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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isomorphism_classes : Cat.{v u} ⥤ Type u | { obj := λ C, quotient (is_isomorphic_setoid C.α),
map := λ C D F, quot.map F.obj $ λ X Y ⟨f⟩, ⟨F.map_iso f⟩ } | def | category_theory.isomorphism_classes | category_theory | src/category_theory/isomorphism_classes.lean | [
"category_theory.category.Cat",
"category_theory.groupoid",
"category_theory.types"
] | [
"quot.map"
] | The functor that sends each category to the quotient space of its objects up to an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
groupoid.is_isomorphic_iff_nonempty_hom {C : Type u} [groupoid.{v} C] {X Y : C} :
is_isomorphic X Y ↔ nonempty (X ⟶ Y) | (groupoid.iso_equiv_hom X Y).nonempty_congr | lemma | category_theory.groupoid.is_isomorphic_iff_nonempty_hom | category_theory | src/category_theory/isomorphism_classes.lean | [
"category_theory.category.Cat",
"category_theory.groupoid",
"category_theory.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected (J : Type u₁) [category.{v₁} J] : Prop | (iso_constant : Π {α : Type u₁} (F : J ⥤ discrete α) (j : J),
nonempty (F ≅ (functor.const J).obj (F.obj j))) | class | category_theory.is_preconnected | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"functor.const",
"is_preconnected"
] | A possibly empty category for which every functor to a discrete category is constant. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_connected (J : Type u₁) [category.{v₁} J] extends is_preconnected J : Prop | [is_nonempty : nonempty J] | class | category_theory.is_connected | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"is_connected",
"is_preconnected"
] | We define a connected category as a _nonempty_ category for which every
functor to a discrete category is constant.
NB. Some authors include the empty category as connected, we do not.
We instead are interested in categories with exactly one 'connected
component'.
This allows us to show that the functor X ⨯ - preserv... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_constant [is_preconnected J] {α : Type u₁} (F : J ⥤ discrete α) (j : J) :
F ≅ (functor.const J).obj (F.obj j) | (is_preconnected.iso_constant F j).some | def | category_theory.iso_constant | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"functor.const",
"is_preconnected"
] | If `J` is connected, any functor `F : J ⥤ discrete α` is isomorphic to
the constant functor with value `F.obj j` (for any choice of `j`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
any_functor_const_on_obj [is_preconnected J]
{α : Type u₁} (F : J ⥤ discrete α) (j j' : J) :
F.obj j = F.obj j' | by { ext, exact ((iso_constant F j').hom.app j).down.1 } | lemma | category_theory.any_functor_const_on_obj | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"is_preconnected"
] | If J is connected, any functor to a discrete category is constant on objects.
The converse is given in `is_connected.of_any_functor_const_on_obj`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_connected.of_any_functor_const_on_obj [nonempty J]
(h : ∀ {α : Type u₁} (F : J ⥤ discrete α), ∀ (j j' : J), F.obj j = F.obj j') :
is_connected J | { iso_constant := λ α F j',
⟨nat_iso.of_components (λ j, eq_to_iso (h F j j')) (λ _ _ _, subsingleton.elim _ _)⟩ } | lemma | category_theory.is_connected.of_any_functor_const_on_obj | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"is_connected"
] | If any functor to a discrete category is constant on objects, J is connected.
The converse of `any_functor_const_on_obj`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
constant_of_preserves_morphisms [is_preconnected J] {α : Type u₁} (F : J → α)
(h : ∀ (j₁ j₂ : J) (f : j₁ ⟶ j₂), F j₁ = F j₂) (j j' : J) :
F j = F j' | by simpa using any_functor_const_on_obj
{ obj := discrete.mk ∘ F,
map := λ _ _ f, eq_to_hom (by { ext, exact (h _ _ f), }) } j j' | lemma | category_theory.constant_of_preserves_morphisms | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"is_preconnected"
] | If `J` is connected, then given any function `F` such that the presence of a
morphism `j₁ ⟶ j₂` implies `F j₁ = F j₂`, we have that `F` is constant.
This can be thought of as a local-to-global property.
The converse is shown in `is_connected.of_constant_of_preserves_morphisms` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_connected.of_constant_of_preserves_morphisms [nonempty J]
(h : ∀ {α : Type u₁} (F : J → α), (∀ {j₁ j₂ : J} (f : j₁ ⟶ j₂), F j₁ = F j₂) →
(∀ j j' : J, F j = F j')) :
is_connected J | is_connected.of_any_functor_const_on_obj
(λ _ F, h F.obj (λ _ _ f, by { ext, exact discrete.eq_of_hom (F.map f) })) | lemma | category_theory.is_connected.of_constant_of_preserves_morphisms | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"is_connected"
] | `J` is connected if: given any function `F : J → α` which is constant for any
`j₁, j₂` for which there is a morphism `j₁ ⟶ j₂`, then `F` is constant.
This can be thought of as a local-to-global property.
