statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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balanced_opposite [balanced C] : balanced Cᵒᵖ | { is_iso_of_mono_of_epi := λ X Y f fmono fepi,
by { rw ← quiver.hom.op_unop f, exactI is_iso_of_op _ } } | lemma | category_theory.balanced_opposite | category_theory | src/category_theory/balanced.lean | [
"category_theory.epi_mono"
] | [
"balanced",
"quiver.hom.op_unop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_sections_of_finite_cofiltered_system.init
{J : Type u} [small_category J] [is_cofiltered_or_empty J] (F : J ⥤ Type u)
[hf : ∀ j, finite (F.obj j)] [hne : ∀ j, nonempty (F.obj j)] :
F.sections.nonempty | begin
let F' : J ⥤ Top := F ⋙ Top.discrete,
haveI : ∀ j, discrete_topology (F'.obj j) := λ _, ⟨rfl⟩,
haveI : ∀ j, finite (F'.obj j) := hf,
haveI : ∀ j, nonempty (F'.obj j) := hne,
obtain ⟨⟨u, hu⟩⟩ := Top.nonempty_limit_cone_of_compact_t2_cofiltered_system F',
exact ⟨u, λ _ _, hu⟩,
end | lemma | nonempty_sections_of_finite_cofiltered_system.init | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [
"Top",
"Top.discrete",
"Top.nonempty_limit_cone_of_compact_t2_cofiltered_system",
"discrete_topology",
"finite"
] | This bootstraps `nonempty_sections_of_finite_inverse_system`. In this version,
the `F` functor is between categories of the same universe, and it is an easy
corollary to `Top.nonempty_limit_cone_of_compact_t2_inverse_system`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_sections_of_finite_cofiltered_system
{J : Type u} [category.{w} J] [is_cofiltered_or_empty J] (F : J ⥤ Type v)
[∀ (j : J), finite (F.obj j)] [∀ (j : J), nonempty (F.obj j)] :
F.sections.nonempty | begin
-- Step 1: lift everything to the `max u v w` universe.
let J' : Type (max w v u) := as_small.{max w v} J,
let down : J' ⥤ J := as_small.down,
let F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ ulift_functor.{(max u w) v},
haveI : ∀ i, nonempty (F'.obj i) := λ i, ⟨⟨classical.arbitrary (F.obj (down.obj i))⟩⟩,
... | theorem | nonempty_sections_of_finite_cofiltered_system | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [
"finite",
"finite.of_equiv",
"is_empty_elim",
"is_empty_or_nonempty",
"nonempty_sections_of_finite_cofiltered_system.init"
] | The cofiltered limit of nonempty finite types is nonempty.
See `nonempty_sections_of_finite_inverse_system` for a specialization to inverse limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_sections_of_finite_inverse_system
{J : Type u} [preorder J] [is_directed J (≤)] (F : Jᵒᵖ ⥤ Type v)
[∀ (j : Jᵒᵖ), finite (F.obj j)] [∀ (j : Jᵒᵖ), nonempty (F.obj j)] :
F.sections.nonempty | begin
casesI is_empty_or_nonempty J,
{ haveI : is_empty Jᵒᵖ := ⟨λ j, is_empty_elim j.unop⟩, -- TODO: this should be a global instance
exact ⟨is_empty_elim, is_empty_elim⟩, },
{ exact nonempty_sections_of_finite_cofiltered_system _, },
end | theorem | nonempty_sections_of_finite_inverse_system | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [
"finite",
"is_directed",
"is_empty",
"is_empty_elim",
"is_empty_or_nonempty",
"nonempty_sections_of_finite_cofiltered_system"
] | The inverse limit of nonempty finite types is nonempty.
See `nonempty_sections_of_finite_cofiltered_system` for a generalization to cofiltered limits.
That version applies in almost all cases, and the only difference is that this version
allows `J` to be empty.
