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pre_map {A₁ A₂ A₃ : C} [exponentiable A₁] [exponentiable A₂] [exponentiable A₃]
(f : A₁ ⟶ A₂) (g : A₂ ⟶ A₃) :
pre (f ≫ g) = pre g ≫ pre f | by rw [pre, pre, pre, transfer_nat_trans_self_comp, prod.functor.map_comp] | lemma | category_theory.pre_map | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
internal_hom [cartesian_closed C] : Cᵒᵖ ⥤ C ⥤ C | { obj := λ X, exp X.unop,
map := λ X Y f, pre f.unop } | def | category_theory.internal_hom | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [
"exp"
] | The internal hom functor given by the cartesian closed structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_mul {I : C} (t : is_initial I) : A ⨯ I ≅ I | { hom := limits.prod.snd,
inv := t.to _,
hom_inv_id' :=
begin
have: (limits.prod.snd : A ⨯ I ⟶ I) = cartesian_closed.uncurry (t.to _),
rw ← curry_eq_iff,
apply t.hom_ext,
rw [this, ← uncurry_natural_right, ← eq_curry_iff],
apply t.hom_ext,
end,
inv_hom_id' := t.hom_ext _ _ } | def | category_theory.zero_mul | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [
"zero_mul"
] | If an initial object `I` exists in a CCC, then `A ⨯ I ≅ I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_zero {I : C} (t : is_initial I) : I ⨯ A ≅ I | limits.prod.braiding _ _ ≪≫ zero_mul t | def | category_theory.mul_zero | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [
"mul_zero",
"zero_mul"
] | If an initial object `0` exists in a CCC, then `0 ⨯ A ≅ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_zero {I : C} (t : is_initial I) [cartesian_closed C] : I ⟹ B ≅ ⊤_ C | { hom := default,
inv := cartesian_closed.curry ((mul_zero t).hom ≫ t.to _),
hom_inv_id' :=
begin
rw [← curry_natural_left, curry_eq_iff, ← cancel_epi (mul_zero t).inv],
{ apply t.hom_ext },
{ apply_instance },
{ apply_instance },
end } | def | category_theory.pow_zero | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [
"mul_zero",
"pow_zero"
] | If an initial object `0` exists in a CCC then `0^B ≅ 1` for any `B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_coprod_distrib [has_binary_coproducts C] [cartesian_closed C] (X Y Z : C) :
(Z ⨯ X) ⨿ (Z ⨯ Y) ≅ Z ⨯ (X ⨿ Y) | { hom := coprod.desc (limits.prod.map (𝟙 _) coprod.inl) (limits.prod.map (𝟙 _) coprod.inr),
inv := cartesian_closed.uncurry
(coprod.desc (cartesian_closed.curry coprod.inl) (cartesian_closed.curry coprod.inr)),
hom_inv_id' :=
begin
apply coprod.hom_ext,
rw [coprod.inl_desc_assoc, comp_id, ←uncurry_n... | def | category_theory.prod_coprod_distrib | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | In a CCC with binary coproducts, the distribution morphism is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_initial {I : C} (t : is_initial I) (f : A ⟶ I) : is_iso f | begin
haveI : mono (limits.prod.lift (𝟙 A) f ≫ (zero_mul t).hom) := mono_comp _ _,
rw [zero_mul_hom, prod.lift_snd] at _inst,
haveI: is_split_epi f := is_split_epi.mk' ⟨t.to _, t.hom_ext _ _⟩,
apply is_iso_of_mono_of_is_split_epi
end | lemma | category_theory.strict_initial | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [
"zero_mul"
] | If an initial object `I` exists in a CCC then it is a strict initial object,
i.e. any morphism to `I` is an iso.
This actually shows a slightly stronger version: any morphism to an initial object from an
exponentiable object is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_initial_is_iso [has_initial C] (f : A ⟶ ⊥_ C) : is_iso f | strict_initial initial_is_initial _ | instance | category_theory.to_initial_is_iso | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
initial_mono {I : C} (B : C) (t : is_initial I) [cartesian_closed C] : mono (t.to B) | ⟨λ B g h _,
begin
haveI := strict_initial t g,
haveI := strict_initial t h,
exact eq_of_inv_eq_inv (t.hom_ext _ _)
end⟩ | lemma | category_theory.initial_mono | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | If an initial object `0` exists in a CCC then every morphism from it is monic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
initial.mono_to [has_initial C] (B : C) [cartesian_closed C] : mono (initial.to B) | initial_mono B initial_is_initial | instance | category_theory.initial.mono_to | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cartesian_closed_of_equiv (e : C ≌ D) [h : cartesian_closed C] : cartesian_closed D | { closed' := λ X,
{ is_adj :=
begin
haveI q : exponentiable (e.inverse.obj X) := infer_instance,
have : is_left_adjoint (prod.functor.obj (e.inverse.obj X)) := q.is_adj,
have : e.functor ⋙ prod.functor.obj X ⋙ e.inverse ≅ prod.functor.obj (e.inverse.obj X),
apply nat_iso.of_components _ _,... | def | category_theory.cartesian_closed_of_equiv | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | Transport the property of being cartesian closed across an equivalence of categories.
Note we didn't require any coherence between the choice of finite products here, since we transport
along the `prod_comparison` isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frobenius_morphism (h : L ⊣ F) (A : C) :
prod.functor.obj (F.obj A) ⋙ L ⟶ L ⋙ prod.functor.obj A | prod_comparison_nat_trans L (F.obj A) ≫ whisker_left _ (prod.functor.map (h.counit.app _)) | def | category_theory.frobenius_morphism | category_theory.closed | src/category_theory/closed/functor.lean | [
"category_theory.closed.cartesian",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.fully_faithful"
] | [] | The Frobenius morphism for an adjunction `L ⊣ F` at `A` is given by the morphism
L(FA ⨯ B) ⟶ LFA ⨯ LB ⟶ A ⨯ LB
natural in `B`, where the first morphism is the product comparison and the latter uses the counit
of the adjunction.
