statement stringlengths 1 1.46k | proof stringlengths 0 10.7k | type stringclasses 18
values | symbolic_name stringlengths 1 52 | library stringclasses 74
values | filename stringclasses 268
values | imports listlengths 1 31 | deps listlengths 0 34 | docstring stringclasses 507
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Adjunction_Hom | := {
hom_adj : Hom C ◯ F^op ∏⟶ Id ≅[[D^op ∏ C, Sets]] Hom D ◯ Id^op ∏⟶ U
}. | Class | Adjunction_Hom | Adjunction | Adjunction/Hom.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Theory.Adjunction",
"Category.Adjunction.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.F... | [
"Hom",
"Id",
"Sets",
"op"
] | Wikipedia: "A hom-set adjunction between two categories C and D consists of
two functors F : C ← D and G : C → D and a natural isomorphism
Φ : homC(F −, −) → homD(−, G −).
"This specifies a family of bijections
Φy,x : homC(F y, x) → homD(y, G x)
for all objects x in C and y in D.
"In this sit... | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb |
hom_unit : Id ⟹ U ◯ F := {|
transform := fun x => @morphism _ _ _ _ (to hom_adj (x, F x)) id
|}. | Next Obligation.
spose (naturality[to hom_adj] (x, F x) (x, F y) (id, fmap[F] f) id) as X.
rewrite id_right in X.
rewrites.
spose (naturality[to hom_adj] (y, F y) (x, F y) (f, id) id) as X.
rewrite fmap_id, id_left in X.
rewrites.
apply proper_morphism; cat.
Qed. | Definition | hom_unit | Adjunction | Adjunction/Hom.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Theory.Adjunction",
"Category.Adjunction.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.F... | [
"Id",
"cat",
"fmap",
"morphism",
"rewrites",
"to",
"transform"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
hom_counit : F ◯ U ⟹ Id := {|
transform := fun x => @morphism _ _ _ _ (from hom_adj (U x, x)) id
|}. | Next Obligation.
spose (naturality[from hom_adj] (U x, x) (U x, y) (id, f) id) as X.
rewrite fmap_id, id_right in X.
rewrites.
spose (naturality[from hom_adj] (U y, y) (U x, y) (fmap[U] f, id) id) as X.
rewrite id_left in X.
rewrites.
apply proper_morphism; cat.
Qed. | Definition | hom_counit | Adjunction | Adjunction/Hom.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Theory.Adjunction",
"Category.Adjunction.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.F... | [
"Id",
"cat",
"fmap",
"from",
"morphism",
"rewrites",
"transform"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
hom_unit_naturality_consequence {x y} (f : F x ~> y) :
to hom_adj (x, y) f ≈ fmap[U] f ∘ hom_unit _. | Proof.
unfold hom_unit; simpl.
spose (naturality[to hom_adj] (x, F x) (x, y) (id, f) id) as X.
rewrite id_right in X.
rewrites.
apply proper_morphism; cat.
Qed. | Theorem | hom_unit_naturality_consequence | Adjunction | Adjunction/Hom.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Theory.Adjunction",
"Category.Adjunction.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.F... | [
"cat",
"fmap",
"hom_unit",
"rewrites",
"to"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
hom_counit_naturality_consequence {x y} (f : x ~> U y) :
from hom_adj (x, y) f ≈ hom_counit _ ∘ fmap[F] f. | Proof.
unfold hom_counit; simpl.
spose (naturality[from hom_adj] (U y, y) (x, y) (f, id) id) as X.
rewrite id_left in X.
rewrites.
apply proper_morphism; cat.
Qed. | Theorem | hom_counit_naturality_consequence | Adjunction | Adjunction/Hom.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Theory.Adjunction",
"Category.Adjunction.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.F... | [
"cat",
"fmap",
"from",
"hom_counit",
"rewrites"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
hom_counit_fmap_unit {x} :
hom_counit (F x) ∘ fmap[F] (hom_unit x) ≈ id. | Proof.
spose (@hom_counit_naturality_consequence x (F x) (hom_unit x)) as X.
rewrites.
unfold hom_unit; simpl.
srewrite (iso_from_to hom_adj (x, F x) id); cat.
Qed. | Theorem | hom_counit_fmap_unit | Adjunction | Adjunction/Hom.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Theory.Adjunction",
"Category.Adjunction.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.F... | [
"cat",
"fmap",
"hom_counit",
"hom_counit_naturality_consequence",
"hom_unit",
"rewrites",
"srewrite"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
hom_fmap_counit_unit {x} :
fmap[U] (hom_counit x) ∘ hom_unit (U x) ≈ id. | Proof.
spose (@hom_unit_naturality_consequence (U x) x (hom_counit x)) as X.
rewrites.
unfold hom_unit; simpl.
srewrite (iso_to_from hom_adj (U x, x) id); cat.