This may be regarded as a generalization of Kőnig's lemm... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventual_range (j : J) | ⋂ i (f : i ⟶ j), range (F.map f) | def | category_theory.functor.eventual_range | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [] | The eventual range of the functor `F : J ⥤ Type v` at index `j : J` is the intersection
of the ranges of all maps `F.map f` with `i : J` and `f : i ⟶ j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_eventual_range_iff {x : F.obj j} :
x ∈ F.eventual_range j ↔ ∀ ⦃i⦄ (f : i ⟶ j), x ∈ range (F.map f) | mem_Inter₂ | lemma | category_theory.functor.mem_eventual_range_iff | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_mittag_leffler : Prop | ∀ j : J, ∃ i (f : i ⟶ j), ∀ ⦃k⦄ (g : k ⟶ j), range (F.map f) ⊆ range (F.map g) | def | category_theory.functor.is_mittag_leffler | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [] | The functor `F : J ⥤ Type v` satisfies the Mittag-Leffler condition if for all `j : J`,
there exists some `i : J` and `f : i ⟶ j` such that for all `k : J` and `g : k ⟶ j`, the range
of `F.map f` is contained in that of `F.map g`;
in other words (see `is_mittag_leffler_iff_eventual_range`), the eventual range at `j` is... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_mittag_leffler_iff_eventual_range : F.is_mittag_leffler ↔
∀ j : J, ∃ i (f : i ⟶ j), F.eventual_range j = range (F.map f) | forall_congr $ λ j, exists₂_congr $ λ i f,
⟨λ h, (Inter₂_subset _ _).antisymm $ subset_Inter₂ h, λ h, h ▸ Inter₂_subset⟩ | lemma | category_theory.functor.is_mittag_leffler_iff_eventual_range | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [
"exists₂_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_mittag_leffler.subset_image_eventual_range (h : F.is_mittag_leffler) (f : j ⟶ i) :
F.eventual_range i ⊆ F.map f '' (F.eventual_range j) | begin
obtain ⟨k, g, hg⟩ := F.is_mittag_leffler_iff_eventual_range.1 h j,
rw hg, intros x hx,
obtain ⟨x, rfl⟩ := F.mem_eventual_range_iff.1 hx (g ≫ f),
refine ⟨_, ⟨x, rfl⟩, by simpa only [F.map_comp]⟩,
end | lemma | category_theory.functor.is_mittag_leffler.subset_image_eventual_range | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventual_range_eq_range_precomp (f : i ⟶ j) (g : j ⟶ k)
(h : F.eventual_range k = range (F.map g)) :
F.eventual_range k = range (F.map $ f ≫ g) | begin
apply subset_antisymm,
{ apply Inter₂_subset, },
{ rw [h, F.map_comp], apply range_comp_subset_range, }
end | lemma | category_theory.functor.eventual_range_eq_range_precomp | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [
"subset_antisymm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_mittag_leffler_of_surjective
(h : ∀ ⦃i j : J⦄ (f :i ⟶ j), (F.map f).surjective) : F.is_mittag_leffler | λ j, ⟨j, 𝟙 j, λ k g, by rw [map_id, types_id, range_id, (h g).range_eq]⟩ | lemma | category_theory.functor.is_mittag_leffler_of_surjective | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_preimages : J ⥤ Type v | { obj := λ j, ⋂ f : j ⟶ i, F.map f ⁻¹' s,
map := λ j k g, maps_to.restrict (F.map g) _ _ $ λ x h, begin
rw [mem_Inter] at h ⊢, intro f,
rw [← mem_preimage, preimage_preimage],
convert h (g ≫ f), rw F.map_comp, refl,
end,
map_id' := λ j, by { simp_rw F.map_id, ext, refl },
map_comp' := λ j k l f g, b... | def | category_theory.functor.to_preimages | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [] | The subfunctor of `F` obtained by restricting to the preimages of a set `s ∈ F.obj i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_preimages_finite [∀ j, finite (F.obj j)] :
∀ j, finite ((F.to_preimages s).obj j) | λ j, subtype.finite | instance | category_theory.functor.to_preimages_finite | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [
"finite",
"subtype.finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventual_range_maps_to (f : j ⟶ i) :
(F.eventual_range j).maps_to (F.map f) (F.eventual_range i) | λ x hx, begin
rw mem_eventual_range_iff at hx ⊢,
intros k f',
obtain ⟨l, g, g', he⟩ := cospan f f',
obtain ⟨x, rfl⟩ := hx g,
rw [← map_comp_apply, he, F.map_comp],
exact ⟨_, rfl⟩,
end | lemma | category_theory.functor.eventual_range_maps_to | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_mittag_leffler.eq_image_eventual_range (h : F.is_mittag_leffler) (f : j ⟶ i) :
F.eventual_range i = F.map f '' (F.eventual_range j) | (h.subset_image_eventual_range F f).antisymm $ maps_to'.1 (F.eventual_range_maps_to f) | lemma | category_theory.functor.is_mittag_leffler.eq_image_eventual_range | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventual_range_eq_iff {f : i ⟶ j} :
F.eventual_range j = range (F.map f) ↔
∀ ⦃k⦄ (g : k ⟶ i), range (F.map f) ⊆ range (F.map $ g ≫ f) | begin
rw [subset_antisymm_iff, eventual_range, and_iff_right (Inter₂_subset _ _), subset_Inter₂_iff],
refine ⟨λ h k g, h _ _, λ h j' f', _⟩,
obtain ⟨k, g, g', he⟩ := cospan f f',