We will show that if `C` and `D` are cartesian closed, then this morphism is an isomo... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frobenius_morphism_iso_of_preserves_binary_products (h : L ⊣ F) (A : C)
[preserves_limits_of_shape (discrete walking_pair) L] [full F] [faithful F] :
is_iso (frobenius_morphism F h A) | begin
apply nat_iso.is_iso_of_is_iso_app _,
intro B,
dsimp [frobenius_morphism],
apply_instance,
end | instance | category_theory.frobenius_morphism_iso_of_preserves_binary_products | category_theory.closed | src/category_theory/closed/functor.lean | [
"category_theory.closed.cartesian",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.fully_faithful"
] | [] | If `F` is full and faithful and has a left adjoint `L` which preserves binary products, then the
Frobenius morphism is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_comparison (A : C) :
exp A ⋙ F ⟶ F ⋙ exp (F.obj A) | transfer_nat_trans (exp.adjunction A) (exp.adjunction (F.obj A)) (prod_comparison_nat_iso F A).inv | def | category_theory.exp_comparison | category_theory.closed | src/category_theory/closed/functor.lean | [
"category_theory.closed.cartesian",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.fully_faithful"
] | [
"exp"
] | The exponential comparison map.
`F` is a cartesian closed functor if this is an iso for all `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_comparison_ev (A B : C) :
limits.prod.map (𝟙 (F.obj A)) ((exp_comparison F A).app B) ≫ (exp.ev (F.obj A)).app (F.obj B) =
inv (prod_comparison F _ _) ≫ F.map ((exp.ev _).app _) | begin
convert transfer_nat_trans_counit _ _ (prod_comparison_nat_iso F A).inv B,
ext,
simp,
end | lemma | category_theory.exp_comparison_ev | category_theory.closed | src/category_theory/closed/functor.lean | [
"category_theory.closed.cartesian",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coev_exp_comparison (A B : C) :
F.map ((exp.coev A).app B) ≫ (exp_comparison F A).app (A ⨯ B) =
(exp.coev _).app (F.obj B) ≫ (exp (F.obj A)).map (inv (prod_comparison F A B)) | begin
convert unit_transfer_nat_trans _ _ (prod_comparison_nat_iso F A).inv B,
ext,
dsimp,
simp,
end | lemma | category_theory.coev_exp_comparison | category_theory.closed | src/category_theory/closed/functor.lean | [
"category_theory.closed.cartesian",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.fully_faithful"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uncurry_exp_comparison (A B : C) :
cartesian_closed.uncurry ((exp_comparison F A).app B) =
inv (prod_comparison F _ _) ≫ F.map ((exp.ev _).app _) | by rw [uncurry_eq, exp_comparison_ev] | lemma | category_theory.uncurry_exp_comparison | category_theory.closed | src/category_theory/closed/functor.lean | [
"category_theory.closed.cartesian",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_comparison_whisker_left {A A' : C} (f : A' ⟶ A) :
exp_comparison F A ≫ whisker_left _ (pre (F.map f)) =
whisker_right (pre f) _ ≫ exp_comparison F A' | begin
ext B,
dsimp,
apply uncurry_injective,
rw [uncurry_natural_left, uncurry_natural_left, uncurry_exp_comparison, uncurry_pre,
prod.map_swap_assoc, ←F.map_id, exp_comparison_ev, ←F.map_id,
←prod_comparison_inv_natural_assoc, ←prod_comparison_inv_natural_assoc, ←F.map_comp,
←F.map_comp, prod_map_p... | lemma | category_theory.exp_comparison_whisker_left | category_theory.closed | src/category_theory/closed/functor.lean | [
"category_theory.closed.cartesian",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.fully_faithful"
] | [] | The exponential comparison map is natural in `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cartesian_closed_functor | (comparison_iso : ∀ A, is_iso (exp_comparison F A)) | class | category_theory.cartesian_closed_functor | category_theory.closed | src/category_theory/closed/functor.lean | [
"category_theory.closed.cartesian",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.fully_faithful"
] | [] | The functor `F` is cartesian closed (ie preserves exponentials) if each natural transformation
`exp_comparison F A` is an isomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frobenius_morphism_mate (h : L ⊣ F) (A : C) :
transfer_nat_trans_self
(h.comp (exp.adjunction A))
((exp.adjunction (F.obj A)).comp h)
(frobenius_morphism F h A) = exp_comparison F A | begin
rw ←equiv.eq_symm_apply,
ext B : 2,
dsimp [frobenius_morphism, transfer_nat_trans_self, transfer_nat_trans, adjunction.comp],
simp only [id_comp, comp_id],
rw [←L.map_comp_assoc, prod.map_id_comp, assoc, exp_comparison_ev, prod.map_id_comp, assoc,
← F.map_id, ← prod_comparison_inv_natura... | lemma | category_theory.frobenius_morphism_mate | category_theory.closed | src/category_theory/closed/functor.lean | [
"category_theory.closed.cartesian",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_morphism_iso_of_exp_comparison_iso (h : L ⊣ F) (A : C)
[i : is_iso (exp_comparison F A)] :
is_iso (frobenius_morphism F h A) | begin
rw ←frobenius_morphism_mate F h at i,
exact @@transfer_nat_trans_self_of_iso _ _ _ _ _ i,
end | lemma | category_theory.frobenius_morphism_iso_of_exp_comparison_iso | category_theory.closed | src/category_theory/closed/functor.lean | [
"category_theory.closed.cartesian",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.fully_faithful"
] | [] | If the exponential comparison transformation (at `A`) is an isomorphism, then the Frobenius morphism
at `A` is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_comparison_iso_of_frobenius_morphism_iso (h : L ⊣ F) (A : C)
[i : is_iso (frobenius_morphism F h A)] :
is_iso (exp_comparison F A) | by { rw ← frobenius_morphism_mate F h, apply_instance } | lemma | category_theory.exp_comparison_iso_of_frobenius_morphism_iso | category_theory.closed | src/category_theory/closed/functor.lean | [
"category_theory.closed.cartesian",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.fully_faithful"
] | [] | If the Frobenius morphism at `A` is an isomorphism, then the exponential comparison transformation
(at `A`) is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cartesian_closed_functor_of_left_adjoint_preserves_binary_products (h : L ⊣ F)
[full F] [faithful F] [preserves_limits_of_shape (discrete walking_pair) L] :
cartesian_closed_functor F | { comparison_iso := λ A, exp_comparison_iso_of_frobenius_morphism_iso F h _ } | def | category_theory.cartesian_closed_functor_of_left_adjoint_preserves_binary_products | category_theory.closed | src/category_theory/closed/functor.lean | [
"category_theory.closed.cartesian",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.fully_faithful"
] | [] | If `F` is full and faithful, and has a left adjoint which preserves binary products, then it is
cartesian closed.