Qed. | Theorem | hom_fmap_counit_unit | Adjunction | Adjunction/Hom.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Theory.Adjunction",
"Category.Adjunction.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.F... | [
"cat",
"fmap",
"hom_counit",
"hom_unit",
"hom_unit_naturality_consequence",
"rewrites",
"srewrite"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Adjunction_Hom_to_Transform : F ∹ U | := {|
unit := hom_unit;
counit := hom_counit;
counit_fmap_unit := @hom_counit_fmap_unit;
fmap_counit_unit := @hom_fmap_counit_unit
|}. | Definition | Adjunction_Hom_to_Transform | Adjunction | Adjunction/Hom.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Theory.Adjunction",
"Category.Adjunction.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.F... | [
"counit",
"counit_fmap_unit",
"fmap_counit_unit",
"hom_counit",
"hom_counit_fmap_unit",
"hom_fmap_counit_unit",
"hom_unit",
"unit"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Adjunction_Transform_to_Hom (A : F ∹ U) : Adjunction_Hom := {|
hom_adj :=
{| to := {| transform := fun _ =>
{| morphism := fun f => fmap[U] f ∘ unit _ |} |}
; from := {| transform := fun _ =>
{| morphism := fun f => counit _ ∘ fmap[F] f |} |} |}
|}. | Next Obligation.
proper; rewrites; reflexivity.
Qed. | Definition | Adjunction_Transform_to_Hom | Adjunction | Adjunction/Hom.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Theory.Adjunction",
"Category.Adjunction.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.F... | [
"Adjunction_Hom",
"counit",
"fmap",
"from",
"morphism",
"proper",
"rewrites",
"to",
"transform",
"unit"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Adjunction_Hom_to_Universal : F ⊣ U := {|
adj := fun a b =>
{| to := transform (to hom_adj) (a, b)
; from := transform (from hom_adj) (a, b) |}
|}. | Next Obligation.
simpl; srewrite (iso_to_from hom_adj (a, b) x); cat.
Qed. | Definition | Adjunction_Hom_to_Universal | Adjunction | Adjunction/Hom.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Theory.Adjunction",
"Category.Adjunction.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.F... | [
"cat",
"from",
"srewrite",
"to",
"transform"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Adjunction_Universal_to_Hom (A : F ⊣ U) : Adjunction_Hom := {|
hom_adj :=
{| to := {| transform := fun _ => {| morphism := to adj |} |}
; from := {| transform := fun _ => {| morphism := from adj |} |} |}
|}. | Next Obligation.
rewrite <- comp_assoc.
rewrite to_adj_nat_l.
rewrite comp_assoc.
rewrite to_adj_nat_r.
reflexivity.
Qed. | Definition | Adjunction_Universal_to_Hom | Adjunction | Adjunction/Hom.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Theory.Adjunction",
"Category.Adjunction.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.F... | [
"Adjunction_Hom",
"from",
"morphism",
"to",
"transform"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Opposite_Adjunction `(F : D ⟶ C) `(U : C ⟶ D)
(A : F ⊣ U) :
U^op ⊣ F^op | := {|
adj := fun x y =>
{| to := from (@adj _ _ _ _ A y x)
; from := to (@adj _ _ _ _ A y x)
; iso_to_from := iso_from_to (@adj _ _ _ _ A y x)
; iso_from_to := iso_to_from (@adj _ _ _ _ A y x) |};
to_adj_nat_l := fun _ _ _ f g => @from_adj_nat_r _ _ _ _ A _ _ _ g f;
to_adj_na... | Definition | Opposite_Adjunction | Adjunction | Adjunction/Opposite.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Adjunction",
"Category.Functor.Opposite"
] | [
"from",
"op",
"to"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
"N ^op" | := (@Opposite_Adjunction _ _ _ _ N)
(at level 7, format "N ^op", left associativity) : adjunction_scope. | Notation | N ^op | Adjunction | Adjunction/Opposite.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Adjunction",
"Category.Functor.Opposite"
] | [
"Opposite_Adjunction",
"left"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Opposite_Adjunction_invol `(F : D ⟶ C) `(U : C ⟶ D) (A : F ⊣ U) :
(A^op)^op = A. | Proof. reflexivity. Qed. | Corollary | Opposite_Adjunction_invol | Adjunction | Adjunction/Opposite.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Adjunction",
"Category.Functor.Opposite"
] | [
"op"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Diagonal_Product_Adjunction (C : Category) `{@Cartesian C} :
Diagonal_Product C ⊣ ×(C) := {
adj := fun _ _ =>
{| to := {| morphism := fun f => fst f △ snd f |}
; from := {| morphism := fun f => (exl ∘ f, exr ∘ f) |} |}
}. | Next Obligation. proper; apply fork_respects; auto. Qed. | Instance | Diagonal_Product_Adjunction | Adjunction.Diagonal | Adjunction/Diagonal/Product.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Adjunction",
"Category.Functor.Diagonal",
"Category.Functor.Product.Internal",
"Category.Structure.Cartesian",
"Category.Instance.Sets"
] | [
"Cartesian",
"Category",
"Diagonal_Product",
"fork_respects",
"from",
"morphism",
"proper",
"to"
