γ ≫ (inv p)) ∈ (S.arrows d d) | λ c d p γ hs, by { convert Sn.conj (inv p) hs, simp, } | lemma | category_theory.subgroupoid.is_normal.conj' | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.conjugation_bij (Sn : is_normal S) {c d} (p : c ⟶ d) :
set.bij_on (λ γ : c ⟶ c, (inv p) ≫ γ ≫ p) (S.arrows c c) (S.arrows d d) | begin
refine ⟨λ γ γS, Sn.conj p γS, λ γ₁ γ₁S γ₂ γ₂S h, _, λ δ δS, ⟨p ≫ δ ≫ (inv p), Sn.conj' p δS, _⟩⟩,
{ simpa only [inv_eq_inv, category.assoc, is_iso.hom_inv_id,
category.comp_id, is_iso.hom_inv_id_assoc] using p ≫= h =≫ inv p },
{ simp only [inv_eq_inv, category.assoc, is_iso.inv_hom_id,
... | lemma | category_theory.subgroupoid.is_normal.conjugation_bij | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"set.bij_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_is_normal : is_normal (⊤ : subgroupoid C) | { wide := (λ c, trivial),
conj := (λ a b c d e, trivial) } | lemma | category_theory.subgroupoid.top_is_normal | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_is_normal (s : set $ subgroupoid C) (sn : ∀ S ∈ s, is_normal S) : is_normal (Inf s) | { wide := by { simp_rw [Inf, mem_Inter₂], exact λ c S Ss, (sn S Ss).wide c },
conj := by { simp_rw [Inf, mem_Inter₂], exact λ c d p γ hγ S Ss, (sn S Ss).conj p (hγ S Ss) } } | lemma | category_theory.subgroupoid.Inf_is_normal | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_is_normal : (@discrete C _).is_normal | { wide := λ c, by { constructor, },
conj := λ c d f γ hγ, by
{ cases hγ, simp only [inv_eq_inv, category.id_comp, is_iso.inv_hom_id], constructor, } } | lemma | category_theory.subgroupoid.discrete_is_normal | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.vertex_subgroup (Sn : is_normal S) (c : C) (cS : c ∈ S.objs) :
(S.vertex_subgroup cS).normal | { conj_mem := λ x hx y, by { rw mul_assoc, exact Sn.conj' y hx } } | lemma | category_theory.subgroupoid.is_normal.vertex_subgroup | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"mul_assoc",
"normal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generated : subgroupoid C | Inf {S : subgroupoid C | ∀ c d, X c d ⊆ S.arrows c d} | def | category_theory.subgroupoid.generated | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | The subgropoid generated by the set of arrows `X` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subset_generated (c d : C) : X c d ⊆ (generated X).arrows c d | begin
dsimp only [generated, Inf],
simp only [subset_Inter₂_iff],
exact λ S hS f fS, hS _ _ fS,
end | lemma | category_theory.subgroupoid.subset_generated | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generated_normal : subgroupoid C | Inf {S : subgroupoid C | (∀ c d, X c d ⊆ S.arrows c d) ∧ S.is_normal} | def | category_theory.subgroupoid.generated_normal | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | The normal sugroupoid generated by the set of arrows `X` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
generated_le_generated_normal : generated X ≤ generated_normal X | begin
apply @Inf_le_Inf (subgroupoid C) _,
exact λ S ⟨h,_⟩, h,
end | lemma | category_theory.subgroupoid.generated_le_generated_normal | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"Inf_le_Inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generated_normal_is_normal : (generated_normal X).is_normal | Inf_is_normal _ (λ S h, h.right) | lemma | category_theory.subgroupoid.generated_normal_is_normal | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.generated_normal_le {S : subgroupoid C} (Sn : S.is_normal) :
generated_normal X ≤ S ↔ ∀ c d, X c d ⊆ S.arrows c d | begin
split,
{ rintro h c d,
let h' := generated_le_generated_normal X,
rw le_iff at h h',
exact ((subset_generated X c d).trans (@h' c d)).trans (@h c d), },