refine (h g).trans _,
rw [he, F.map_comp],
apply range_comp_subset_range,
end | lemma | category_theory.functor.eventual_range_eq_iff | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [
"subset_antisymm_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_mittag_leffler_iff_subset_range_comp : F.is_mittag_leffler ↔
∀ j : J, ∃ i (f : i ⟶ j), ∀ ⦃k⦄ (g : k ⟶ i), range (F.map f) ⊆ range (F.map $ g ≫ f) | by simp_rw [is_mittag_leffler_iff_eventual_range, eventual_range_eq_iff] | lemma | category_theory.functor.is_mittag_leffler_iff_subset_range_comp | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_mittag_leffler.to_preimages (h : F.is_mittag_leffler) :
(F.to_preimages s).is_mittag_leffler | (is_mittag_leffler_iff_subset_range_comp _).2 $ λ j, begin
obtain ⟨j₁, g₁, f₁, -⟩ := cone_objs i j,
obtain ⟨j₂, f₂, h₂⟩ := F.is_mittag_leffler_iff_eventual_range.1 h j₁,
refine ⟨j₂, f₂ ≫ f₁, λ j₃ f₃, _⟩,
rintro _ ⟨⟨x, hx⟩, rfl⟩,
have : F.map f₂ x ∈ F.eventual_range j₁, { rw h₂, exact ⟨_, rfl⟩ },
obtain ⟨y, ... | lemma | category_theory.functor.is_mittag_leffler.to_preimages | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [
"subtype.coe_mk",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_mittag_leffler_of_exists_finite_range
(h : ∀ (j : J), ∃ i (f : i ⟶ j), (range $ F.map f).finite) :
F.is_mittag_leffler | λ j, begin
obtain ⟨i, hi, hf⟩ := h j,
obtain ⟨m, ⟨i, f, hm⟩, hmin⟩ := finset.is_well_founded_lt.wf.has_min
{s : finset (F.obj j) | ∃ i (f : i ⟶ j), ↑s = range (F.map f)} ⟨_, i, hi, hf.coe_to_finset⟩,
refine ⟨i, f, λ k g,
(directed_on_range.mp $ F.ranges_directed j).is_bot_of_is_min ⟨⟨i, f⟩, rfl⟩ _ _ ⟨⟨k, ... | lemma | category_theory.functor.is_mittag_leffler_of_exists_finite_range | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [
"eq_of_le_of_not_lt",
"finite",
"finset",
"finset.coe_ssubset",
"finset.lt_iff_ssubset",
"set.finite.coe_to_finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_eventual_ranges : J ⥤ Type v | { obj := λ j, F.eventual_range j,
map := λ i j f, (F.eventual_range_maps_to f).restrict _ _ _,
map_id' := λ i, by { simp_rw F.map_id, ext, refl },
map_comp' := λ _ _ _ _ _, by { simp_rw F.map_comp, refl } } | def | category_theory.functor.to_eventual_ranges | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [] | The subfunctor of `F` obtained by restricting to the eventual range at each index. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_eventual_ranges_finite [∀ j, finite (F.obj j)] :
∀ j, finite (F.to_eventual_ranges.obj j) | λ j, subtype.finite | instance | category_theory.functor.to_eventual_ranges_finite | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [
"finite",
"subtype.finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_eventual_ranges_sections_equiv : F.to_eventual_ranges.sections ≃ F.sections | { to_fun := λ s, ⟨_, λ i j f, subtype.coe_inj.2 $ s.prop f⟩,
inv_fun := λ s, ⟨λ j, ⟨_, mem_Inter₂.2 $ λ i f, ⟨_, s.prop f⟩⟩, λ i j f, subtype.ext $ s.prop f⟩,
left_inv := λ _, by { ext, refl },
right_inv := λ _, by { ext, refl } } | def | category_theory.functor.to_eventual_ranges_sections_equiv | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [
"inv_fun",
"subtype.ext"
] | The sections of the functor `F : J ⥤ Type v` are in bijection with the sections of
`F.eventual_ranges`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
surjective_to_eventual_ranges (h : F.is_mittag_leffler) ⦃i j⦄ (f : i ⟶ j) :
(F.to_eventual_ranges.map f).surjective | λ ⟨x, hx⟩, by { obtain ⟨y, hy, rfl⟩ := h.subset_image_eventual_range F f hx, exact ⟨⟨y, hy⟩, rfl⟩ } | lemma | category_theory.functor.surjective_to_eventual_ranges | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [] | If `F` satisfies the Mittag-Leffler condition, its restriction to eventual ranges is a surjective
functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_eventual_ranges_nonempty (h : F.is_mittag_leffler) [∀ (j : J), nonempty (F.obj j)]
(j : J) : nonempty (F.to_eventual_ranges.obj j) | let ⟨i, f, h⟩ := F.is_mittag_leffler_iff_eventual_range.1 h j in
by { rw [to_eventual_ranges_obj, h], apply_instance } | lemma | category_theory.functor.to_eventual_ranges_nonempty | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [] | If `F` is nonempty at each index and Mittag-Leffler, then so is `F.to_eventual_ranges`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
thin_diagram_of_surjective (Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), (F.map f).surjective)
{i j} (f g : i ⟶ j) : F.map f = F.map g | let ⟨k, φ, hφ⟩ := cone_maps f g in
(Fsur φ).injective_comp_right $ by simp_rw [← types_comp, ← F.map_comp, hφ] | lemma | category_theory.functor.thin_diagram_of_surjective | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [] | If `F` has all arrows surjective, then it "factors through a poset". | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_preimages_nonempty_of_surjective [hFn : ∀ (j : J), nonempty (F.obj j)]
(Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), (F.map f).surjective)
(hs : s.nonempty) (j) : nonempty ((F.to_preimages s).obj j) | begin
simp only [to_preimages_obj, nonempty_coe_sort, nonempty_Inter, mem_preimage],
obtain (h|⟨⟨ji⟩⟩) := is_empty_or_nonempty (j ⟶ i),
{ exact ⟨(hFn j).some, λ ji, h.elim ji⟩, },
{ obtain ⟨y, ys⟩ := hs,
obtain ⟨x, rfl⟩ := Fsur ji y,
exact ⟨x, λ ji', (F.thin_diagram_of_surjective Fsur ji' ji).symm ▸ ys⟩... | lemma | category_theory.functor.to_preimages_nonempty_of_surjective | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [
"is_empty_or_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval_section_injective_of_eventually_injective
{j} (Finj : ∀ i (f : i ⟶ j), (F.map f).injective) (i) (f : i ⟶ j) :
(λ s : F.sections, s.val j).injective | begin
refine λ s₀ s₁ h, subtype.ext $ funext $ λ k, _,
obtain ⟨m, mi, mk, _⟩ := cone_objs i k,
dsimp at h,
rw [←s₀.prop (mi ≫ f), ←s₁.prop (mi ≫ f)] at h,
rw [←s₀.prop mk, ←s₁.prop mk],
refine congr_arg _ (Finj m (mi ≫ f) h),
end | lemma | category_theory.functor.eval_section_injective_of_eventually_injective | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval_section_surjective_of_surjective (i : J) :
(λ s : F.sections, s.val i).surjective | λ x,
begin
let s : set (F.obj i) := {x},
haveI := F.to_preimages_nonempty_of_surjective s Fsur (singleton_nonempty x),
obtain ⟨sec, h⟩ := nonempty_sections_of_finite_cofiltered_system (F.to_preimages s),
refine ⟨⟨λ j, (sec j).val, λ j k jk, by simpa [subtype.ext_iff] using h jk⟩, _⟩,
{ have := (sec i).prop,
... | lemma | category_theory.functor.eval_section_surjective_of_surjective | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [
"nonempty_sections_of_finite_cofiltered_system",
"subtype.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_injective [nonempty J] [finite F.sections] :
∃ j, ∀ i (f : i ⟶ j), (F.map f).injective | begin
haveI : ∀ j, fintype (F.obj j) := λ j, fintype.of_finite (F.obj j),
haveI : fintype F.sections := fintype.of_finite F.sections,
have card_le : ∀ j, fintype.card (F.obj j) ≤ fintype.card F.sections :=
λ j, fintype.card_le_of_surjective _ (F.eval_section_surjective_of_surjective Fsur j),
let fn := λ j, ... | lemma | category_theory.functor.eventually_injective | category_theory | src/category_theory/cofiltered_system.lean | [
"category_theory.filtered",
"data.set.finite",
"topology.category.Top.limits.konig"
] | [
"finite",
"fintype",
"fintype.bijective_iff_surjective_and_card",
"fintype.card",
"fintype.card_le_of_surjective",
"fintype.of_finite",
"tsub_le_iff_tsub_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comma (L : A ⥤ T) (R : B ⥤ T) : Type (max u₁ u₂ v₃) | (left : A)
(right : B)
(hom : L.obj left ⟶ R.obj right) | structure | category_theory.comma | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | The objects of the comma category are triples of an object `left : A`, an object
`right : B` and a morphism `hom : L.obj left ⟶ R.obj right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comma.inhabited [inhabited T] : inhabited (comma (𝟭 T) (𝟭 T)) | { default :=
{ left := default,
right := default,
hom := 𝟙 default } } | instance | category_theory.comma.inhabited | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comma_morphism (X Y : comma L R) | (left : X.left ⟶ Y.left)
(right : X.right ⟶ Y.right)
(w' : L.map left ≫ Y.hom = X.hom ≫ R.map right . obviously) | structure | category_theory.comma_morphism | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | A morphism between two objects in the comma category is a commutative square connecting the
morphisms coming from the two objects using morphisms in the image of the functors `L` and `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comma_morphism.inhabited [inhabited (comma L R)] :
inhabited (comma_morphism (default : comma L R) default) | ⟨⟨𝟙 _, 𝟙 _⟩⟩ | instance | category_theory.comma_morphism.inhabited | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comma_category : category (comma L R) | { hom := comma_morphism,
id := λ X,
{ left := 𝟙 X.left,
right := 𝟙 X.right },
comp := λ X Y Z f g,
{ left := f.left ≫ g.left,
right := f.right ≫ g.right } } | instance | category_theory.comma_category | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_left : ((𝟙 X) : comma_morphism X X).left = 𝟙 X.left | rfl | lemma | category_theory.comma.id_left | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_right : ((𝟙 X) : comma_morphism X X).right = 𝟙 X.right | rfl | lemma | category_theory.comma.id_right | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_left : (f ≫ g).left = f.left ≫ g.left | rfl | lemma | category_theory.comma.comp_left | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_right : (f ≫ g).right = f.right ≫ g.right | rfl | lemma | category_theory.comma.comp_right | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst : comma L R ⥤ A | { obj := λ X, X.left,