] | jww (2021-08-04): Is it right to use Diagonal_Product here? | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb |
Adjunction_Transform | := {
unit : Id ⟹ U ◯ F;
counit : F ◯ U ⟹ Id;
counit_fmap_unit {X} :
transform[counit] (F X) ∘ fmap[F] (transform[unit] X) ≈ id;
fmap_counit_unit {X} :
fmap[U] (transform[counit] X) ∘ transform[unit] (U X) ≈ id
}. | Class | Adjunction_Transform | Adjunction.Natural | Adjunction/Natural/Transformation.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation"
] | [
"Id",
"counit",
"counit_fmap_unit",
"fmap",
"fmap_counit_unit",
"transform",
"unit"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
"F ∹ G" | := (@Adjunction_Transform _ _ F G)
(at level 59) : adjunction_type_scope. | Notation | F ∹ G | Adjunction.Natural | Adjunction/Natural/Transformation.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation"
] | [
"Adjunction_Transform"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
"unit[ A ]" | := (@unit _ _ _ _ A)
(at level 9, format "unit[ A ]") : morphism_scope. | Notation | unit[ A ] | Adjunction.Natural | Adjunction/Natural/Transformation.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation"
] | [
"unit"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
"counit[ A ]" | := (@counit _ _ _ _ A)
(at level 9, format "counit[ A ]") : morphism_scope. | Notation | counit[ A ] | Adjunction.Natural | Adjunction/Natural/Transformation.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation"
] | [
"counit"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Opposite_Adjunction_Transform
`(F : D ⟶ C) `(U : C ⟶ D) (A : F ∹ U) :
U^op ∹ F^op := {|
unit := _;
counit := _
|}. | Next Obligation.
transform; simpl; intros.
- apply counit.
- apply (@naturality_sym _ _ _ _ counit).
- apply (@naturality _ _ _ _ counit).
Defined. | Definition | Opposite_Adjunction_Transform | Adjunction.Natural.Transformation | Adjunction/Natural/Transformation/Opposite.v | [
"Category.Lib",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Functor.Opposite",
"Category.Adjunction.Natural.Transformation"
] | [
"counit",
"op",
"transform",
"unit"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Opposite_Adjunction_Transform_invol
`(F : D ⟶ C) `(U : C ⟶ D) (A : F ∹ U) :
Opposite_Adjunction_Transform
(U^op) (F^op) (Opposite_Adjunction_Transform F U A) = A. | Proof. reflexivity. Qed. | Corollary | Opposite_Adjunction_Transform_invol | Adjunction.Natural.Transformation | Adjunction/Natural/Transformation/Opposite.v | [
"Category.Lib",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Functor.Opposite",
"Category.Adjunction.Natural.Transformation"
] | [
"Opposite_Adjunction_Transform",
"op"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Adjunction_from_Transform (A : F ∹ U) : F ⊣ U := {|
adj := fun a b =>
{| to := {| morphism := fun f => fmap f ∘ transform[unit[A]] a |}
; from := {| morphism := fun f => transform[counit[A]] b ∘ fmap f |} |}
|}. | Next Obligation. proper; now rewrites. Qed. | Definition | Adjunction_from_Transform | Adjunction.Natural.Transformation | Adjunction/Natural/Transformation/Universal.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Adjunction.Natural.Transformation",
"Category.Theory.Adjunction",
"Category.Instance.Sets"
] | [
"counit",
"fmap",
"from",
"morphism",
"proper",
"rewrites",
"to",
"transform",
"unit"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Adjunction_to_Transform {A : F ⊣ U} : F ∹ U := {|
Transformation.unit := {| transform := fun _ => unit |};
Transformation.counit := {| transform := fun _ => counit |}
|}. | Next Obligation.
unfold unit.
rewrite <- to_adj_nat_r, <- to_adj_nat_l; cat.
Qed. | Definition | Adjunction_to_Transform | Adjunction.Natural.Transformation | Adjunction/Natural/Transformation/Universal.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Adjunction.Natural.Transformation",
"Category.Theory.Adjunction",
"Category.Instance.Sets"
] | [
"cat",
"counit",
"transform",
"unit"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Arrow {C : Category} : Category | := (Id[C] ↓ Id[C]). | Definition | Arrow | Construction | Construction/Arrow.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Functor",
"Category.Construction.Comma"
] | [
"Category",
"Id"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
"C ⃗" | := (@Arrow C) (at level 90) : category_scope. | Notation | C ⃗ | Construction | Construction/Arrow.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Functor",
"Category.Construction.Comma"
] | [
"Arrow"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Cayley : Category := {
obj := C;
hom := fun x y =>
{ f : ∀ r, (y ~> r) → (x ~> r)
& Proper (forall_relation (fun _ => respectful equiv equiv)) f ∧
∀ (r : C) (k : y ~> r), f r k ≈ k ∘ f _ id };
homset := fun x y => {| equiv := fun f g => ∀ r k, `1 f r k ≈ `1 g r k |};
id := fun _ => (... | Next Obligation.
equivalence.
now rewrite X, X0.