The converse of `constant_of_preserves_morphisms`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
induct_on_objects [is_preconnected J] (p : set J) {j₀ : J} (h0 : j₀ ∈ p)
(h1 : ∀ {j₁ j₂ : J} (f : j₁ ⟶ j₂), j₁ ∈ p ↔ j₂ ∈ p) (j : J) :
j ∈ p | begin
injection (constant_of_preserves_morphisms (λ k, ulift.up (k ∈ p)) (λ j₁ j₂ f, _) j j₀) with i,
rwa i,
dsimp,
exact congr_arg ulift.up (propext (h1 f)),
end | lemma | category_theory.induct_on_objects | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"is_preconnected"
] | An inductive-like property for the objects of a connected category.
If the set `p` is nonempty, and `p` is closed under morphisms of `J`,
then `p` contains all of `J`.
The converse is given in `is_connected.of_induct`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_connected.of_induct [nonempty J] {j₀ : J}
(h : ∀ (p : set J), j₀ ∈ p → (∀ {j₁ j₂ : J} (f : j₁ ⟶ j₂), j₁ ∈ p ↔ j₂ ∈ p) → ∀ (j : J), j ∈ p) :
is_connected J | is_connected.of_constant_of_preserves_morphisms (λ α F a,
begin
have w := h {j | F j = F j₀} rfl (λ _ _ f, by simp [a f]),
dsimp at w,
intros j j',
rw [w j, w j'],
end) | lemma | category_theory.is_connected.of_induct | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"is_connected"
] | If any maximal connected component containing some element j₀ of J is all of J, then J is connected.
The converse of `induct_on_objects`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected_induction [is_preconnected J] (Z : J → Sort*)
(h₁ : Π {j₁ j₂ : J} (f : j₁ ⟶ j₂), Z j₁ → Z j₂)
(h₂ : Π {j₁ j₂ : J} (f : j₁ ⟶ j₂), Z j₂ → Z j₁)
{j₀ : J} (x : Z j₀) (j : J) : nonempty (Z j) | (induct_on_objects {j | nonempty (Z j)} ⟨x⟩
(λ j₁ j₂ f, ⟨by { rintro ⟨y⟩, exact ⟨h₁ f y⟩, }, by { rintro ⟨y⟩, exact ⟨h₂ f y⟩, }⟩) j : _) | lemma | category_theory.is_preconnected_induction | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"is_preconnected"
] | Another induction principle for `is_preconnected J`:
given a type family `Z : J → Sort*` and
a rule for transporting in *both* directions along a morphism in `J`,
we can transport an `x : Z j₀` to a point in `Z j` for any `j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected_of_equivalent {K : Type u₁} [category.{v₂} K] [is_preconnected J]
(e : J ≌ K) :
is_preconnected K | { iso_constant := λ α F k, ⟨
calc F ≅ e.inverse ⋙ e.functor ⋙ F : (e.inv_fun_id_assoc F).symm
... ≅ e.inverse ⋙ (functor.const J).obj ((e.functor ⋙ F).obj (e.inverse.obj k)) :
iso_whisker_left e.inverse (iso_constant (e.functor ⋙ F) (e.inverse.obj k))
... ≅ e.inverse ⋙ (functor.const... | lemma | category_theory.is_preconnected_of_equivalent | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"functor.const",
"is_preconnected"
] | If `J` and `K` are equivalent, then if `J` is preconnected then `K` is as well. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_connected_of_equivalent {K : Type u₁} [category.{v₂} K]
(e : J ≌ K) [is_connected J] :
is_connected K | { is_nonempty := nonempty.map e.functor.obj (by apply_instance),
to_is_preconnected := is_preconnected_of_equivalent e } | lemma | category_theory.is_connected_of_equivalent | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"is_connected",
"nonempty.map"
] | If `J` and `K` are equivalent, then if `J` is connected then `K` is as well. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected_op [is_preconnected J] : is_preconnected Jᵒᵖ | { iso_constant := λ α F X, ⟨nat_iso.of_components
(λ Y, eq_to_iso (discrete.ext _ _ (discrete.eq_of_hom ((nonempty.some
(is_preconnected.iso_constant (F.right_op ⋙ (discrete.opposite α).functor) (unop X))).app
(unop Y)).hom)))
(λ Y Z f, subsingleton.elim _ _)⟩ } | instance | category_theory.is_preconnected_op | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"is_preconnected",
"nonempty.some"
] | If `J` is preconnected, then `Jᵒᵖ` is preconnected as well. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_connected_op [is_connected J] : is_connected Jᵒᵖ | { is_nonempty := nonempty.intro (op (classical.arbitrary J)) } | instance | category_theory.is_connected_op | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"classical.arbitrary",
"is_connected"
] | If `J` is connected, then `Jᵒᵖ` is connected as well. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected_of_is_preconnected_op [is_preconnected Jᵒᵖ] : is_preconnected J | is_preconnected_of_equivalent (op_op_equivalence J) | lemma | category_theory.is_preconnected_of_is_preconnected_op | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_of_is_connected_op [is_connected Jᵒᵖ] : is_connected J | is_connected_of_equivalent (op_op_equivalence J) | lemma | category_theory.is_connected_of_is_connected_op | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"is_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zag (j₁ j₂ : J) : Prop | nonempty (j₁ ⟶ j₂) ∨ nonempty (j₂ ⟶ j₁) | def | category_theory.zag | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [] | j₁ and j₂ are related by `zag` if there is a morphism between them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zag_symmetric : symmetric (@zag J _) | λ j₂ j₁ h, h.swap | lemma | category_theory.zag_symmetric | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zigzag : J → J → Prop | relation.refl_trans_gen zag | def | category_theory.zigzag | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"relation.refl_trans_gen"