{ rintro h,
apply @Inf_le (subgroupoid C) _,
exact ⟨h,Sn⟩, },
end | lemma | category_theory.subgroupoid.is_normal.generated_normal_le | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"Inf_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap (S : subgroupoid D) : subgroupoid C | { arrows := λ c d, {f : c ⟶ d | φ.map f ∈ S.arrows (φ.obj c) (φ.obj d)},
inv := λ c d p hp, by { rw [mem_set_of, inv_eq_inv, φ.map_inv p, ← inv_eq_inv], exact S.inv hp },
mul := begin
rintros,
simp only [mem_set_of, functor.map_comp],
apply S.mul; assumption,
end } | def | category_theory.subgroupoid.comap | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | A functor between groupoid defines a map of subgroupoids in the reverse direction
by taking preimages. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_mono (S T : subgroupoid D) :
S ≤ T → comap φ S ≤ comap φ T | λ ST ⟨c,d,p⟩, @ST ⟨_,_,_⟩ | lemma | category_theory.subgroupoid.comap_mono | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal_comap {S : subgroupoid D} (Sn : is_normal S) : is_normal (comap φ S) | { wide := λ c, by { rw [comap, mem_set_of, functor.map_id], apply Sn.wide, },
conj := λ c d f γ hγ, by
{ simp_rw [inv_eq_inv f, comap, mem_set_of, functor.map_comp, functor.map_inv, ←inv_eq_inv],
exact Sn.conj _ hγ, } } | lemma | category_theory.subgroupoid.is_normal_comap | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"functor.map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comp {E : Type*} [groupoid E] (ψ : D ⥤ E) :
comap (φ ⋙ ψ) = (comap φ) ∘ (comap ψ) | rfl | lemma | category_theory.subgroupoid.comap_comp | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker : subgroupoid C | comap φ discrete | def | category_theory.subgroupoid.ker | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | The kernel of a functor between subgroupoid is the preimage. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_ker_iff {c d : C} (f : c ⟶ d) :
f ∈ (ker φ).arrows c d ↔ ∃ (h : φ.obj c = φ.obj d), φ.map f = eq_to_hom h | mem_discrete_iff (φ.map f) | lemma | category_theory.subgroupoid.mem_ker_iff | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_is_normal : (ker φ).is_normal | is_normal_comap φ (discrete_is_normal) | lemma | category_theory.subgroupoid.ker_is_normal | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_comp {E : Type*} [groupoid E] (ψ : D ⥤ E) : ker (φ ⋙ ψ) = comap φ (ker ψ) | rfl | lemma | category_theory.subgroupoid.ker_comp | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map.arrows (hφ : function.injective φ.obj) (S : subgroupoid C) :
Π (c d : D), (c ⟶ d) → Prop
| im {c d : C} (f : c ⟶ d) (hf : f ∈ S.arrows c d) : map.arrows (φ.obj c) (φ.obj d) (φ.map f) | inductive | category_theory.subgroupoid.map.arrows | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | The family of arrows of the image of a subgroupoid under a functor injective on objects | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map.arrows_iff (hφ : function.injective φ.obj) (S : subgroupoid C) {c d : D} (f : c ⟶ d) :
map.arrows φ hφ S c d f ↔
∃ (a b : C) (g : a ⟶ b) (ha : φ.obj a = c) (hb : φ.obj b = d) (hg : g ∈ S.arrows a b),
f = (eq_to_hom ha.symm) ≫ φ.map g ≫ (eq_to_hom hb) | begin
split,
{ rintro ⟨g,hg⟩, exact ⟨_,_,g,rfl,rfl,hg, eq_conj_eq_to_hom _⟩ },
{ rintro ⟨a,b,g,rfl,rfl,hg,rfl⟩, rw ← eq_conj_eq_to_hom, split, exact hg },
end | lemma | category_theory.subgroupoid.map.arrows_iff | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (hφ : function.injective φ.obj) (S : subgroupoid C) : subgroupoid D | { arrows := map.arrows φ hφ S,
inv := begin
rintro _ _ _ ⟨⟩,
rw [inv_eq_inv, ←functor.map_inv, ←inv_eq_inv],
split, apply S.inv, assumption,
end,
mul := begin
rintro _ _ _ _ ⟨f,hf⟩ q hq,