Qed. | Instance | Cayley | Construction | Construction/Cayley.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Structure.Cartesian",
"Category.Functor.Hom.Yoneda",
"Category.Instance.Sets",
"Category.Functor.Structure.Cartesian"
] | [
"Category",
"equivalence",
"obj"
] | Given any category, the Cayley representation forces all associations to
the left. | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb |
To_Cayley : C ⟶ Cayley := {
fobj := fun x => x;
fmap := fun _ _ f => (fun _ k => k ∘ f; _);
}. | Next Obligation.
proper.
proper.
Defined. | Instance | To_Cayley | Construction | Construction/Cayley.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Structure.Cartesian",
"Category.Functor.Hom.Yoneda",
"Category.Instance.Sets",
"Category.Functor.Structure.Cartesian"
] | [
"Cayley",
"fmap",
"fobj",
"proper"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
From_Cayley : Cayley ⟶ C | := {
fobj := fun x => x;
fmap := fun _ y f => `1 f y (@id C y);
}. | Instance | From_Cayley | Construction | Construction/Cayley.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Structure.Cartesian",
"Category.Functor.Hom.Yoneda",
"Category.Instance.Sets",
"Category.Functor.Structure.Cartesian"
] | [
"Cayley",
"fmap",
"fobj"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Cayley_Right (x y z w : C) (f : z ~> w) (g : y ~> z) (h : x ~> y) :
(∀ a b (k : a ~{C}~> b), id[b] ∘ k = k) ->
f ∘ g ∘ h = fmap[From_Cayley]
(fmap[To_Cayley] f ∘ (fmap[To_Cayley] g
∘ fmap[To_Cayley] h)). | Proof.
intros.
simpl.
rewrite H0.
reflexivity.
Qed. | Lemma | Cayley_Right | Construction | Construction/Cayley.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Structure.Cartesian",
"Category.Functor.Hom.Yoneda",
"Category.Instance.Sets",
"Category.Functor.Structure.Cartesian"
] | [
"From_Cayley",
"To_Cayley",
"fmap"
] | No matter how we associate the mapped morphisms, the functor back from
Cayley yields them left-associated. | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb |
Cayley_Left (x y z w : C) (f : z ~> w) (g : y ~> z) (h : x ~> y) :
(∀ a b (k : a ~{C}~> b), id[b] ∘ k = k) ->
f ∘ g ∘ h = fmap[From_Cayley]
(((fmap[To_Cayley] f ∘ fmap[To_Cayley] g)
∘ fmap[To_Cayley] h)). | Proof.
intros.
simpl.
rewrite H0.
reflexivity.
Qed. | Lemma | Cayley_Left | Construction | Construction/Cayley.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Structure.Cartesian",
"Category.Functor.Hom.Yoneda",
"Category.Instance.Sets",
"Category.Functor.Structure.Cartesian"
] | [
"From_Cayley",
"To_Cayley",
"fmap"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Cayley_Cartesian `{CA : @Cartesian C} : @Cartesian Cayley := {
product_obj := @product_obj C CA;
fork := fun x y z f g =>
let f' := to (Covariant_Yoneda_Embedding C x y) (_ f) in
let g' := to (Covariant_Yoneda_Embedding C x z) (_ g) in
_ f' g';
exl := fun x y =>
let f' := from (Covariant_Yoneda_Em... | Next Obligation.
construct.
- construct.
+ apply f.
exact X.
+ proper.
rewrite e1.
rewrite X.
rewrite <- e1.
reflexivity.
- simpl.
rewrite e1.
rewrite comp_assoc.
rewrite <- e1.
reflexivity.
- simpl.
rewrite e1.
rewrite <- comp_assoc.
rewrite <- ... | Instance | Cayley_Cartesian | Construction | Construction/Cayley.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Structure.Cartesian",
"Category.Functor.Hom.Yoneda",
"Category.Instance.Sets",
"Category.Functor.Structure.Cartesian"
] | [
"Cartesian",
"Cayley",
"Covariant_Yoneda_Embedding",
"construct",
"fork",
"from",
"proper",
"to"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
To_Cayley_CartesianFunctor `{@Cartesian C} :
@CartesianFunctor _ _ To_Cayley _ Cayley_Cartesian. | Instance | To_Cayley_CartesianFunctor | Construction | Construction/Cayley.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Structure.Cartesian",
"Category.Functor.Hom.Yoneda",
"Category.Instance.Sets",
"Category.Functor.Structure.Cartesian"
] | [
"Cartesian",
"CartesianFunctor",
"Cayley_Cartesian",
"To_Cayley"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | ||
From_Cayley_CartesianFunctor `{@Cartesian C} :
@CartesianFunctor _ _ From_Cayley Cayley_Cartesian _. | Instance | From_Cayley_CartesianFunctor | Construction | Construction/Cayley.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Structure.Cartesian",
"Category.Functor.Hom.Yoneda",
"Category.Instance.Sets",
"Category.Functor.Structure.Cartesian"
] | [
"Cartesian",
"CartesianFunctor",
"Cayley_Cartesian",
"From_Cayley"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | ||
ColouredPROP : Type | := {
cprop_cat : Category;
cprop_strict : @StrictMonoidal cprop_cat;
cprop_symmetric : @SymmetricMonoidal cprop_cat;
(** Coherence between the two [Monoidal] paths — see the analogous
[prop_monoidal_coherence] in [Construction/PROP.v] for
motivation. *)