] | `j₁` and `j₂` are related by `zigzag` if there is a chain of
morphisms from `j₁` to `j₂`, with backward morphisms allowed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zigzag_symmetric : symmetric (@zigzag J _) | relation.refl_trans_gen.symmetric zag_symmetric | lemma | category_theory.zigzag_symmetric | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"relation.refl_trans_gen.symmetric"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zigzag_equivalence : _root_.equivalence (@zigzag J _) | mk_equivalence _
relation.reflexive_refl_trans_gen
zigzag_symmetric
relation.transitive_refl_trans_gen | lemma | category_theory.zigzag_equivalence | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"relation.reflexive_refl_trans_gen",
"relation.transitive_refl_trans_gen"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zigzag.setoid (J : Type u₂) [category.{v₁} J] : setoid J | { r := zigzag,
iseqv := zigzag_equivalence } | def | category_theory.zigzag.setoid | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [] | The setoid given by the equivalence relation `zigzag`. A quotient for this
setoid is a connected component of the category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zigzag_obj_of_zigzag (F : J ⥤ K) {j₁ j₂ : J} (h : zigzag j₁ j₂) :
zigzag (F.obj j₁) (F.obj j₂) | h.lift _ $ λ j k, or.imp (nonempty.map (λ f, F.map f)) (nonempty.map (λ f, F.map f)) | lemma | category_theory.zigzag_obj_of_zigzag | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"nonempty.map"
] | If there is a zigzag from `j₁` to `j₂`, then there is a zigzag from `F j₁` to
`F j₂` as long as `F` is a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zag_of_zag_obj (F : J ⥤ K) [full F] {j₁ j₂ : J} (h : zag (F.obj j₁) (F.obj j₂)) :
zag j₁ j₂ | or.imp (nonempty.map F.preimage) (nonempty.map F.preimage) h | lemma | category_theory.zag_of_zag_obj | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"nonempty.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_relation [is_connected J] (r : J → J → Prop) (hr : _root_.equivalence r)
(h : ∀ {j₁ j₂ : J} (f : j₁ ⟶ j₂), r j₁ j₂) :
∀ (j₁ j₂ : J), r j₁ j₂ | begin
have z : ∀ (j : J), r (classical.arbitrary J) j :=
induct_on_objects (λ k, r (classical.arbitrary J) k)
(hr.1 (classical.arbitrary J)) (λ _ _ f, ⟨λ t, hr.2.2 t (h f), λ t, hr.2.2 t (hr.2.1 (h f))⟩),
intros, apply hr.2.2 (hr.2.1 (z _)) (z _)
end | lemma | category_theory.equiv_relation | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"classical.arbitrary",
"is_connected"
] | Any equivalence relation containing (⟶) holds for all pairs of a connected category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_connected_zigzag [is_connected J] (j₁ j₂ : J) : zigzag j₁ j₂ | equiv_relation _ zigzag_equivalence
(λ _ _ f, relation.refl_trans_gen.single (or.inl (nonempty.intro f))) _ _ | lemma | category_theory.is_connected_zigzag | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"is_connected",
"relation.refl_trans_gen.single"
] | In a connected category, any two objects are related by `zigzag`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zigzag_is_connected [nonempty J] (h : ∀ (j₁ j₂ : J), zigzag j₁ j₂) : is_connected J | begin
apply is_connected.of_induct,
intros p hp hjp j,
have: ∀ (j₁ j₂ : J), zigzag j₁ j₂ → (j₁ ∈ p ↔ j₂ ∈ p),
{ introv k,
induction k with _ _ rt_zag zag,
{ refl },
{ rw k_ih,
rcases zag with ⟨⟨_⟩⟩ | ⟨⟨_⟩⟩,
apply hjp zag,
apply (hjp zag).symm } },
rwa this j (classical.arbitrary ... | lemma | category_theory.zigzag_is_connected | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"classical.arbitrary",
"is_connected"
] | If any two objects in an nonempty category are related by `zigzag`, the category is connected. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_zigzag' [is_connected J] (j₁ j₂ : J) :
∃ l, list.chain zag j₁ l ∧ list.last (j₁ :: l) (list.cons_ne_nil _ _) = j₂ | list.exists_chain_of_relation_refl_trans_gen (is_connected_zigzag _ _) | lemma | category_theory.exists_zigzag' | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"is_connected",
"list.chain",
"list.cons_ne_nil",
"list.exists_chain_of_relation_refl_trans_gen"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_of_zigzag [nonempty J]
(h : ∀ (j₁ j₂ : J), ∃ l, list.chain zag j₁ l ∧ list.last (j₁ :: l) (list.cons_ne_nil _ _) = j₂) :
is_connected J | begin
apply zigzag_is_connected,
intros j₁ j₂,
rcases h j₁ j₂ with ⟨l, hl₁, hl₂⟩,
apply list.relation_refl_trans_gen_of_exists_chain l hl₁ hl₂,
end | lemma | category_theory.is_connected_of_zigzag | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"is_connected",
"list.chain",
"list.cons_ne_nil",
"list.relation_refl_trans_gen_of_exists_chain"
] | If any two objects in an nonempty category are linked by a sequence of (potentially reversed)
morphisms, then J is connected.