TODO: Show the converse, that if `F` is cartesian closed and its left adjoint preserves binary
products, then it is full and faithful. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_ihom (F : D ⥤ C) : (D ⥤ C) ⥤ (D ⥤ C) | ((whiskering_right₂ D Cᵒᵖ C C).obj internal_hom).obj (groupoid.inv_functor D ⋙ F.op) | def | category_theory.functor.closed_ihom | category_theory.closed | src/category_theory/closed/functor_category.lean | [
"category_theory.closed.monoidal",
"category_theory.monoidal.functor_category"
] | [] | Auxiliary definition for `category_theory.monoidal_closed.functor_closed`.
The internal hom functor `F ⟶[C] -` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_unit (F : D ⥤ C) : 𝟭 (D ⥤ C) ⟶ (tensor_left F) ⋙ (closed_ihom F) | { app := λ G,
{ app := λ X, (ihom.coev (F.obj X)).app (G.obj X),
naturality' := begin
intros X Y f,
dsimp,
simp only [ihom.coev_naturality, closed_ihom_obj_map, monoidal.tensor_obj_map],
dsimp,
rw [coev_app_comp_pre_app_assoc, ←functor.map_comp],
simp,
end } } | def | category_theory.functor.closed_unit | category_theory.closed | src/category_theory/closed/functor_category.lean | [
"category_theory.closed.monoidal",
"category_theory.monoidal.functor_category"
] | [] | Auxiliary definition for `category_theory.monoidal_closed.functor_closed`.
The unit for the adjunction `(tensor_left F) ⊣ (ihom F)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_counit (F : D ⥤ C) : (closed_ihom F) ⋙ (tensor_left F) ⟶ 𝟭 (D ⥤ C) | { app := λ G,
{ app := λ X, (ihom.ev (F.obj X)).app (G.obj X),
naturality' := begin
intros X Y f,
dsimp,
simp only [closed_ihom_obj_map, pre_comm_ihom_map],
rw [←tensor_id_comp_id_tensor, id_tensor_comp],
simp,
end } } | def | category_theory.functor.closed_counit | category_theory.closed | src/category_theory/closed/functor_category.lean | [
"category_theory.closed.monoidal",
"category_theory.monoidal.functor_category"
] | [] | Auxiliary definition for `category_theory.monoidal_closed.functor_closed`.
The counit for the adjunction `(tensor_left F) ⊣ (ihom F)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed (F : D ⥤ C) : closed F | { is_adj :=
{ right := closed_ihom F,
adj := adjunction.mk_of_unit_counit
{ unit := closed_unit F,
counit := closed_counit F } } } | instance | category_theory.functor.closed | category_theory.closed | src/category_theory/closed/functor_category.lean | [
"category_theory.closed.monoidal",
"category_theory.monoidal.functor_category"
] | [
"adj"
] | If `C` is a monoidal closed category and `D` is groupoid, then every functor `F : D ⥤ C` is
closed in the functor category `F : D ⥤ C` with the pointwise monoidal structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoidal_closed : monoidal_closed (D ⥤ C) | { closed' := by apply_instance } | instance | category_theory.functor.monoidal_closed | category_theory.closed | src/category_theory/closed/functor_category.lean | [
"category_theory.closed.monoidal",
"category_theory.monoidal.functor_category"
] | [] | If `C` is a monoidal closed category and `D` is groupoid, then the functor category `D ⥤ C`,
with the pointwise monoidal structure, is monoidal closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ihom_map (F : D ⥤ C) {G H : D ⥤ C} (f : G ⟶ H) :
(ihom F).map f = (closed_ihom F).map f | rfl | lemma | category_theory.functor.ihom_map | category_theory.closed | src/category_theory/closed/functor_category.lean | [
"category_theory.closed.monoidal",
"category_theory.monoidal.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ihom_ev_app (F G : D ⥤ C) :
(ihom.ev F).app G = (closed_counit F).app G | rfl | lemma | category_theory.functor.ihom_ev_app | category_theory.closed | src/category_theory/closed/functor_category.lean | [
"category_theory.closed.monoidal",
"category_theory.monoidal.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ihom_coev_app (F G : D ⥤ C) :
(ihom.coev F).app G = (closed_unit F).app G | rfl | lemma | category_theory.functor.ihom_coev_app | category_theory.closed | src/category_theory/closed/functor_category.lean | [
"category_theory.closed.monoidal",
"category_theory.monoidal.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exponential_ideal : Prop | (exp_closed : ∀ {B}, B ∈ i.ess_image → ∀ A, (A ⟹ B) ∈ i.ess_image) | class | category_theory.exponential_ideal | category_theory.closed | src/category_theory/closed/ideal.lean | [
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.constructions.finite_products_of_binary_products",
"category_theory.monad.limits",
"category_theory.adjunction.fully_faithful",
"category_theory.adjunction.reflective",
"category_theory.closed.cartesian",
"category_theory... | [] | The subcategory `D` of `C` expressed as an inclusion functor is an *exponential ideal* if
`B ∈ D` implies `A ⟹ B ∈ D` for all `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exponential_ideal.mk' (h : ∀ (B : D) (A : C), (A ⟹ i.obj B) ∈ i.ess_image) :
exponential_ideal i | ⟨λ B hB A,
begin
rcases hB with ⟨B', ⟨iB'⟩⟩,
exact functor.ess_image.of_iso ((exp A).map_iso iB') (h B' A),
end⟩ | lemma | category_theory.exponential_ideal.mk' | category_theory.closed | src/category_theory/closed/ideal.lean | [
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.constructions.finite_products_of_binary_products",
"category_theory.monad.limits",
"category_theory.adjunction.fully_faithful",
"category_theory.adjunction.reflective",
"category_theory.closed.cartesian",
"category_theory... | [
"exp"
] | To show `i` is an exponential ideal it suffices to show that `A ⟹ iB` is "in" `D` for any `A` in
`C` and `B` in `D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exponential_ideal_reflective (A : C) [reflective i] [exponential_ideal i] :
i ⋙ exp A ⋙ left_adjoint i ⋙ i ≅ i ⋙ exp A | begin
symmetry,
apply nat_iso.of_components _ _,
{ intro X,
haveI := (exponential_ideal.exp_closed (i.obj_mem_ess_image X) A).unit_is_iso,
apply as_iso ((adjunction.of_right_adjoint i).unit.app (A ⟹ i.obj X)) },
{ simp }
end | def | category_theory.exponential_ideal_reflective | category_theory.closed | src/category_theory/closed/ideal.lean | [
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.constructions.finite_products_of_binary_products",
"category_theory.monad.limits",
"category_theory.adjunction.fully_faithful",
"category_theory.adjunction.reflective",
"category_theory.closed.cartesian",
"category_theory... | [
"exp"
] | If `D` is a reflective subcategory, the property of being an exponential ideal is equivalent to
the presence of a natural isomorphism `i ⋙ exp A ⋙ left_adjoint i ⋙ i ≅ i ⋙ exp A`, that is:
`(A ⟹ iB) ≅ i L (A ⟹ iB)`, naturally in `B`.