cprop_monoidal_coherence :
(@strict_is... | Class | ColouredPROP | Construction | Construction/ColouredPROP.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Functor.Bifunctor",
"Category.Structure.Monoidal",
"Category.Structure.Monoidal.Braided",
"Category.Structure.Monoidal.Symmetric",
"Category.Structure.Monoidal.Strict",
"Category.Structu... | [
"Category",
"StrictMonoidal",
"SymmetricMonoidal",
"braided_is_monoidal",
"cprop_cat",
"obj",
"object",
"strict_is_monoidal",
"symmetric_is_braided"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
cprop_cat : ColouredPROP >-> Category. | Coercion | cprop_cat | Construction | Construction/ColouredPROP.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Functor.Bifunctor",
"Category.Structure.Monoidal",
"Category.Structure.Monoidal.Braided",
"Category.Structure.Monoidal.Symmetric",
"Category.Structure.Monoidal.Strict",
"Category.Structu... | [
"Category",
"ColouredPROP"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | ||
"'⟦' cs '⟧c'" | := (cprop_of_list (ColouredPROP := P) cs)
(at level 0, format "⟦ cs ⟧c"). | Notation | '⟦' cs '⟧c' | Construction | Construction/ColouredPROP.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Functor.Bifunctor",
"Category.Structure.Monoidal",
"Category.Structure.Monoidal.Braided",
"Category.Structure.Monoidal.Symmetric",
"Category.Structure.Monoidal.Strict",
"Category.Structu... | [
"ColouredPROP"
] | The Coloured-PROP-object indexed by colour list [cs]. | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb |
cprop_I_eq_nil : @I P _ = ⟦nil⟧c. | Proof. exact (@cprop_unit_nil Colour P). Qed. | Lemma | cprop_I_eq_nil | Construction | Construction/ColouredPROP.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Functor.Bifunctor",
"Category.Structure.Monoidal",
"Category.Structure.Monoidal.Braided",
"Category.Structure.Monoidal.Symmetric",
"Category.Structure.Monoidal.Strict",
"Category.Structu... | [] | Sanity: the unit object of [P] is [⟦[]⟧c]. | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb |
cprop_tensor_eq_app (cs ds : list Colour) :
(⟦cs⟧c ⨂ ⟦ds⟧c)%object = ⟦cs ++ ds⟧c. | Proof. exact (cprop_tensor_app cs ds). Qed. | Lemma | cprop_tensor_eq_app | Construction | Construction/ColouredPROP.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Functor.Bifunctor",
"Category.Structure.Monoidal",
"Category.Structure.Monoidal.Braided",
"Category.Structure.Monoidal.Symmetric",
"Category.Structure.Monoidal.Strict",
"Category.Structu... | [
"object"
] | Sanity: tensor on objects is list concatenation. | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb |
wire (c : Colour) : @obj P | := ⟦[c]⟧c. | Definition | wire | Construction | Construction/ColouredPROP.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Functor.Bifunctor",
"Category.Structure.Monoidal",
"Category.Structure.Monoidal.Braided",
"Category.Structure.Monoidal.Symmetric",
"Category.Structure.Monoidal.Strict",
"Category.Structu... | [
"obj"
] | Singleton-colour wire: [⟦[c]⟧c] is the canonical 1-colour object. | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb |
HypergraphColouredPROP {Colour : Type} : Type | := {
hcprop : ColouredPROP Colour;
hcprop_hyper : @Hypergraph
(@cprop_cat Colour hcprop)
(@cprop_symmetric Colour hcprop)
}. | Class | HypergraphColouredPROP | Construction | Construction/ColouredPROP.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Functor.Bifunctor",
"Category.Structure.Monoidal",
"Category.Structure.Monoidal.Braided",
"Category.Structure.Monoidal.Symmetric",
"Category.Structure.Monoidal.Strict",
"Category.Structu... | [
"ColouredPROP",
"Hypergraph",
"cprop_cat",
"hcprop"
] | ** Hypergraph Coloured PROPs
Analogous to [HypergraphPROP]: a [ColouredPROP] whose underlying
symmetric monoidal category carries a [Hypergraph] instance.
Every wire-list carries a special commutative Frobenius algebra
coherent under concatenation. | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb |
hcprop : HypergraphColouredPROP >-> ColouredPROP. | Coercion | hcprop | Construction | Construction/ColouredPROP.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Functor.Bifunctor",
"Category.Structure.Monoidal",
"Category.Structure.Monoidal.Braided",
"Category.Structure.Monoidal.Symmetric",
"Category.Structure.Monoidal.Strict",
"Category.Structu... | [
"ColouredPROP",
"HypergraphColouredPROP"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | ||
Comma : Category := {|
obj := ∃ p : A ∏ B, S (fst p) ~{C}~> T (snd p);
hom := fun x y =>
∃ f : (fst (`1 x) ~{A}~> fst (`1 y)) * (snd (`1 x) ~{B}~> snd (`1 y)),
`2 y ∘ fmap[S] (fst f) ≈ fmap[T] (snd f) ∘ `2 x;
homset := fun _ _ =>
{| equiv := fun f g => (fst `1 f ≈ fst `1 g) * (snd `1 f ≈ sn... | Next Obligation.
intros [[]] [[]]; simpl in *; equivalence.