The converse of `exists_zigzag'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discrete_is_connected_equiv_punit {α : Type u₁} [is_connected (discrete α)] : α ≃ punit | discrete.equiv_of_equivalence.{u₁ u₁}
{ functor := functor.star (discrete α),
inverse := discrete.functor (λ _, classical.arbitrary _),
unit_iso := by { exact (iso_constant _ (classical.arbitrary _)), },
counit_iso := functor.punit_ext _ _ } | def | category_theory.discrete_is_connected_equiv_punit | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"classical.arbitrary",
"is_connected"
] | If `discrete α` is connected, then `α` is (type-)equivalent to `punit`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans_from_is_connected [is_preconnected J] {X Y : C}
(α : (functor.const J).obj X ⟶ (functor.const J).obj Y) :
∀ (j j' : J), α.app j = (α.app j' : X ⟶ Y) | @constant_of_preserves_morphisms _ _ _
(X ⟶ Y)
(λ j, α.app j)
(λ _ _ f, (by { have := α.naturality f, erw [id_comp, comp_id] at this, exact this.symm })) | lemma | category_theory.nat_trans_from_is_connected | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"functor.const",
"is_preconnected"
] | For objects `X Y : C`, any natural transformation `α : const X ⟶ const Y` from a connected
category must be constant.
This is the key property of connected categories which we use to establish properties about limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_hom_of_connected_groupoid {G} [groupoid G] [is_connected G] :
∀ (x y : G), nonempty (x ⟶ y) | begin
refine equiv_relation _ _ (λ j₁ j₂, nonempty.intro),
exact ⟨λ j, ⟨𝟙 _⟩, λ j₁ j₂, nonempty.map (λ f, inv f), λ _ _ _, nonempty.map2 (≫)⟩,
end | instance | category_theory.nonempty_hom_of_connected_groupoid | category_theory | src/category_theory/is_connected.lean | [
"data.list.chain",
"category_theory.punit",
"category_theory.groupoid",
"category_theory.category.ulift"
] | [
"is_connected",
"nonempty.map",
"nonempty.map2"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
morphism_property | ∀ ⦃X Y : C⦄ (f : X ⟶ Y), Prop | def | category_theory.morphism_property | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | A `morphism_property C` is a class of morphisms between objects in `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op (P : morphism_property C) : morphism_property Cᵒᵖ | λ X Y f, P f.unop | def | category_theory.morphism_property.op | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | The morphism property in `Cᵒᵖ` associated to a morphism property in `C` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop (P : morphism_property Cᵒᵖ) : morphism_property C | λ X Y f, P f.op | def | category_theory.morphism_property.unop | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | The morphism property in `C` associated to a morphism property in `Cᵒᵖ` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_op (P : morphism_property C) : P.op.unop = P | rfl | lemma | category_theory.morphism_property.unop_op | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_unop (P : morphism_property Cᵒᵖ) : P.unop.op = P | rfl | lemma | category_theory.morphism_property.op_unop | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inverse_image (P : morphism_property D) (F : C ⥤ D) : morphism_property C | λ X Y f, P (F.map f) | def | category_theory.morphism_property.inverse_image | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | The inverse image of a `morphism_property D` by a functor `C ⥤ D` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
respects_iso (P : morphism_property C) : Prop | (∀ {X Y Z} (e : X ≅ Y) (f : Y ⟶ Z), P f → P (e.hom ≫ f)) ∧
(∀ {X Y Z} (e : Y ≅ Z) (f : X ⟶ Y), P f → P (f ≫ e.hom)) | def | category_theory.morphism_property.respects_iso | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | A morphism property `respects_iso` if it still holds when composed with an isomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
respects_iso.op {P : morphism_property C} (h : respects_iso P) : respects_iso P.op | ⟨λ X Y Z e f hf, h.2 e.unop f.unop hf, λ X Y Z e f hf, h.1 e.unop f.unop hf⟩ | lemma | category_theory.morphism_property.respects_iso.op | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
respects_iso.unop {P : morphism_property Cᵒᵖ} (h : respects_iso P) : respects_iso P.unop | ⟨λ X Y Z e f hf, h.2 e.op f.op hf, λ X Y Z e f hf, h.1 e.op f.op hf⟩ | lemma | category_theory.morphism_property.respects_iso.unop | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_composition (P : morphism_property C) : Prop | ∀ ⦃X Y Z⦄ (f : X ⟶ Y) (g : Y ⟶ Z), P f → P g → P (f ≫ g) | def | category_theory.morphism_property.stable_under_composition | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | A morphism property is `stable_under_composition` if the composition of two such morphisms