map := λ _ _ f, f.left } | def | category_theory.comma.fst | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | The functor sending an object `X` in the comma category to `X.left`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd : comma L R ⥤ B | { obj := λ X, X.right,
map := λ _ _ f, f.right } | def | category_theory.comma.snd | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | The functor sending an object `X` in the comma category to `X.right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans : fst L R ⋙ L ⟶ snd L R ⋙ R | { app := λ X, X.hom } | def | category_theory.comma.nat_trans | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | We can interpret the commutative square constituting a morphism in the comma category as a
natural transformation between the functors `fst ⋙ L` and `snd ⋙ R` from the comma category
to `T`, where the components are given by the morphism that constitutes an object of the comma
category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_to_hom_left (X Y : comma L R) (H : X = Y) :
comma_morphism.left (eq_to_hom H) = eq_to_hom (by { cases H, refl }) | by { cases H, refl } | lemma | category_theory.comma.eq_to_hom_left | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_to_hom_right (X Y : comma L R) (H : X = Y) :
comma_morphism.right (eq_to_hom H) = eq_to_hom (by { cases H, refl }) | by { cases H, refl } | lemma | category_theory.comma.eq_to_hom_right | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_mk {X Y : comma L₁ R₁} (l : X.left ≅ Y.left) (r : X.right ≅ Y.right)
(h : L₁.map l.hom ≫ Y.hom = X.hom ≫ R₁.map r.hom) : X ≅ Y | { hom := { left := l.hom, right := r.hom },
inv :=
{ left := l.inv,
right := r.inv,
w' := begin
rw [←L₁.map_iso_inv l, iso.inv_comp_eq, L₁.map_iso_hom, reassoc_of h, ← R₁.map_comp],
simp
end, } } | def | category_theory.comma.iso_mk | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | Construct an isomorphism in the comma category given isomorphisms of the objects whose forward
directions give a commutative square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_left (l : L₁ ⟶ L₂) : comma L₂ R ⥤ comma L₁ R | { obj := λ X,
{ left := X.left,
right := X.right,
hom := l.app X.left ≫ X.hom },
map := λ X Y f,
{ left := f.left,
right := f.right } } | def | category_theory.comma.map_left | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | A natural transformation `L₁ ⟶ L₂` induces a functor `comma L₂ R ⥤ comma L₁ R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_left_id : map_left R (𝟙 L) ≅ 𝟭 _ | { hom :=
{ app := λ X, { left := 𝟙 _, right := 𝟙 _ } },
inv :=
{ app := λ X, { left := 𝟙 _, right := 𝟙 _ } } } | def | category_theory.comma.map_left_id | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | The functor `comma L R ⥤ comma L R` induced by the identity natural transformation on `L` is
naturally isomorphic to the identity functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_left_comp (l : L₁ ⟶ L₂) (l' : L₂ ⟶ L₃) :
(map_left R (l ≫ l')) ≅ (map_left R l') ⋙ (map_left R l) | { hom :=
{ app := λ X, { left := 𝟙 _, right := 𝟙 _ } },
inv :=
{ app := λ X, { left := 𝟙 _, right := 𝟙 _ } } } | def | category_theory.comma.map_left_comp | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | The functor `comma L₁ R ⥤ comma L₃ R` induced by the composition of two natural transformations
`l : L₁ ⟶ L₂` and `l' : L₂ ⟶ L₃` is naturally isomorphic to the composition of the two functors
induced by these natural transformations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_right (r : R₁ ⟶ R₂) : comma L R₁ ⥤ comma L R₂ | { obj := λ X,
{ left := X.left,
right := X.right,
hom := X.hom ≫ r.app X.right },
map := λ X Y f,
{ left := f.left,
right := f.right } } | def | category_theory.comma.map_right | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | A natural transformation `R₁ ⟶ R₂` induces a functor `comma L R₁ ⥤ comma L R₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_right_id : map_right L (𝟙 R) ≅ 𝟭 _ | { hom :=
{ app := λ X, { left := 𝟙 _, right := 𝟙 _ } },
inv :=
{ app := λ X, { left := 𝟙 _, right := 𝟙 _ } } } | def | category_theory.comma.map_right_id | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | The functor `comma L R ⥤ comma L R` induced by the identity natural transformation on `R` is
naturally isomorphic to the identity functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_right_comp (r : R₁ ⟶ R₂) (r' : R₂ ⟶ R₃) :
(map_right L (r ≫ r')) ≅ (map_right L r) ⋙ (map_right L r') | { hom :=
{ app := λ X, { left := 𝟙 _, right := 𝟙 _ } },
inv :=
{ app := λ X, { left := 𝟙 _, right := 𝟙 _ } } } | def | category_theory.comma.map_right_comp | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | The functor `comma L R₁ ⥤ comma L R₃` induced by the composition of the natural transformations