obtain ⟨c₃,c₄,g,he,rfl,hg,gq⟩ := (map.arrows_iff φ hφ S q).mp hq,
cases hφ he, rw [gq, ← eq_conj_eq_to_... | def | category_theory.subgroupoid.map | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | The "forward" image of a subgroupoid under a functor injective on objects | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_map_iff (hφ : function.injective φ.obj) (S : subgroupoid C) {c d : D} (f : c ⟶ d) :
f ∈ (map φ hφ S).arrows c d ↔
∃ (a b : C) (g : a ⟶ b) (ha : φ.obj a = c) (hb : φ.obj b = d) (hg : g ∈ S.arrows a b),
f = (eq_to_hom ha.symm) ≫ φ.map g ≫ (eq_to_hom hb) | map.arrows_iff φ hφ S f | lemma | category_theory.subgroupoid.mem_map_iff | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
galois_connection_map_comap (hφ : function.injective φ.obj) :
galois_connection (map φ hφ) (comap φ) | begin
rintro S T, simp_rw [le_iff], split,
{ exact λ h c d f fS, h (map.arrows.im f fS), },
{ rintros h _ _ g ⟨a,gφS⟩,
exact h gφS, },
end | lemma | category_theory.subgroupoid.galois_connection_map_comap | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"galois_connection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mono (hφ : function.injective φ.obj) (S T : subgroupoid C) :
S ≤ T → map φ hφ S ≤ map φ hφ T | λ h, (galois_connection_map_comap φ hφ).monotone_l h | lemma | category_theory.subgroupoid.map_mono | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_comap_map (hφ : function.injective φ.obj) (S : subgroupoid C) :
S ≤ comap φ (map φ hφ S) | (galois_connection_map_comap φ hφ).le_u_l S | lemma | category_theory.subgroupoid.le_comap_map | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comap_le (hφ : function.injective φ.obj) (T : subgroupoid D) :
map φ hφ (comap φ T) ≤ T | (galois_connection_map_comap φ hφ).l_u_le T | lemma | category_theory.subgroupoid.map_comap_le | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le_iff_le_comap (hφ : function.injective φ.obj)
(S : subgroupoid C) (T : subgroupoid D) :
map φ hφ S ≤ T ↔ S ≤ comap φ T | (galois_connection_map_comap φ hφ).le_iff_le | lemma | category_theory.subgroupoid.map_le_iff_le_comap | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map_objs_iff (hφ : function.injective φ.obj) (d : D) :
d ∈ (map φ hφ S).objs ↔ ∃ c ∈ S.objs, φ.obj c = d | begin
dsimp [objs, map],
split,
{ rintro ⟨f,hf⟩,
change map.arrows φ hφ S d d f at hf, rw map.arrows_iff at hf,
obtain ⟨c,d,g,ec,ed,eg,gS,eg⟩ := hf,
exact ⟨c, ⟨mem_objs_of_src S eg, ec⟩⟩, },
{ rintros ⟨c,⟨γ,γS⟩,rfl⟩,
exact ⟨φ.map γ,⟨γ,γS⟩⟩, }
end | lemma | category_theory.subgroupoid.mem_map_objs_iff | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_objs_eq (hφ : function.injective φ.obj) : (map φ hφ S).objs = φ.obj '' S.objs | by { ext, convert mem_map_objs_iff S φ hφ x, simp only [mem_image, exists_prop], } | lemma | category_theory.subgroupoid.map_objs_eq | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"exists_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im (hφ : function.injective φ.obj) | map φ hφ (⊤) | def | category_theory.subgroupoid.im | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | The image of a functor injective on objects | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_im_iff (hφ : function.injective φ.obj) {c d : D} (f : c ⟶ d) :
f ∈ (im φ hφ).arrows c d ↔
∃ (a b : C) (g : a ⟶ b) (ha : φ.obj a = c) (hb : φ.obj b = d),
f = (eq_to_hom ha.symm) ≫ φ.map g ≫ (eq_to_hom hb) | by { convert map.arrows_iff φ hφ ⊤ f, simp only [has_top.top, mem_univ, exists_true_left] } | lemma | category_theory.subgroupoid.mem_im_iff | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"exists_true_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_im_objs_iff (hφ : function.injective φ.obj) (d : D) :