still falls in the class. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stable_under_composition.op {P : morphism_property C} (h : stable_under_composition P) :
stable_under_composition P.op | λ X Y Z f g hf hg, h g.unop f.unop hg hf | lemma | category_theory.morphism_property.stable_under_composition.op | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_composition.unop {P : morphism_property Cᵒᵖ} (h : stable_under_composition P) :
stable_under_composition P.unop | λ X Y Z f g hf hg, h g.op f.op hg hf | lemma | category_theory.morphism_property.stable_under_composition.unop | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_inverse (P : morphism_property C) : Prop | ∀ ⦃X Y⦄ (e : X ≅ Y), P e.hom → P e.inv | def | category_theory.morphism_property.stable_under_inverse | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | A morphism property is `stable_under_inverse` if the inverse of a morphism satisfying
the property still falls in the class. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stable_under_inverse.op {P : morphism_property C} (h : stable_under_inverse P) :
stable_under_inverse P.op | λ X Y e he, h e.unop he | lemma | category_theory.morphism_property.stable_under_inverse.op | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_inverse.unop {P : morphism_property Cᵒᵖ} (h : stable_under_inverse P) :
stable_under_inverse P.unop | λ X Y e he, h e.op he | lemma | category_theory.morphism_property.stable_under_inverse.unop | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_base_change (P : morphism_property C) : Prop | ∀ ⦃X Y Y' S : C⦄ ⦃f : X ⟶ S⦄ ⦃g : Y ⟶ S⦄ ⦃f' : Y' ⟶ Y⦄ ⦃g' : Y' ⟶ X⦄
(sq : is_pullback f' g' g f) (hg : P g), P g' | def | category_theory.morphism_property.stable_under_base_change | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | A morphism property is `stable_under_base_change` if the base change of such a morphism
still falls in the class. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stable_under_cobase_change (P : morphism_property C) : Prop | ∀ ⦃A A' B B' : C⦄ ⦃f : A ⟶ A'⦄ ⦃g : A ⟶ B⦄ ⦃f' : B ⟶ B'⦄ ⦃g' : A' ⟶ B'⦄
(sq : is_pushout g f f' g') (hf : P f), P f' | def | category_theory.morphism_property.stable_under_cobase_change | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | A morphism property is `stable_under_cobase_change` if the cobase change of such a morphism
still falls in the class. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stable_under_composition.respects_iso {P : morphism_property C}
(hP : stable_under_composition P) (hP' : ∀ {X Y} (e : X ≅ Y), P e.hom) : respects_iso P | ⟨λ X Y Z e f hf, hP _ _ (hP' e) hf, λ X Y Z e f hf, hP _ _ hf (hP' e)⟩ | lemma | category_theory.morphism_property.stable_under_composition.respects_iso | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
respects_iso.cancel_left_is_iso {P : morphism_property C}
(hP : respects_iso P) {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [is_iso f] :
P (f ≫ g) ↔ P g | ⟨λ h, by simpa using hP.1 (as_iso f).symm (f ≫ g) h, hP.1 (as_iso f) g⟩ | lemma | category_theory.morphism_property.respects_iso.cancel_left_is_iso | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
respects_iso.cancel_right_is_iso {P : morphism_property C}
(hP : respects_iso P) {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [is_iso g] :
P (f ≫ g) ↔ P f | ⟨λ h, by simpa using hP.2 (as_iso g).symm (f ≫ g) h, hP.2 (as_iso g) f⟩ | lemma | category_theory.morphism_property.respects_iso.cancel_right_is_iso | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
respects_iso.arrow_iso_iff {P : morphism_property C}
(hP : respects_iso P) {f g : arrow C} (e : f ≅ g) : P f.hom ↔ P g.hom | by { rw [← arrow.inv_left_hom_right e.hom, hP.cancel_left_is_iso, hP.cancel_right_is_iso], refl } | lemma | category_theory.morphism_property.respects_iso.arrow_iso_iff | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
respects_iso.arrow_mk_iso_iff {P : morphism_property C}
(hP : respects_iso P) {W X Y Z : C} {f : W ⟶ X} {g : Y ⟶ Z} (e : arrow.mk f ≅ arrow.mk g) :
P f ↔ P g | hP.arrow_iso_iff e | lemma | category_theory.morphism_property.respects_iso.arrow_mk_iso_iff | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
respects_iso.of_respects_arrow_iso (P : morphism_property C)
(hP : ∀ (f g : arrow C) (e : f ≅ g) (hf : P f.hom), P g.hom) : respects_iso P | begin
split,
{ intros X Y Z e f hf,
refine hP (arrow.mk f) (arrow.mk (e.hom ≫ f)) (arrow.iso_mk e.symm (iso.refl _) _) hf,
dsimp,
simp only [iso.inv_hom_id_assoc, category.comp_id], },
{ intros X Y Z e f hf,
refine hP (arrow.mk f) (arrow.mk (f ≫ e.hom)) (arrow.iso_mk (iso.refl _) e _) hf,
dsim... | lemma | category_theory.morphism_property.respects_iso.of_respects_arrow_iso | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_base_change.mk {P : morphism_property C} [has_pullbacks C]
(hP₁ : respects_iso P)
(hP₂ : ∀ (X Y S : C) (f : X ⟶ S) (g : Y ⟶ S) (hg : P g), P (pullback.fst : pullback f g ⟶ X)) :