`r : R₁ ⟶ R₂` and `r' : R₂ ⟶ R₃` is naturally isomorphic to the composition of the functors
induced by these natural transformations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pre_left (F: C ⥤ A) (L : A ⥤ T) (R : B ⥤ T) : comma (F ⋙ L) R ⥤ comma L R | { obj := λ X, { left := F.obj X.left, right := X.right, hom := X.hom },
map := λ X Y f, { left := F.map f.left, right := f.right, w' := by simpa using f.w } } | def | category_theory.comma.pre_left | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | The functor `(F ⋙ L, R) ⥤ (L, R)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pre_right (L : A ⥤ T) (F: C ⥤ B) (R : B ⥤ T) : comma L (F ⋙ R) ⥤ comma L R | { obj := λ X, { left := X.left, right := F.obj X.right, hom := X.hom },
map := λ X Y f, { left := f.left, right := F.map f.right, w' := by simp } } | def | category_theory.comma.pre_right | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | The functor `(F ⋙ L, R) ⥤ (L, R)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
post (L : A ⥤ T) (R : B ⥤ T) (F: T ⥤ C) : comma L R ⥤ comma (L ⋙ F) (R ⋙ F) | { obj := λ X, { left := X.left, right := X.right, hom := F.map X.hom },
map := λ X Y f, { left := f.left, right := f.right, w' :=
by { simp only [functor.comp_map, ←F.map_comp, f.w] } } } | def | category_theory.comma.post | category_theory | src/category_theory/comma.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.eq_to_hom"
] | [] | The functor `(L, R) ⥤ (L ⋙ F, R ⋙ F)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_sq {W X Y Z : C} (f : W ⟶ X) (g : W ⟶ Y) (h : X ⟶ Z) (i : Y ⟶ Z) : Prop | (w : f ≫ h = g ≫ i) | structure | category_theory.comm_sq | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [] | The proposition that a square
```
W ---f---> X
| |
g h
| |
v v
Y ---i---> Z
```
is a commuting square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
flip (p : comm_sq f g h i) : comm_sq g f i h | ⟨p.w.symm⟩ | lemma | category_theory.comm_sq.flip | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_arrow {f g : arrow C} (h : f ⟶ g) : comm_sq f.hom h.left h.right g.hom | ⟨h.w.symm⟩ | lemma | category_theory.comm_sq.of_arrow | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op (p : comm_sq f g h i) : comm_sq i.op h.op g.op f.op | ⟨by simp only [← op_comp, p.w]⟩ | lemma | category_theory.comm_sq.op | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [] | The commutative square in the opposite category associated to a commutative square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop {W X Y Z : Cᵒᵖ} {f : W ⟶ X} {g : W ⟶ Y} {h : X ⟶ Z} {i : Y ⟶ Z}
(p : comm_sq f g h i) : comm_sq i.unop h.unop g.unop f.unop | ⟨by simp only [← unop_comp, p.w]⟩ | lemma | category_theory.comm_sq.unop | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [] | The commutative square associated to a commutative square in the opposite category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_comm_sq (s : comm_sq f g h i) : comm_sq (F.map f) (F.map g) (F.map h) (F.map i) | ⟨by simpa using congr_arg (λ k : W ⟶ Z, F.map k) s.w⟩ | lemma | category_theory.functor.map_comm_sq | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_struct (sq : comm_sq f i p g) | (l : B ⟶ X) (fac_left' : i ≫ l = f) (fac_right' : l ≫ p = g) | structure | category_theory.comm_sq.lift_struct | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [] | The datum of a lift in a commutative square, i.e. a up-right-diagonal
morphism which makes both triangles commute. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op {sq : comm_sq f i p g} (l : lift_struct sq) : lift_struct sq.op | { l := l.l.op,
fac_left' := by rw [← op_comp, l.fac_right],
fac_right' := by rw [← op_comp, l.fac_left], } | def | category_theory.comm_sq.lift_struct.op | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [] | A `lift_struct` for a commutative square gives a `lift_struct` for the
corresponding square in the opposite category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop {A B X Y : Cᵒᵖ} {f : A ⟶ X} {i : A ⟶ B} {p : X ⟶ Y} {g : B ⟶ Y} {sq : comm_sq f i p g}
(l : lift_struct sq) : lift_struct sq.unop | { l := l.l.unop,
fac_left' := by rw [← unop_comp, l.fac_right],
fac_right' := by rw [← unop_comp, l.fac_left], } | def | category_theory.comm_sq.lift_struct.unop | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [] | A `lift_struct` for a commutative square in the opposite category
gives a `lift_struct` for the corresponding square in the original category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_equiv (sq : comm_sq f i p g) : lift_struct sq ≃ lift_struct sq.op | { to_fun := op,
inv_fun := unop,
left_inv := by tidy,
right_inv := by tidy, } | def | category_theory.comm_sq.lift_struct.op_equiv | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [
"inv_fun"