d ∈ (im φ hφ).objs ↔ ∃ c : C, φ.obj c = d | by
{ simp only [im, mem_map_objs_iff, mem_top_objs, exists_true_left], } | lemma | category_theory.subgroupoid.mem_im_objs_iff | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"exists_true_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
obj_surjective_of_im_eq_top (hφ : function.injective φ.obj) (hφ' : im φ hφ = ⊤) :
function.surjective φ.obj | begin
rintro d,
rw [←mem_im_objs_iff, hφ'],
apply mem_top_objs,
end | lemma | category_theory.subgroupoid.obj_surjective_of_im_eq_top | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal_map (hφ : function.injective φ.obj) (hφ' : im φ hφ = ⊤) (Sn : S.is_normal) :
(map φ hφ S).is_normal | { wide := λ d, by
{ obtain ⟨c,rfl⟩ := obj_surjective_of_im_eq_top φ hφ hφ' d,
change map.arrows φ hφ S _ _ (𝟙 _), rw ←functor.map_id,
constructor, exact Sn.wide c, },
conj := λ d d' g δ hδ, by
{ rw mem_map_iff at hδ,
obtain ⟨c,c',γ,cd,cd',γS,hγ⟩ := hδ, subst_vars, cases hφ cd',
have : d' ∈ (im φ ... | lemma | category_theory.subgroupoid.is_normal_map | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_thin | quiver.is_thin S.objs | abbreviation | category_theory.subgroupoid.is_thin | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"quiver.is_thin"
] | A subgroupoid `is_thin` if it has at most one arrow between any two vertices. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_thin_iff : S.is_thin ↔ ∀ (c : S.objs), subsingleton (S.arrows c c) | by apply is_thin_iff | lemma | category_theory.subgroupoid.is_thin_iff | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_totally_disconnected | is_totally_disconnected S.objs | abbreviation | category_theory.subgroupoid.is_totally_disconnected | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"is_totally_disconnected"
] | A subgroupoid `is_totally_disconnected` if it has only isotropy arrows. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_totally_disconnected_iff :
S.is_totally_disconnected ↔ ∀ c d, (S.arrows c d).nonempty → c = d | begin
split,
{ rintro h c d ⟨f,fS⟩,
rw ←@subtype.mk_eq_mk _ _ c (mem_objs_of_src S fS) d (mem_objs_of_tgt S fS),
exact h ⟨c, mem_objs_of_src S fS⟩ ⟨d, mem_objs_of_tgt S fS⟩ ⟨f, fS⟩, },
{ rintros h ⟨c, hc⟩ ⟨d, hd⟩ ⟨f, fS⟩,
simp only [subtype.mk_eq_mk],
exact h c d ⟨f, fS⟩, },
end | lemma | category_theory.subgroupoid.is_totally_disconnected_iff | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"subtype.mk_eq_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disconnect : subgroupoid C | { arrows := λ c d f, c = d ∧ f ∈ S.arrows c d,
inv := by { rintros _ _ _ ⟨rfl, h⟩, exact ⟨rfl, S.inv h⟩, },
mul := by { rintros _ _ _ _ ⟨rfl, h⟩ _ ⟨rfl, h'⟩, exact ⟨rfl, S.mul h h'⟩, } } | def | category_theory.subgroupoid.disconnect | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | The isotropy subgroupoid of `S` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
disconnect_le : S.disconnect ≤ S | by { rw le_iff, rintros _ _ _ ⟨⟩, assumption, } | lemma | category_theory.subgroupoid.disconnect_le | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disconnect_normal (Sn : S.is_normal) : S.disconnect.is_normal | { wide := λ c, ⟨rfl, Sn.wide c⟩,
conj := λ c d p γ ⟨_,h'⟩, ⟨rfl, Sn.conj _ h'⟩ } | lemma | category_theory.subgroupoid.disconnect_normal | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_disconnect_objs_iff {c : C} : c ∈ S.disconnect.objs ↔ c ∈ S.objs | ⟨λ ⟨γ, h, γS⟩, ⟨γ, γS⟩, λ ⟨γ, γS⟩, ⟨γ, rfl, γS⟩⟩ | lemma | category_theory.subgroupoid.mem_disconnect_objs_iff | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disconnect_objs : S.disconnect.objs = S.objs | by { apply set.ext, apply mem_disconnect_objs_iff, } | lemma | category_theory.subgroupoid.disconnect_objs | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disconnect_is_totally_disconnected : S.disconnect.is_totally_disconnected | by { rw is_totally_disconnected_iff, exact λ c d ⟨f, h, fS⟩, h } | lemma | category_theory.subgroupoid.disconnect_is_totally_disconnected | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full : subgroupoid C | { arrows := λ c d _, c ∈ D ∧ d ∈ D,