The converse is given in `exponential_ideal.mk_of_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exponential_ideal.mk_of_iso [reflective i]
(h : Π (A : C), i ⋙ exp A ⋙ left_adjoint i ⋙ i ≅ i ⋙ exp A) :
exponential_ideal i | begin
apply exponential_ideal.mk',
intros B A,
exact ⟨_, ⟨(h A).app B⟩⟩,
end | lemma | category_theory.exponential_ideal.mk_of_iso | category_theory.closed | src/category_theory/closed/ideal.lean | [
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.constructions.finite_products_of_binary_products",
"category_theory.monad.limits",
"category_theory.adjunction.fully_faithful",
"category_theory.adjunction.reflective",
"category_theory.closed.cartesian",
"category_theory... | [
"exp"
] | Given a natural isomorphism `i ⋙ exp A ⋙ left_adjoint i ⋙ i ≅ i ⋙ exp A`, we can show `i`
is an exponential ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflective_products [has_finite_products C] [reflective i] : has_finite_products D | ⟨λ n, has_limits_of_shape_of_reflective i⟩ | lemma | category_theory.reflective_products | category_theory.closed | src/category_theory/closed/ideal.lean | [
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.constructions.finite_products_of_binary_products",
"category_theory.monad.limits",
"category_theory.adjunction.fully_faithful",
"category_theory.adjunction.reflective",
"category_theory.closed.cartesian",
"category_theory... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exponential_ideal_of_preserves_binary_products
[preserves_limits_of_shape (discrete walking_pair) (left_adjoint i)] :
exponential_ideal i | begin
let ir := adjunction.of_right_adjoint i,
let L : C ⥤ D := left_adjoint i,
let η : 𝟭 C ⟶ L ⋙ i := ir.unit,
let ε : i ⋙ L ⟶ 𝟭 D := ir.counit,
apply exponential_ideal.mk',
intros B A,
let q : i.obj (L.obj (A ⟹ i.obj B)) ⟶ A ⟹ i.obj B,
apply cartesian_closed.curry (ir.hom_equiv _ _ _),
apply _... | instance | category_theory.exponential_ideal_of_preserves_binary_products | category_theory.closed | src/category_theory/closed/ideal.lean | [
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.constructions.finite_products_of_binary_products",
"category_theory.monad.limits",
"category_theory.adjunction.fully_faithful",
"category_theory.adjunction.reflective",
"category_theory.closed.cartesian",
"category_theory... | [] | If the reflector preserves binary products, the subcategory is an exponential ideal.
This is the converse of `preserves_binary_products_of_exponential_ideal`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cartesian_closed_of_reflective : cartesian_closed D | { closed' := λ B,
{ is_adj :=
{ right := i ⋙ exp (i.obj B) ⋙ left_adjoint i,
adj :=
begin
apply adjunction.restrict_fully_faithful i i (exp.adjunction (i.obj B)),
{ symmetry,
apply nat_iso.of_components _ _,
{ intro X,
haveI :=
adjunction.r... | def | category_theory.cartesian_closed_of_reflective | category_theory.closed | src/category_theory/closed/ideal.lean | [
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.constructions.finite_products_of_binary_products",
"category_theory.monad.limits",
"category_theory.adjunction.fully_faithful",
"category_theory.adjunction.reflective",
"category_theory.closed.cartesian",
"category_theory... | [
"adj",
"exp"
] | If `i` witnesses that `D` is a reflective subcategory and an exponential ideal, then `D` is
itself cartesian closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bijection (A B : C) (X : D) :
((left_adjoint i).obj (A ⨯ B) ⟶ X) ≃ ((left_adjoint i).obj A ⨯ (left_adjoint i).obj B ⟶ X) | calc _ ≃ (A ⨯ B ⟶ i.obj X) :
(adjunction.of_right_adjoint i).hom_equiv _ _
... ≃ (B ⨯ A ⟶ i.obj X) :
(limits.prod.braiding _ _).hom_congr (iso.refl _)
... ≃ (A ⟶ B ⟹ i.obj X) :
(exp.adjunction _).hom_equiv _ _
... ≃ (i.obj ((left_adjoint i).obj A) ⟶ B ⟹ i.obj X) :
... | def | category_theory.bijection | category_theory.closed | src/category_theory/closed/ideal.lean | [
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.constructions.finite_products_of_binary_products",
"category_theory.monad.limits",
"category_theory.adjunction.fully_faithful",
"category_theory.adjunction.reflective",
"category_theory.closed.cartesian",
"category_theory... | [] | We construct a bijection between morphisms `L(A ⨯ B) ⟶ X` and morphisms `LA ⨯ LB ⟶ X`.