stable_under_base_change P | λ X Y Y' S f g f' g' sq hg,
begin
let e := sq.flip.iso_pullback,
rw [← hP₁.cancel_left_is_iso e.inv, sq.flip.iso_pullback_inv_fst],
exact hP₂ _ _ _ f g hg,
end | lemma | category_theory.morphism_property.stable_under_base_change.mk | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_base_change.respects_iso {P : morphism_property C}
(hP : stable_under_base_change P) : respects_iso P | begin
apply respects_iso.of_respects_arrow_iso,
intros f g e,
exact hP (is_pullback.of_horiz_is_iso (comm_sq.mk e.inv.w)),
end | lemma | category_theory.morphism_property.stable_under_base_change.respects_iso | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_base_change.fst {P : morphism_property C}
(hP : stable_under_base_change P) {X Y S : C} (f : X ⟶ S) (g : Y ⟶ S) [has_pullback f g]
(H : P g) : P (pullback.fst : pullback f g ⟶ X) | hP (is_pullback.of_has_pullback f g).flip H | lemma | category_theory.morphism_property.stable_under_base_change.fst | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_base_change.snd {P : morphism_property C}
(hP : stable_under_base_change P) {X Y S : C} (f : X ⟶ S) (g : Y ⟶ S) [has_pullback f g]
(H : P f) : P (pullback.snd : pullback f g ⟶ Y) | hP (is_pullback.of_has_pullback f g) H | lemma | category_theory.morphism_property.stable_under_base_change.snd | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_base_change.base_change_obj [has_pullbacks C] {P : morphism_property C}
(hP : stable_under_base_change P) {S S' : C} (f : S' ⟶ S)
(X : over S) (H : P X.hom) : P ((base_change f).obj X).hom | hP.snd X.hom f H | lemma | category_theory.morphism_property.stable_under_base_change.base_change_obj | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_base_change.base_change_map [has_pullbacks C] {P : morphism_property C}
(hP : stable_under_base_change P) {S S' : C} (f : S' ⟶ S)
{X Y : over S} (g : X ⟶ Y) (H : P g.left) : P ((base_change f).map g).left | begin
let e := pullback_right_pullback_fst_iso Y.hom f g.left ≪≫
pullback.congr_hom (g.w.trans (category.comp_id _)) rfl,
have : e.inv ≫ pullback.snd = ((base_change f).map g).left,
{ apply pullback.hom_ext; dsimp; simp },
rw [← this, hP.respects_iso.cancel_left_is_iso],
exact hP.snd _ _ H,
end | lemma | category_theory.morphism_property.stable_under_base_change.base_change_map | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_base_change.pullback_map [has_pullbacks C] {P : morphism_property C}
(hP : stable_under_base_change P) (hP' : stable_under_composition P) {S X X' Y Y' : C}
{f : X ⟶ S} {g : Y ⟶ S} {f' : X' ⟶ S} {g' : Y' ⟶ S} {i₁ : X ⟶ X'} {i₂ : Y ⟶ Y'}
(h₁ : P i₁) (h₂ : P i₂) (e₁ : f = i₁ ≫ f') (e₂ : g = i₂ ≫ g') :
... | begin
have : pullback.map f g f' g' i₁ i₂ (𝟙 _)
((category.comp_id _).trans e₁) ((category.comp_id _).trans e₂) =
((pullback_symmetry _ _).hom ≫
((base_change _).map (over.hom_mk _ e₂.symm : over.mk g ⟶ over.mk g')).left) ≫
(pullback_symmetry _ _).hom ≫
((base_change g').map (over.hom_mk ... | lemma | category_theory.morphism_property.stable_under_base_change.pullback_map | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_cobase_change.mk {P : morphism_property C} [has_pushouts C]
(hP₁ : respects_iso P)
(hP₂ : ∀ (A B A' : C) (f : A ⟶ A') (g : A ⟶ B) (hf : P f), P (pushout.inr : B ⟶ pushout f g)) :
stable_under_cobase_change P | λ A A' B B' f g f' g' sq hf,
begin
let e := sq.flip.iso_pushout,
rw [← hP₁.cancel_right_is_iso _ e.hom, sq.flip.inr_iso_pushout_hom],
exact hP₂ _ _ _ f g hf,
end | lemma | category_theory.morphism_property.stable_under_cobase_change.mk | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_cobase_change.respects_iso {P : morphism_property C}
(hP : stable_under_cobase_change P) : respects_iso P | respects_iso.of_respects_arrow_iso _ (λ f g e, hP (is_pushout.of_horiz_is_iso (comm_sq.mk e.hom.w))) | lemma | category_theory.morphism_property.stable_under_cobase_change.respects_iso | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_cobase_change.inl {P : morphism_property C}
(hP : stable_under_cobase_change P) {A B A' : C} (f : A ⟶ A') (g : A ⟶ B) [has_pushout f g]
(H : P g) : P (pushout.inl : A' ⟶ pushout f g) | hP (is_pushout.of_has_pushout f g) H | lemma | category_theory.morphism_property.stable_under_cobase_change.inl | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_cobase_change.inr {P : morphism_property C}
(hP : stable_under_cobase_change P) {A B A' : C} (f : A ⟶ A') (g : A ⟶ B) [has_pushout f g]
(H : P f) : P (pushout.inr : B ⟶ pushout f g) | hP (is_pushout.of_has_pushout f g).flip H | lemma | category_theory.morphism_property.stable_under_cobase_change.inr | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_cobase_change.op {P : morphism_property C}