inv := by { rintros _ _ _ ⟨⟩, constructor; assumption, },
mul := by { rintros _ _ _ _ ⟨⟩ _ ⟨⟩, constructor; assumption,} } | def | category_theory.subgroupoid.full | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | The full subgroupoid on a set `D : set C` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
full_objs : (full D).objs = D | set.ext $ λ _, ⟨λ ⟨f, h, _⟩, h , λ h, ⟨𝟙 _, h, h⟩⟩ | lemma | category_theory.subgroupoid.full_objs | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_full_iff {c d : C} {f : c ⟶ d} : f ∈ (full D).arrows c d ↔ c ∈ D ∧ d ∈ D | iff.rfl | lemma | category_theory.subgroupoid.mem_full_iff | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_full_objs_iff {c : C} : c ∈ (full D).objs ↔ c ∈ D | by rw full_objs | lemma | category_theory.subgroupoid.mem_full_objs_iff | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full_empty : full ∅ = (⊥ : subgroupoid C) | by { ext, simp only [has_bot.bot, mem_full_iff, mem_empty_iff_false, and_self], } | lemma | category_theory.subgroupoid.full_empty | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full_univ : full set.univ = (⊤ : subgroupoid C) | by { ext, simp only [mem_full_iff, mem_univ, and_self, true_iff], } | lemma | category_theory.subgroupoid.full_univ | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full_mono {D E : set C} (h : D ≤ E) : full D ≤ full E | begin
rw le_iff,
rintro c d f,
simp only [mem_full_iff],
exact λ ⟨hc, hd⟩, ⟨h hc, h hd⟩,
end | lemma | category_theory.subgroupoid.full_mono | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full_arrow_eq_iff {c d : (full D).objs} {f g : c ⟶ d} :
f = g ↔ (↑f : c.val ⟶ d.val) = ↑g | by apply subtype.ext_iff | lemma | category_theory.subgroupoid.full_arrow_eq_iff | category_theory.groupoid | src/category_theory/groupoid/subgroupoid.lean | [
"category_theory.groupoid.vertex_group",
"category_theory.groupoid.basic",
"category_theory.groupoid",
"algebra.group.defs",
"data.set.lattice",
"group_theory.subgroup.basic",
"order.galois_connection"
] | [
"subtype.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vertex_group (c : C): group (c ⟶ c) | { mul := λ (x y : c ⟶ c), x ≫ y,
mul_assoc := category.assoc,
one := 𝟙 c,
one_mul := category.id_comp,
mul_one := category.comp_id,
inv := groupoid.inv,
mul_left_inv := inv_comp } | instance | category_theory.groupoid.vertex_group | category_theory.groupoid | src/category_theory/groupoid/vertex_group.lean | [
"category_theory.groupoid",
"category_theory.path_category",
"algebra.group.defs",
"algebra.hom.group",
"algebra.hom.equiv.basic",
"combinatorics.quiver.path"
] | [
"group",
"mul_assoc",
"mul_left_inv",
"mul_one",
"one_mul"
] | The vertex group at `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vertex_group.inv_eq_inv (c : C) (γ : c ⟶ c) :
γ ⁻¹ = category_theory.inv γ | groupoid.inv_eq_inv γ | lemma | category_theory.groupoid.vertex_group.inv_eq_inv | category_theory.groupoid | src/category_theory/groupoid/vertex_group.lean | [
"category_theory.groupoid",
"category_theory.path_category",
"algebra.group.defs",
"algebra.hom.group",
"algebra.hom.equiv.basic",
"combinatorics.quiver.path"
] | [
"category_theory.inv"
] | The inverse in the group is equal to the inverse given by `category_theory.inv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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