This bijection has two key properties:
* It is natural in `X`: See `bijection_natural`.
* When `X = LA ⨯ LB`, then the backwards direction sends the identity morphism to the product
comparison morphism: See `bijection_symm_apply_id... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bijection_symm_apply_id (A B : C) :
(bijection i A B _).symm (𝟙 _) = prod_comparison _ _ _ | begin
dsimp [bijection],
rw [comp_id, comp_id, comp_id, i.map_id, comp_id, unit_comp_partial_bijective_symm_apply,
unit_comp_partial_bijective_symm_apply, uncurry_natural_left, uncurry_curry,
uncurry_natural_left, uncurry_curry, prod.lift_map_assoc, comp_id, prod.lift_map_assoc,
comp_id, prod.comp... | lemma | category_theory.bijection_symm_apply_id | category_theory.closed | src/category_theory/closed/ideal.lean | [
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.constructions.finite_products_of_binary_products",
"category_theory.monad.limits",
"category_theory.adjunction.fully_faithful",
"category_theory.adjunction.reflective",
"category_theory.closed.cartesian",
"category_theory... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bijection_natural
(A B : C) (X X' : D) (f : ((left_adjoint i).obj (A ⨯ B) ⟶ X)) (g : X ⟶ X') :
bijection i _ _ _ (f ≫ g) = bijection i _ _ _ f ≫ g | begin
dsimp [bijection],
apply i.map_injective,
rw [i.image_preimage, i.map_comp, i.image_preimage, comp_id, comp_id, comp_id, comp_id, comp_id,
comp_id, adjunction.hom_equiv_naturality_right, ← assoc, curry_natural_right _ (i.map g),
unit_comp_partial_bijective_natural, uncurry_natural_right, ← assoc... | lemma | category_theory.bijection_natural | category_theory.closed | src/category_theory/closed/ideal.lean | [
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.constructions.finite_products_of_binary_products",
"category_theory.monad.limits",
"category_theory.adjunction.fully_faithful",
"category_theory.adjunction.reflective",
"category_theory.closed.cartesian",
"category_theory... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_comparison_iso (A B : C) :
is_iso (prod_comparison (left_adjoint i) A B) | ⟨⟨bijection i _ _ _ (𝟙 _),
by rw [←(bijection i _ _ _).injective.eq_iff, bijection_natural, ← bijection_symm_apply_id,
equiv.apply_symm_apply, id_comp],
by rw [←bijection_natural, id_comp, ←bijection_symm_apply_id, equiv.apply_symm_apply]⟩⟩ | lemma | category_theory.prod_comparison_iso | category_theory.closed | src/category_theory/closed/ideal.lean | [
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.constructions.finite_products_of_binary_products",
"category_theory.monad.limits",
"category_theory.adjunction.fully_faithful",
"category_theory.adjunction.reflective",
"category_theory.closed.cartesian",
"category_theory... | [
"equiv.apply_symm_apply"
] | The bijection allows us to show that `prod_comparison L A B` is an isomorphism, where the inverse
is the forward map of the identity morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_binary_products_of_exponential_ideal :
preserves_limits_of_shape (discrete walking_pair) (left_adjoint i) | { preserves_limit := λ K,
begin
apply limits.preserves_limit_of_iso_diagram _ (diagram_iso_pair K).symm,
apply preserves_limit_pair.of_iso_prod_comparison,
end } | def | category_theory.preserves_binary_products_of_exponential_ideal | category_theory.closed | src/category_theory/closed/ideal.lean | [
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.constructions.finite_products_of_binary_products",
"category_theory.monad.limits",
"category_theory.adjunction.fully_faithful",
"category_theory.adjunction.reflective",
"category_theory.closed.cartesian",
"category_theory... | [] | If a reflective subcategory is an exponential ideal, then the reflector preserves binary products.
This is the converse of `exponential_ideal_of_preserves_binary_products`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_products_of_exponential_ideal (J : Type) [fintype J] :
preserves_limits_of_shape (discrete J) (left_adjoint i) | begin
letI := preserves_binary_products_of_exponential_ideal i,
letI := left_adjoint_preserves_terminal_of_reflective.{0} i,
apply preserves_finite_products_of_preserves_binary_and_terminal (left_adjoint i) J,
end | def | category_theory.preserves_finite_products_of_exponential_ideal | category_theory.closed | src/category_theory/closed/ideal.lean | [
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.constructions.finite_products_of_binary_products",
"category_theory.monad.limits",
"category_theory.adjunction.fully_faithful",
"category_theory.adjunction.reflective",
"category_theory.closed.cartesian",
"category_theory... | [
"fintype"
] | If a reflective subcategory is an exponential ideal, then the reflector preserves finite products. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed {C : Type u} [category.{v} C] [monoidal_category.{v} C] (X : C) | (is_adj : is_left_adjoint (tensor_left X)) | class | category_theory.closed | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoidal_closed (C : Type u) [category.{v} C] [monoidal_category.{v} C] | (closed' : Π (X : C), closed X) | class | category_theory.monoidal_closed | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | A monoidal category `C` is (right) monoidal closed if every object is (right) closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tensor_closed {X Y : C}
(hX : closed X) (hY : closed Y) : closed (X ⊗ Y) | { is_adj :=
begin
haveI := hX.is_adj,
haveI := hY.is_adj,
exact adjunction.left_adjoint_of_nat_iso (monoidal_category.tensor_left_tensor _ _).symm
end } | def | category_theory.tensor_closed | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | If `X` and `Y` are closed then `X ⊗ Y` is.