(hP : stable_under_cobase_change P) : stable_under_base_change P.op | λ X Y Y' S f g f' g' sq hg, hP sq.unop hg | lemma | category_theory.morphism_property.stable_under_cobase_change.op | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_cobase_change.unop {P : morphism_property Cᵒᵖ}
(hP : stable_under_cobase_change P) : stable_under_base_change P.unop | λ X Y Y' S f g f' g' sq hg, hP sq.op hg | lemma | category_theory.morphism_property.stable_under_cobase_change.unop | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_base_change.op {P : morphism_property C}
(hP : stable_under_base_change P) : stable_under_cobase_change P.op | λ A A' B B' f g f' g' sq hf, hP sq.unop hf | lemma | category_theory.morphism_property.stable_under_base_change.op | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_base_change.unop {P : morphism_property Cᵒᵖ}
(hP : stable_under_base_change P) : stable_under_cobase_change P.unop | λ A A' B B' f g f' g' sq hf, hP sq.op hf | lemma | category_theory.morphism_property.stable_under_base_change.unop | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_inverted_by (P : morphism_property C) (F : C ⥤ D) : Prop | ∀ ⦃X Y : C⦄ (f : X ⟶ Y) (hf : P f), is_iso (F.map f) | def | category_theory.morphism_property.is_inverted_by | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | If `P : morphism_property C` and `F : C ⥤ D`, then
`P.is_inverted_by F` means that all morphisms in `P` are mapped by `F`
to isomorphisms in `D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_comp {C₁ C₂ C₃ : Type*} [category C₁] [category C₂] [category C₃]
(W : morphism_property C₁) (F : C₁ ⥤ C₂) (hF : W.is_inverted_by F) (G : C₂ ⥤ C₃) :
W.is_inverted_by (F ⋙ G) | λ X Y f hf, by { haveI := hF f hf, dsimp, apply_instance, } | lemma | category_theory.morphism_property.is_inverted_by.of_comp | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op {W : morphism_property C} {L : C ⥤ D} (h : W.is_inverted_by L) :
W.op.is_inverted_by L.op | λ X Y f hf, by { haveI := h f.unop hf, dsimp, apply_instance, } | lemma | category_theory.morphism_property.is_inverted_by.op | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_op {W : morphism_property C} {L : Cᵒᵖ ⥤ D} (h : W.op.is_inverted_by L) :
W.is_inverted_by L.right_op | λ X Y f hf, by { haveI := h f.op hf, dsimp, apply_instance, } | lemma | category_theory.morphism_property.is_inverted_by.right_op | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_op {W : morphism_property C} {L : C ⥤ Dᵒᵖ} (h : W.is_inverted_by L) :
W.op.is_inverted_by L.left_op | λ X Y f hf, by { haveI := h f.unop hf, dsimp, apply_instance, } | lemma | category_theory.morphism_property.is_inverted_by.left_op | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop {W : morphism_property C} {L : Cᵒᵖ ⥤ Dᵒᵖ} (h : W.op.is_inverted_by L) :
W.is_inverted_by L.unop | λ X Y f hf, by { haveI := h f.op hf, dsimp, apply_instance, } | lemma | category_theory.morphism_property.is_inverted_by.unop | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
naturality_property {F₁ F₂ : C ⥤ D} (app : Π X, F₁.obj X ⟶ F₂.obj X) :
morphism_property C | λ X Y f, F₁.map f ≫ app Y = app X ≫ F₂.map f | def | category_theory.morphism_property.naturality_property | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | Given `app : Π X, F₁.obj X ⟶ F₂.obj X` where `F₁` and `F₂` are two functors,
this is the `morphism_property C` satisfied by the morphisms in `C` with respect
to whom `app` is natural. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_stable_under_composition {F₁ F₂ : C ⥤ D} (app : Π X, F₁.obj X ⟶ F₂.obj X) :
(naturality_property app).stable_under_composition | λ X Y Z f g hf hg,
begin
simp only [naturality_property] at ⊢ hf hg,
simp only [functor.map_comp, category.assoc, hg],
slice_lhs 1 2 { rw hf },
rw category.assoc,
end | lemma | category_theory.morphism_property.naturality_property.is_stable_under_composition | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_stable_under_inverse {F₁ F₂ : C ⥤ D} (app : Π X, F₁.obj X ⟶ F₂.obj X) :
(naturality_property app).stable_under_inverse | λ X Y e he,
begin
simp only [naturality_property] at ⊢ he,
rw ← cancel_epi (F₁.map e.hom),
slice_rhs 1 2 { rw he },
simp only [category.assoc, ← F₁.map_comp_assoc, ← F₂.map_comp,
e.hom_inv_id, functor.map_id, category.id_comp, category.comp_id],
end | lemma | category_theory.morphism_property.naturality_property.is_stable_under_inverse | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [
"functor.map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
respects_iso.inverse_image {P : morphism_property D} (h : respects_iso P) (F : C ⥤ D) :
respects_iso (P.inverse_image F) | begin
split,
all_goals
{ intros X Y Z e f hf,
dsimp [inverse_image],
rw F.map_comp, },
exacts [h.1 (F.map_iso e) (F.map f) hf, h.2 (F.map_iso e) (F.map f) hf],
end | lemma | category_theory.morphism_property.respects_iso.inverse_image | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_composition.inverse_image {P : morphism_property D}