] | Equivalences of `lift_struct` for a square and the corresponding square
in the opposite category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_equiv {A B X Y : Cᵒᵖ} {f : A ⟶ X} {i : A ⟶ B} {p : X ⟶ Y} {g : B ⟶ Y}
(sq : comm_sq f i p g) : lift_struct sq ≃ lift_struct sq.unop | { to_fun := unop,
inv_fun := op,
left_inv := by tidy,
right_inv := by tidy, } | def | category_theory.comm_sq.lift_struct.unop_equiv | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [
"inv_fun"
] | Equivalences of `lift_struct` for a square in the oppositive category and
the corresponding square in the original category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsingleton_lift_struct_of_epi (sq : comm_sq f i p g) [epi i] :
subsingleton (lift_struct sq) | ⟨λ l₁ l₂, by { ext, simp only [← cancel_epi i, lift_struct.fac_left], }⟩ | instance | category_theory.comm_sq.subsingleton_lift_struct_of_epi | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_lift_struct_of_mono (sq : comm_sq f i p g) [mono p] :
subsingleton (lift_struct sq) | ⟨λ l₁ l₂, by { ext, simp only [← cancel_mono p, lift_struct.fac_right], }⟩ | instance | category_theory.comm_sq.subsingleton_lift_struct_of_mono | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_lift : Prop | (exists_lift : nonempty sq.lift_struct) | class | category_theory.comm_sq.has_lift | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [] | The assertion that a square has a `lift_struct`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk' (l : sq.lift_struct) : has_lift sq | ⟨nonempty.intro l⟩ | lemma | category_theory.comm_sq.has_lift.mk' | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff : has_lift sq ↔ nonempty sq.lift_struct | by { split, exacts [λ h, h.exists_lift, λ h, mk h], } | lemma | category_theory.comm_sq.has_lift.iff | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_op : has_lift sq ↔ has_lift sq.op | begin
rw [iff, iff],
exact nonempty.congr (lift_struct.op_equiv sq).to_fun (lift_struct.op_equiv sq).inv_fun,
end | lemma | category_theory.comm_sq.has_lift.iff_op | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [
"inv_fun",
"nonempty.congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_unop {A B X Y : Cᵒᵖ} {f : A ⟶ X} {i : A ⟶ B} {p : X ⟶ Y} {g : B ⟶ Y}
(sq : comm_sq f i p g) : has_lift sq ↔ has_lift sq.unop | begin
rw [iff, iff],
exact nonempty.congr (lift_struct.unop_equiv sq).to_fun (lift_struct.unop_equiv sq).inv_fun,
end | lemma | category_theory.comm_sq.has_lift.iff_unop | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [
"inv_fun",
"nonempty.congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift [hsq : has_lift sq] : B ⟶ X | hsq.exists_lift.some.l | def | category_theory.comm_sq.lift | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [
"lift"
] | A choice of a diagonal morphism that is part of a `lift_struct` when
the square has a lift. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fac_left [hsq : has_lift sq] : i ≫ sq.lift = f | hsq.exists_lift.some.fac_left | lemma | category_theory.comm_sq.fac_left | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fac_right [hsq : has_lift sq] : sq.lift ≫ p = g | hsq.exists_lift.some.fac_right | lemma | category_theory.comm_sq.fac_right | category_theory | src/category_theory/comm_sq.lean | [
"category_theory.arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_congr {X Y X₁ Y₁ : C} (α : X ≅ X₁) (β : Y ≅ Y₁) :
(X ⟶ Y) ≃ (X₁ ⟶ Y₁) | { to_fun := λ f, α.inv ≫ f ≫ β.hom,
inv_fun := λ f, α.hom ≫ f ≫ β.inv,
left_inv := λ f, show α.hom ≫ (α.inv ≫ f ≫ β.hom) ≫ β.inv = f,
by rw [category.assoc, category.assoc, β.hom_inv_id, α.hom_inv_id_assoc, category.comp_id],
right_inv := λ f, show α.inv ≫ (α.hom ≫ f ≫ β.inv) ≫ β.hom = f,
by rw [category.... | def | category_theory.iso.hom_congr | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [
"inv_fun"
] | If `X` is isomorphic to `X₁` and `Y` is isomorphic to `Y₁`, then
there is a natural bijection between `X ⟶ Y` and `X₁ ⟶ Y₁`. See also `equiv.arrow_congr`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_congr_apply {X Y X₁ Y₁ : C} (α : X ≅ X₁) (β : Y ≅ Y₁) (f : X ⟶ Y) :
α.hom_congr β f = α.inv ≫ f ≫ β.hom | rfl | lemma | category_theory.iso.hom_congr_apply | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_congr_comp {X Y Z X₁ Y₁ Z₁ : C} (α : X ≅ X₁) (β : Y ≅ Y₁) (γ : Z ≅ Z₁)
(f : X ⟶ Y) (g : Y ⟶ Z) :
α.hom_congr γ (f ≫ g) = α.hom_congr β f ≫ β.hom_congr γ g | by simp | lemma | category_theory.iso.hom_congr_comp | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_congr_refl {X Y : C} (f : X ⟶ Y) :
(iso.refl X).hom_congr (iso.refl Y) f = f | by simp | lemma | category_theory.iso.hom_congr_refl | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_congr_trans {X₁ Y₁ X₂ Y₂ X₃ Y₃ : C}
(α₁ : X₁ ≅ X₂) (β₁ : Y₁ ≅ Y₂) (α₂ : X₂ ≅ X₃) (β₂ : Y₂ ≅ Y₃) (f : X₁ ⟶ Y₁) :