This isn't an instance because it's not usually how we want to construct internal homs,
we'll usually prove all objects are closed uniformly. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_closed : closed (𝟙_ C) | { is_adj :=
{ right := 𝟭 C,
adj := adjunction.mk_of_hom_equiv
{ hom_equiv := λ X _,
{ to_fun := λ a, (left_unitor X).inv ≫ a,
inv_fun := λ a, (left_unitor X).hom ≫ a,
left_inv := by tidy,
right_inv := by tidy },
hom_equiv_naturality_left_symm' := λ X' X Y f g,
by { d... | def | category_theory.unit_closed | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [
"adj",
"inv_fun"
] | The unit object is always closed.
This isn't an instance because most of the time we'll prove closedness for all objects at once,
rather than just for this one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ihom : C ⥤ C | (@closed.is_adj _ _ _ A _).right | def | category_theory.ihom | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | This is the internal hom `A ⟶[C] -`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjunction : tensor_left A ⊣ ihom A | closed.is_adj.adj | def | category_theory.ihom.adjunction | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | The adjunction between `A ⊗ -` and `A ⟹ -`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ev : ihom A ⋙ tensor_left A ⟶ 𝟭 C | (ihom.adjunction A).counit | def | category_theory.ihom.ev | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | The evaluation natural transformation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coev : 𝟭 C ⟶ tensor_left A ⋙ ihom A | (ihom.adjunction A).unit | def | category_theory.ihom.coev | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | The coevaluation natural transformation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ihom_adjunction_counit : (ihom.adjunction A).counit = ev A | rfl | lemma | category_theory.ihom.ihom_adjunction_counit | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ihom_adjunction_unit : (ihom.adjunction A).unit = coev A | rfl | lemma | category_theory.ihom.ihom_adjunction_unit | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ev_naturality {X Y : C} (f : X ⟶ Y) :
((𝟙 A) ⊗ ((ihom A).map f)) ≫ (ev A).app Y = (ev A).app X ≫ f | (ev A).naturality f | lemma | category_theory.ihom.ev_naturality | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coev_naturality {X Y : C} (f : X ⟶ Y) :
f ≫ (coev A).app Y = (coev A).app X ≫ (ihom A).map ((𝟙 A) ⊗ f) | (coev A).naturality f | lemma | category_theory.ihom.coev_naturality | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ev_coev :
((𝟙 A) ⊗ ((coev A).app B)) ≫ (ev A).app (A ⊗ B) = 𝟙 (A ⊗ B) | adjunction.left_triangle_components (ihom.adjunction A) | lemma | category_theory.ihom.ev_coev | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coev_ev :
(coev A).app (A ⟶[C] B) ≫ (ihom A).map ((ev A).app B) = 𝟙 (A ⟶[C] B) | adjunction.right_triangle_components (ihom.adjunction A) | lemma | category_theory.ihom.coev_ev | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry : (A ⊗ Y ⟶ X) → (Y ⟶ (A ⟶[C] X)) | (ihom.adjunction A).hom_equiv _ _ | def | category_theory.monoidal_closed.curry | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | Currying in a monoidal closed category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uncurry : (Y ⟶ (A ⟶[C] X)) → (A ⊗ Y ⟶ X) | ((ihom.adjunction A).hom_equiv _ _).symm | def | category_theory.monoidal_closed.uncurry | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | Uncurrying in a monoidal closed category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_equiv_apply_eq (f : A ⊗ Y ⟶ X) :
(ihom.adjunction A).hom_equiv _ _ f = curry f | rfl | lemma | category_theory.monoidal_closed.hom_equiv_apply_eq | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_equiv_symm_apply_eq (f : Y ⟶ (A ⟶[C] X)) :
((ihom.adjunction A).hom_equiv _ _).symm f = uncurry f | rfl | lemma | category_theory.monoidal_closed.hom_equiv_symm_apply_eq | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry_natural_left (f : X ⟶ X') (g : A ⊗ X' ⟶ Y) :
curry (((𝟙 _) ⊗ f) ≫ g) = f ≫ curry g | adjunction.hom_equiv_naturality_left _ _ _ | lemma | category_theory.monoidal_closed.curry_natural_left | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry_natural_right (f : A ⊗ X ⟶ Y) (g : Y ⟶ Y') :
curry (f ≫ g) = curry f ≫ (ihom _).map g | adjunction.hom_equiv_naturality_right _ _ _ | lemma | category_theory.monoidal_closed.curry_natural_right | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uncurry_natural_right (f : X ⟶ (A ⟶[C] Y)) (g : Y ⟶ Y') :
uncurry (f ≫ (ihom _).map g) = uncurry f ≫ g | adjunction.hom_equiv_naturality_right_symm _ _ _ | lemma | category_theory.monoidal_closed.uncurry_natural_right | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uncurry_natural_left (f : X ⟶ X') (g : X' ⟶ (A ⟶[C] Y)) :
uncurry (f ≫ g) = ((𝟙 _) ⊗ f) ≫ uncurry g | adjunction.hom_equiv_naturality_left_symm _ _ _ | lemma | category_theory.monoidal_closed.uncurry_natural_left | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uncurry_curry (f : A ⊗ X ⟶ Y) : uncurry (curry f) = f | (closed.is_adj.adj.hom_equiv _ _).left_inv f | lemma | category_theory.monoidal_closed.uncurry_curry | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry_uncurry (f : X ⟶ (A ⟶[C] Y)) : curry (uncurry f) = f | (closed.is_adj.adj.hom_equiv _ _).right_inv f | lemma | category_theory.monoidal_closed.curry_uncurry | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry_eq_iff (f : A ⊗ Y ⟶ X) (g : Y ⟶ (A ⟶[C] X)) :
curry f = g ↔ f = uncurry g | adjunction.hom_equiv_apply_eq _ f g | lemma | category_theory.monoidal_closed.curry_eq_iff | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_curry_iff (f : A ⊗ Y ⟶ X) (g : Y ⟶ (A ⟶[C] X)) :