Qed. | Definition | Comma | Construction | Construction/Comma.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.Functor.Opposite"
] | [
"Category",
"equivalence",
"fmap",
"obj"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
comma_proj : Comma ⟶ A ∏ B := {|
fobj := fun x => ``x;
fmap := fun _ _ f => ``f
|}. | Next Obligation.
intros ? ? ? ? [e0 e1].
now split.
Qed. | Instance | comma_proj | Construction | Construction/Comma.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.Functor.Opposite"
] | [
"Comma",
"fmap",
"fobj",
"split"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
comma_proj1 : Comma ⟶ A := {|
fobj := fun x => fst ``x;
fmap := fun _ _ f => fst ``f
|}. | Next Obligation. now intros ? ? ? ? [e0 e1]. Qed. | Instance | comma_proj1 | Construction | Construction/Comma.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.Functor.Opposite"
] | [
"Comma",
"fmap",
"fobj"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
comma_proj2 : Comma ⟶ B := {|
fobj := fun x => snd ``x;
fmap := fun _ _ f => snd ``f
|}. | Next Obligation. now intros ? ? ? ? [e0 e1]. Qed. | Instance | comma_proj2 | Construction | Construction/Comma.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.Functor.Opposite"
] | [
"Comma",
"fmap",
"fobj"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
comma_proj_nat : S ◯ comma_proj1 ⟹ T ◯ comma_proj2. | Instance | comma_proj_nat | Construction | Construction/Comma.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.Functor.Opposite"
] | [
"comma_proj1",
"comma_proj2"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | ||
"S ↓ T" | := (@Comma _ _ _ S T) (at level 90) : category_scope. | Notation | S ↓ T | Construction | Construction/Comma.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.Functor.Opposite"
] | [
"Comma"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
comma_proj_mor_iso A B C (S : A ⟶ C) (T : B ⟶ C) (x y : S ↓ T) :
x ≅ y → `1 x ≅[A ∏ B] `1 y. | Proof.
destruct 1; simpl.
isomorphism.
- exact (`1 to).
- exact (`1 from).
- apply iso_to_from.
- apply iso_from_to.
Defined. | Theorem | comma_proj_mor_iso | Construction | Construction/Comma.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.Functor.Opposite"
] | [
"from",
"isomorphism",
"to"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
comma_proj_com_iso A B C (S : A ⟶ C) (T : B ⟶ C) (x y : S ↓ T) :
∀ iso : x ≅ y,
`2 x ≈ fmap[T] (snd `1 (from iso)) ∘ `2 y ∘ fmap[S] (fst `1 (to iso)). | Proof.
intros.
pose proof (iso_from_to iso); simpl in X.
destruct (from iso), x0; simpl in *.
rewrite <- e.
rewrite <- comp_assoc.
rewrite <- fmap_comp.
rewrite (fst X).
cat.
Qed. | Theorem | comma_proj_com_iso | Construction | Construction/Comma.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.Functor.Opposite"
] | [
"cat",
"fmap",
"from",
"to"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Cocomma {A : Category} {B : Category} {C : Category}
{S : A ⟶ C} {T : B ⟶ C} | := @Comma (B^op) (A^op) (C^op) (T^op) (S^op). | Definition | Cocomma | Construction | Construction/Comma.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.Functor.Opposite"
] | [
"Category",
"Comma",
"op"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
"S ↑ T" | := (@Cocomma _ _ _ S T) (at level 90) : category_scope. | Notation | S ↑ T | Construction | Construction/Comma.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Construction.Product",
"Category.Construction.Opposite",
"Category.Functor.Opposite"
] | [
"Cocomma"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Coproduct : Category := {|
obj := C + D;
hom := fun x y =>
match x return Type with
| Datatypes.inl x =>
match y with
| Datatypes.inl y => x ~> y
| Datatypes.inr _ => False
end
| Datatypes.inr x ... | Next Obligation.
destruct x.
- destruct y.
+ exact (f ≈ g).
+ contradiction.
- destruct y.
+ contradiction.
+ exact (f ≈ g).