(α₁ ≪≫ α₂).hom_congr (β₁ ≪≫ β₂) f = (α₁.hom_congr β₁).trans (α₂.hom_congr β₂) f | by simp | lemma | category_theory.iso.hom_congr_trans | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_congr_symm {X₁ Y₁ X₂ Y₂ : C} (α : X₁ ≅ X₂) (β : Y₁ ≅ Y₂) :
(α.hom_congr β).symm = α.symm.hom_congr β.symm | rfl | lemma | category_theory.iso.hom_congr_symm | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj : End X ≃* End Y | { map_mul' := λ f g, hom_congr_comp α α α g f,
.. hom_congr α α } | def | category_theory.iso.conj | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | An isomorphism between two objects defines a monoid isomorphism between their
monoid of endomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conj_apply (f : End X) : α.conj f = α.inv ≫ f ≫ α.hom | rfl | lemma | category_theory.iso.conj_apply | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_comp (f g : End X) : α.conj (f ≫ g) = (α.conj f) ≫ (α.conj g) | α.conj.map_mul g f | lemma | category_theory.iso.conj_comp | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_id : α.conj (𝟙 X) = 𝟙 Y | α.conj.map_one | lemma | category_theory.iso.conj_id | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_conj (f : End X) : (iso.refl X).conj f = f | by rw [conj_apply, iso.refl_inv, iso.refl_hom, category.id_comp, category.comp_id] | lemma | category_theory.iso.refl_conj | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_conj {Z : C} (β : Y ≅ Z) (f : End X) : (α ≪≫ β).conj f = β.conj (α.conj f) | hom_congr_trans α α β β f | lemma | category_theory.iso.trans_conj | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_self_conj (f : End X) : α.symm.conj (α.conj f) = f | by rw [← trans_conj, α.self_symm_id, refl_conj] | lemma | category_theory.iso.symm_self_conj | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_symm_conj (f : End Y) : α.conj (α.symm.conj f) = f | α.symm.symm_self_conj f | lemma | category_theory.iso.self_symm_conj | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_pow (f : End X) (n : ℕ) : α.conj (f^n) = (α.conj f)^n | α.conj.to_monoid_hom.map_pow f n | lemma | category_theory.iso.conj_pow | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [
"conj_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_Aut : Aut X ≃* Aut Y | (Aut.units_End_equiv_Aut X).symm.trans $
(units.map_equiv α.conj).trans $
Aut.units_End_equiv_Aut Y | def | category_theory.iso.conj_Aut | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [
"units.map_equiv"
] | `conj` defines a group isomorphisms between groups of automorphisms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conj_Aut_apply (f : Aut X) : α.conj_Aut f = α.symm ≪≫ f ≪≫ α | by cases f; cases α; ext; refl | lemma | category_theory.iso.conj_Aut_apply | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_Aut_hom (f : Aut X) : (α.conj_Aut f).hom = α.conj f.hom | rfl | lemma | category_theory.iso.conj_Aut_hom | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_conj_Aut {Z : C} (β : Y ≅ Z) (f : Aut X) :
(α ≪≫ β).conj_Aut f = β.conj_Aut (α.conj_Aut f) | by simp only [conj_Aut_apply, iso.trans_symm, iso.trans_assoc] | lemma | category_theory.iso.trans_conj_Aut | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_Aut_mul (f g : Aut X) : α.conj_Aut (f * g) = α.conj_Aut f * α.conj_Aut g | α.conj_Aut.map_mul f g | lemma | category_theory.iso.conj_Aut_mul | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_Aut_trans (f g : Aut X) : α.conj_Aut (f ≪≫ g) = α.conj_Aut f ≪≫ α.conj_Aut g | conj_Aut_mul α g f | lemma | category_theory.iso.conj_Aut_trans | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_Aut_pow (f : Aut X) (n : ℕ) : α.conj_Aut (f^n) = (α.conj_Aut f)^n | α.conj_Aut.to_monoid_hom.map_pow f n | lemma | category_theory.iso.conj_Aut_pow | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_Aut_zpow (f : Aut X) (n : ℤ) : α.conj_Aut (f^n) = (α.conj_Aut f)^n | α.conj_Aut.to_monoid_hom.map_zpow f n | lemma | category_theory.iso.conj_Aut_zpow | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_hom_congr {X Y X₁ Y₁ : C} (α : X ≅ X₁) (β : Y ≅ Y₁) (f : X ⟶ Y) :
F.map (iso.hom_congr α β f) = iso.hom_congr (F.map_iso α) (F.map_iso β) (F.map f) | by simp | lemma | category_theory.functor.map_hom_congr | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_conj {X Y : C} (α : X ≅ Y) (f : End X) :
F.map (α.conj f) = (F.map_iso α).conj (F.map f) | map_hom_congr F α α f | lemma | category_theory.functor.map_conj | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_conj_Aut (F : C ⥤ D) {X Y : C} (α : X ≅ Y) (f : Aut X) :
F.map_iso (α.conj_Aut f) = (F.map_iso α).conj_Aut (F.map_iso f) | by ext; simp only [map_iso_hom, iso.conj_Aut_hom, F.map_conj] | lemma | category_theory.functor.map_conj_Aut | category_theory | src/category_theory/conj.lean | [
"algebra.hom.equiv.units.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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