g = curry f ↔ uncurry g = f | adjunction.eq_hom_equiv_apply _ f g | lemma | category_theory.monoidal_closed.eq_curry_iff | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uncurry_eq (g : Y ⟶ (A ⟶[C] X)) : uncurry g = ((𝟙 A) ⊗ g) ≫ (ihom.ev A).app X | adjunction.hom_equiv_counit _ | lemma | category_theory.monoidal_closed.uncurry_eq | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry_eq (g : A ⊗ Y ⟶ X) : curry g = (ihom.coev A).app Y ≫ (ihom A).map g | adjunction.hom_equiv_unit _ | lemma | category_theory.monoidal_closed.curry_eq | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry_injective : function.injective (curry : (A ⊗ Y ⟶ X) → (Y ⟶ (A ⟶[C] X))) | (closed.is_adj.adj.hom_equiv _ _).injective | lemma | category_theory.monoidal_closed.curry_injective | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uncurry_injective : function.injective (uncurry : (Y ⟶ (A ⟶[C] X)) → (A ⊗ Y ⟶ X)) | (closed.is_adj.adj.hom_equiv _ _).symm.injective | lemma | category_theory.monoidal_closed.uncurry_injective | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uncurry_id_eq_ev : uncurry (𝟙 (A ⟶[C] X)) = (ihom.ev A).app X | by rw [uncurry_eq, tensor_id, id_comp] | lemma | category_theory.monoidal_closed.uncurry_id_eq_ev | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry_id_eq_coev : curry (𝟙 _) = (ihom.coev A).app X | by { rw [curry_eq, (ihom A).map_id (A ⊗ _)], apply comp_id } | lemma | category_theory.monoidal_closed.curry_id_eq_coev | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pre (f : B ⟶ A) : ihom A ⟶ ihom B | transfer_nat_trans_self (ihom.adjunction _) (ihom.adjunction _) ((tensoring_left C).map f) | def | category_theory.monoidal_closed.pre | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | Pre-compose an internal hom with an external hom. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_tensor_pre_app_comp_ev (f : B ⟶ A) (X : C) :
(𝟙 B ⊗ ((pre f).app X)) ≫ (ihom.ev B).app X =
(f ⊗ (𝟙 (A ⟶[C] X))) ≫ (ihom.ev A).app X | transfer_nat_trans_self_counit _ _ ((tensoring_left C).map f) X | lemma | category_theory.monoidal_closed.id_tensor_pre_app_comp_ev | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uncurry_pre (f : B ⟶ A) (X : C) :
monoidal_closed.uncurry ((pre f).app X) = (f ⊗ 𝟙 _) ≫ (ihom.ev A).app X | by rw [uncurry_eq, id_tensor_pre_app_comp_ev] | lemma | category_theory.monoidal_closed.uncurry_pre | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coev_app_comp_pre_app (f : B ⟶ A) :
(ihom.coev A).app X ≫ (pre f).app (A ⊗ X) =
(ihom.coev B).app X ≫ (ihom B).map (f ⊗ (𝟙 _)) | unit_transfer_nat_trans_self _ _ ((tensoring_left C).map f) X | lemma | category_theory.monoidal_closed.coev_app_comp_pre_app | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pre_id (A : C) [closed A] : pre (𝟙 A) = 𝟙 _ | by { simp only [pre, functor.map_id], dsimp, simp, } | lemma | category_theory.monoidal_closed.pre_id | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [
"functor.map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pre_map {A₁ A₂ A₃ : C} [closed A₁] [closed A₂] [closed A₃]
(f : A₁ ⟶ A₂) (g : A₂ ⟶ A₃) :
pre (f ≫ g) = pre g ≫ pre f | by rw [pre, pre, pre, transfer_nat_trans_self_comp, (tensoring_left C).map_comp] | lemma | category_theory.monoidal_closed.pre_map | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [
"map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pre_comm_ihom_map {W X Y Z : C} [closed W] [closed X]
(f : W ⟶ X) (g : Y ⟶ Z) :
(pre f).app Y ≫ (ihom W).map g = (ihom X).map g ≫ (pre f).app Z | by simp | lemma | category_theory.monoidal_closed.pre_comm_ihom_map | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
internal_hom [monoidal_closed C] : Cᵒᵖ ⥤ C ⥤ C | { obj := λ X, ihom X.unop,
map := λ X Y f, pre f.unop } | def | category_theory.monoidal_closed.internal_hom | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | The internal hom functor given by the monoidal closed structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_equiv (F : monoidal_functor C D) [is_equivalence F.to_functor] [h : monoidal_closed D] :
monoidal_closed C | { closed' := λ X,
{ is_adj := begin
haveI q : closed (F.to_functor.obj X) := infer_instance,
haveI : is_left_adjoint (tensor_left (F.to_functor.obj X)) := q.is_adj,
have i := comp_inv_iso (monoidal_functor.comm_tensor_left F X),
exact adjunction.left_adjoint_of_nat_iso i,
end } } | def | category_theory.monoidal_closed.of_equiv | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | Transport the property of being monoidal closed across a monoidal equivalence of categories | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_equiv_curry_def (F : monoidal_functor C D) [is_equivalence F.to_functor]
[h : monoidal_closed D] {X Y Z : C} (f : X ⊗ Y ⟶ Z) :
@monoidal_closed.curry _ _ _ _ _ _ ((monoidal_closed.of_equiv F).1 _) f =
(F.1.1.adjunction.hom_equiv Y ((ihom _).obj _)) (monoidal_closed.curry
(F.1.1.inv.adjunction.hom_equiv (F.1.... | rfl | lemma | category_theory.monoidal_closed.of_equiv_curry_def | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | Suppose we have a monoidal equivalence `F : C ≌ D`, with `D` monoidal closed. We can pull the
monoidal closed instance back along the equivalence. For `X, Y, Z : C`, this lemma describes the
resulting currying map `Hom(X ⊗ Y, Z) → Hom(Y, (X ⟶[C] Z))`. (`X ⟶[C] Z` is defined to be
`F⁻¹(F(X) ⟶[D] F(Z))`, so currying in `... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_equiv_uncurry_def
(F : monoidal_functor C D) [is_equivalence F.to_functor] [h : monoidal_closed D] {X Y Z : C}
(f : Y ⟶ (@ihom _ _ _ X $ (monoidal_closed.of_equiv F).1 X).obj Z) :
@monoidal_closed.uncurry _ _ _ _ _ _ ((monoidal_closed.of_equiv F).1 _) f =
(comp_inv_iso (F.comm_tensor_left X)).inv.app Y ≫ (F.... | rfl | lemma | category_theory.monoidal_closed.of_equiv_uncurry_def | category_theory.closed | src/category_theory/closed/monoidal.lean | [
"category_theory.monoidal.functor",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.functor.inv_isos"