(h : stable_under_composition P) (F : C ⥤ D) : stable_under_composition (P.inverse_image F) | λ X Y Z f g hf hg, by simpa only [← F.map_comp] using h (F.map f) (F.map g) hf hg | lemma | category_theory.morphism_property.stable_under_composition.inverse_image | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isomorphisms : morphism_property C | λ X Y f, is_iso f | def | category_theory.morphism_property.isomorphisms | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | The `morphism_property C` satisfied by isomorphisms in `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monomorphisms : morphism_property C | λ X Y f, mono f | def | category_theory.morphism_property.monomorphisms | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | The `morphism_property C` satisfied by monomorphisms in `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epimorphisms : morphism_property C | λ X Y f, epi f | def | category_theory.morphism_property.epimorphisms | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | The `morphism_property C` satisfied by epimorphisms in `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
isomorphisms.iff : (isomorphisms C) f ↔ is_iso f | by refl | lemma | category_theory.morphism_property.isomorphisms.iff | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monomorphisms.iff : (monomorphisms C) f ↔ mono f | by refl | lemma | category_theory.morphism_property.monomorphisms.iff | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epimorphisms.iff : (epimorphisms C) f ↔ epi f | by refl | lemma | category_theory.morphism_property.epimorphisms.iff | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isomorphisms.infer_property [hf : is_iso f] : (isomorphisms C) f | hf | lemma | category_theory.morphism_property.isomorphisms.infer_property | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monomorphisms.infer_property [hf : mono f] : (monomorphisms C) f | hf | lemma | category_theory.morphism_property.monomorphisms.infer_property | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epimorphisms.infer_property [hf : epi f] : (epimorphisms C) f | hf | lemma | category_theory.morphism_property.epimorphisms.infer_property | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
respects_iso.monomorphisms : respects_iso (monomorphisms C) | by { split; { intros X Y Z e f, simp only [monomorphisms.iff], introI, apply mono_comp, }, } | lemma | category_theory.morphism_property.respects_iso.monomorphisms | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
respects_iso.epimorphisms : respects_iso (epimorphisms C) | by { split; { intros X Y Z e f, simp only [epimorphisms.iff], introI, apply epi_comp, }, } | lemma | category_theory.morphism_property.respects_iso.epimorphisms | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
respects_iso.isomorphisms : respects_iso (isomorphisms C) | by { split; { intros X Y Z e f, simp only [isomorphisms.iff], introI, apply_instance, }, } | lemma | category_theory.morphism_property.respects_iso.isomorphisms | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_composition.isomorphisms : stable_under_composition (isomorphisms C) | λ X Y Z f g hf hg, begin
rw isomorphisms.iff at hf hg ⊢,
haveI := hf,
haveI := hg,
apply_instance,
end | lemma | category_theory.morphism_property.stable_under_composition.isomorphisms | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_composition.monomorphisms : stable_under_composition (monomorphisms C) | λ X Y Z f g hf hg, begin
rw monomorphisms.iff at hf hg ⊢,
haveI := hf,
haveI := hg,
apply mono_comp,
end | lemma | category_theory.morphism_property.stable_under_composition.monomorphisms | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_composition.epimorphisms : stable_under_composition (epimorphisms C) | λ X Y Z f g hf hg, begin
rw epimorphisms.iff at hf hg ⊢,
haveI := hf,
haveI := hg,
apply epi_comp,
end | lemma | category_theory.morphism_property.stable_under_composition.epimorphisms | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functors_inverting (W : morphism_property C) (D : Type*) [category D] | full_subcategory (λ (F : C ⥤ D), W.is_inverted_by F) | def | category_theory.morphism_property.functors_inverting | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | The full subcategory of `C ⥤ D` consisting of functors inverting morphisms in `W` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functors_inverting.mk {W : morphism_property C} {D : Type*} [category D]
(F : C ⥤ D) (hF : W.is_inverted_by F) : W.functors_inverting D | ⟨F, hF⟩ | def | category_theory.morphism_property.functors_inverting.mk | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | A constructor for `W.functors_inverting D` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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