Defined. | Definition | Coproduct | Construction | Construction/Coproduct.v | [
"Category.Lib",
"Category.Theory.Category"
] | [
"Category",
"inl",
"inr",
"obj"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
"C ∐ D" | := (@Coproduct C D) (at level 90) : category_scope. | Notation | C ∐ D | Construction | Construction/Coproduct.v | [
"Category.Lib",
"Category.Theory.Category"
] | [
"Coproduct"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
DecoratedCospanArrow (X Y : C) : Type | := {
dc_cospan : CospanArrow X Y;
dc_decoration : @I D _ ~{D}~> F (cospan_apex dc_cospan)
}. | Record | DecoratedCospanArrow | Construction | Construction/DecoratedCospan.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Functor.Bifunctor",
"Category.Construction.Product",
"Category.Structure.Cocartesian",
"Category.Structure.Initial",
"Category.Structure.Termina... | [
"CospanArrow"
] | ** Decorated cospan arrows
A decorated cospan from [X] to [Y] is a cospan together with a
decoration of its apex. | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb |
dec_cospan_equiv {X Y : C} (f g : DecoratedCospanArrow X Y) : Type | := {
dce_cospan_eq : cospan_equiv (dc_cospan f) (dc_cospan g);
dce_dec_eq :
fmap[F] (to (projT1 dce_cospan_eq)) ∘ dc_decoration f
≈ dc_decoration g
}. | Record | dec_cospan_equiv | Construction | Construction/DecoratedCospan.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Functor.Bifunctor",
"Category.Construction.Product",
"Category.Structure.Cocartesian",
"Category.Structure.Initial",
"Category.Structure.Termina... | [
"DecoratedCospanArrow",
"cospan_equiv",
"fmap",
"to"
] | ** Equivalence of decorated cospans
Two decorated cospans are equivalent if their underlying cospans are
equivalent via an apex isomorphism [phi] AND their decorations agree
up to transport along [F phi]:
F(to phi) ∘ d_f ≈ d_g.
This is the categorical analogue of "the decoration on the LHS apex... | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb |
dec_cospan_equiv_refl {X Y : C} (f : DecoratedCospanArrow X Y) :
dec_cospan_equiv f f. | Proof.
unshelve econstructor.
- apply cospan_equiv_refl.
- simpl. rewrite fmap_id, id_left. reflexivity.
Defined. | Lemma | dec_cospan_equiv_refl | Construction | Construction/DecoratedCospan.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Functor.Bifunctor",
"Category.Construction.Product",
"Category.Structure.Cocartesian",
"Category.Structure.Initial",
"Category.Structure.Termina... | [
"DecoratedCospanArrow",
"cospan_equiv_refl",
"dec_cospan_equiv"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
dec_cospan_equiv_sym {X Y : C} (f g : DecoratedCospanArrow X Y) :
dec_cospan_equiv f g -> dec_cospan_equiv g f. | Proof.
intros [E He].
unshelve econstructor.
- apply cospan_equiv_sym; exact E.
- destruct E as [phi [E1 E2]]; simpl in *.
(* Need: fmap[F] (from phi) ∘ dc_decoration g ≈ dc_decoration f.
Have: fmap[F] (to phi) ∘ dc_decoration f ≈ dc_decoration g.
Apply fmap[F] (from phi) ∘ - to both sides. *)... | Lemma | dec_cospan_equiv_sym | Construction | Construction/DecoratedCospan.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Functor.Bifunctor",
"Category.Construction.Product",
"Category.Structure.Cocartesian",
"Category.Structure.Initial",
"Category.Structure.Termina... | [
"DecoratedCospanArrow",
"cospan_equiv_sym",
"dec_cospan_equiv"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
dec_cospan_equiv_trans {X Y : C} (f g h : DecoratedCospanArrow X Y) :
dec_cospan_equiv f g -> dec_cospan_equiv g h -> dec_cospan_equiv f h. | Proof.
intros [E He] [E' He'].
unshelve econstructor.
- eapply cospan_equiv_trans; eassumption.
- destruct E as [phi [E1 E2]].
destruct E' as [psi [F1 F2]].
simpl in *.
(* The transitive cospan-equiv has apex iso (psi ∘ phi);
fmap[F] (to (psi ∘ phi)) = fmap[F] (to psi ∘ to phi)
... | Lemma | dec_cospan_equiv_trans | Construction | Construction/DecoratedCospan.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Functor.Bifunctor",
"Category.Construction.Product",
"Category.Structure.Cocartesian",
"Category.Structure.Initial",
"Category.Structure.Termina... | [
"DecoratedCospanArrow",
"cospan_equiv_trans",
"dec_cospan_equiv"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
DecoratedCospanArrow_Setoid {X Y : C} :
Setoid (DecoratedCospanArrow X Y) := {|
equiv := fun f g => dec_cospan_equiv f g
|}. | Next Obligation.
constructor.
- intros f; apply dec_cospan_equiv_refl.
- intros f g; apply dec_cospan_equiv_sym.
- intros f g h; apply dec_cospan_equiv_trans.