] | [] | Suppose we have a monoidal equivalence `F : C ≌ D`, with `D` monoidal closed. We can pull the
monoidal closed instance back along the equivalence. For `X, Y, Z : C`, this lemma describes the
resulting uncurrying map `Hom(Y, (X ⟶[C] Z)) → Hom(X ⊗ Y ⟶ Z)`. (`X ⟶[C] Z` is
defined to be `F⁻¹(F(X) ⟶[D] F(Z))`, so uncurrying... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_homset_of_initial_iso_terminal [has_initial C] (i : ⊥_ C ≅ ⊤_ C) (X Y : C) :
unique (X ⟶ Y) | equiv.unique $
calc (X ⟶ Y) ≃ (X ⨯ ⊤_ C ⟶ Y) : iso.hom_congr (prod.right_unitor _).symm (iso.refl _)
... ≃ (X ⨯ ⊥_ C ⟶ Y) : iso.hom_congr (prod.map_iso (iso.refl _) i.symm) (iso.refl _)
... ≃ (⊥_ C ⟶ Y ^^ X) : (exp.adjunction _).hom_equiv _ _ | def | category_theory.unique_homset_of_initial_iso_terminal | category_theory.closed | src/category_theory/closed/zero.lean | [
"category_theory.closed.cartesian",
"category_theory.punit",
"category_theory.conj",
"category_theory.limits.shapes.zero_objects"
] | [
"equiv.unique",
"unique"
] | If a cartesian closed category has an initial object which is isomorphic to the terminal object,
then each homset has exactly one element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_homset_of_zero [has_zero_object C] (X Y : C) :
unique (X ⟶ Y) | begin
haveI : has_initial C := has_zero_object.has_initial,
apply unique_homset_of_initial_iso_terminal _ X Y,
refine ⟨default, (default : ⊤_ C ⟶ 0) ≫ default, _, _⟩; simp
end | def | category_theory.unique_homset_of_zero | category_theory.closed | src/category_theory/closed/zero.lean | [
"category_theory.closed.cartesian",
"category_theory.punit",
"category_theory.conj",
"category_theory.limits.shapes.zero_objects"
] | [
"unique"
] | If a cartesian closed category has a zero object, each homset has exactly one element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_punit [has_zero_object C] : C ≌ discrete punit | equivalence.mk
(functor.star C)
(functor.from_punit 0)
(nat_iso.of_components
(λ X, { hom := default, inv := default })
(λ X Y f, dec_trivial))
(functor.punit_ext _ _) | def | category_theory.equiv_punit | category_theory.closed | src/category_theory/closed/zero.lean | [
"category_theory.closed.cartesian",
"category_theory.punit",
"category_theory.conj",
"category_theory.limits.shapes.zero_objects"
] | [] | A cartesian closed category with a zero object is equivalent to the category with one object and
one morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concrete_category (C : Type u) [category.{v} C] | (forget [] : C ⥤ Type w)
[forget_faithful : faithful forget] | class | category_theory.concrete_category | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | A concrete category is a category `C` with a fixed faithful functor `forget : C ⥤ Type`.
Note that `concrete_category` potentially depends on three independent universe levels,
* the universe level `w` appearing in `forget : C ⥤ Type w`
* the universe level `v` of the morphisms (i.e. we have a `category.{v} C`)
* the ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget (C : Type u) [category.{v} C] [concrete_category.{w} C] : C ⥤ Type w | concrete_category.forget C | def | category_theory.forget | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | The forgetful functor from a concrete category to `Type u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concrete_category.types : concrete_category (Type u) | { forget := 𝟭 _ } | instance | category_theory.concrete_category.types | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete_category.has_coe_to_sort (C : Type u) [category.{v} C] [concrete_category.{w} C] :
has_coe_to_sort C (Type w) | ⟨(concrete_category.forget C).obj⟩ | def | category_theory.concrete_category.has_coe_to_sort | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | Provide a coercion to `Type u` for a concrete category. This is not marked as an instance
as it could potentially apply to every type, and so is too expensive in typeclass search.
You can use it on particular examples as:
```
instance : has_coe_to_sort X := concrete_category.has_coe_to_sort X
``` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_obj_eq_coe {X : C} : (forget C).obj X = X | rfl | lemma | category_theory.forget_obj_eq_coe | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete_category.has_coe_to_fun {X Y : C} : has_coe_to_fun (X ⟶ Y) (λ f, X → Y) | ⟨λ f, (forget _).map f⟩ | def | category_theory.concrete_category.has_coe_to_fun | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | Usually a bundled hom structure already has a coercion to function
that works with different universes. So we don't use this as a global instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concrete_category.hom_ext {X Y : C} (f g : X ⟶ Y) (w : ∀ x : X, f x = g x) : f = g | begin
apply faithful.map_injective (forget C),
ext,
exact w x,
end | lemma | category_theory.concrete_category.hom_ext | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | In any concrete category, we can test equality of morphisms by pointwise evaluations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_map_eq_coe {X Y : C} (f : X ⟶ Y) : (forget C).map f = f | rfl | lemma | category_theory.forget_map_eq_coe | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_hom {X Y : C} {f g : X ⟶ Y} (h : f = g) (x : X) : f x = g x | congr_fun (congr_arg (λ k : X ⟶ Y, (k : X → Y)) h) x | lemma | category_theory.congr_hom | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | Analogue of `congr_fun h x`,
when `h : f = g` is an equality between morphisms in a concrete category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id {X : C} : ((𝟙 X) : X → X) = id | (forget _).map_id X | lemma | category_theory.coe_id | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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