Defined. | Instance | DecoratedCospanArrow_Setoid | Construction | Construction/DecoratedCospan.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Functor.Bifunctor",
"Category.Construction.Product",
"Category.Structure.Cocartesian",
"Category.Structure.Initial",
"Category.Structure.Termina... | [
"DecoratedCospanArrow",
"Setoid",
"dec_cospan_equiv",
"dec_cospan_equiv_refl",
"dec_cospan_equiv_sym",
"dec_cospan_equiv_trans"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
dec_cospan_id (X : C) : DecoratedCospanArrow X X | := {|
dc_cospan := cospan_id X;
dc_decoration := id_decoration X
|}. | Definition | dec_cospan_id | Construction | Construction/DecoratedCospan.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Functor.Bifunctor",
"Category.Construction.Product",
"Category.Structure.Cocartesian",
"Category.Structure.Initial",
"Category.Structure.Termina... | [
"DecoratedCospanArrow",
"cospan_id"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
dec_compose_decoration
{X Y Z : C} (g : DecoratedCospanArrow Y Z) (f : DecoratedCospanArrow X Y)
: @I D _ ~{D}~> F (cospan_apex (cospan_compose HP (dc_cospan g) (dc_cospan f))) | :=
let N := cospan_apex (dc_cospan f) in
let M := cospan_apex (dc_cospan g) in
let P := pushout (cospan_in2 (dc_cospan f)) (cospan_in1 (dc_cospan g)) in
fmap[F] (pushout_in1 P ▽ pushout_in2 P)
∘ fmap[F] (cospan_merge N M)
∘ lax_ap[F]
∘ bimap (dc_decoration f) (dc_decoration g)
∘ from (@unit_left... | Definition | dec_compose_decoration | Construction | Construction/DecoratedCospan.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Functor.Bifunctor",
"Category.Construction.Product",
"Category.Structure.Cocartesian",
"Category.Structure.Initial",
"Category.Structure.Termina... | [
"DecoratedCospanArrow",
"bimap",
"cospan_compose",
"fmap",
"from",
"pushout_in1",
"pushout_in2"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
dec_cospan_compose
{X Y Z : C} (g : DecoratedCospanArrow Y Z) (f : DecoratedCospanArrow X Y)
: DecoratedCospanArrow X Z | := {|
dc_cospan := cospan_compose HP (dc_cospan g) (dc_cospan f);
dc_decoration := dec_compose_decoration g f
|}. | Definition | dec_cospan_compose | Construction | Construction/DecoratedCospan.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Natural.Transformation",
"Category.Functor.Bifunctor",
"Category.Construction.Product",
"Category.Structure.Cocartesian",
"Category.Structure.Initial",
"Category.Structure.Termina... | [
"DecoratedCospanArrow",
"cospan_compose",
"dec_compose_decoration"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Enriched (K : Category) `{@Monoidal K} | := {
eobj : Type;
ehom : eobj → eobj → K where "a ⟿ b" := (ehom a b);
eid {x} : I ~{K}~> (x ⟿ x);
ecompose {x y z} : (y ⟿ z) ⨂ (x ⟿ y) ~{K}~> (x ⟿ z);
eid_left {x y} :
ecompose ∘ eid ⨂ id << I ⨂ (x ⟿ y) ~~> (x ⟿ y) >> unit_left;
eid_right {x y} :
ecompose ∘ id ⨂ eid << (x ⟿ y) ⨂ I ~~> (x ⟿ y) >>... | Class | Enriched | Construction | Construction/Enriched.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Structure.Monoidal",
"Category.Instance.Sets"
] | [
"Category",
"Monoidal",
"eobj"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
eobj : Enriched >-> Sortclass. | Coercion | eobj | Construction | Construction/Enriched.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Structure.Monoidal",
"Category.Instance.Sets"
] | [
"Enriched"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | ||
EnrichedFunctor
(K : Category) `{@Monoidal K}
(C : Enriched K) (D : Enriched K) | := {
efobj : C → D;
efmap {x y} : (x ⟿ y) ~{K}~> (efobj x ⟿ efobj y);
efmap_id : ∀ x,
efmap ∘ eid << I ~~> (efobj x ⟿ efobj x) >> eid;
efmap_comp : ∀ x y z,
ecompose ∘ efmap ⨂ efmap
<< (y ⟿ z) ⨂ (x ⟿ y) ~~> (efobj x ⟿ efobj z) >>
efmap ∘ ecompose
}. | Class | EnrichedFunctor | Construction | Construction/Enriched.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Structure.Monoidal",
"Category.Instance.Sets"
] | [
"Category",
"Enriched",
"Monoidal"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Category_is_Enriched_over_Set : Enriched Sets ↔ Category. | Proof.
split; intros.
- unshelve refine
{| obj := eobj
; hom := @ehom _ _ X
; homset := @ehom _ _ X
; id := fun x => @eid _ _ X x ttt
; compose := fun x y z f g => @ecompose _ _ X x y z (f, g) |}.
+ intros.
proper.
destruct X.
simpl in *.
d... | Theorem | Category_is_Enriched_over_Set | Construction | Construction/Enriched.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Structure.Monoidal",
"Category.Instance.Sets"
] | [
"Category",
"Enriched",
"Sets",
"Sets_Product_Monoidal",
"cat",
"eobj",
"morphism",
"obj",
"proper",
"sapply",
"split"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Functor_is_Enriched_over_Set (C D : Category) :
EnrichedFunctor
Sets
(snd Category_is_Enriched_over_Set C)
(snd Category_is_Enriched_over_Set D)
↔ (C ⟶ D). | Proof.
split; intros.
- destruct X; simpl in *.
construct.
+ now apply efobj0.
+ now apply efmap0.
+ proper.
now apply efmap0.
+ now apply efmap_id0.
+ simpl in *.
now srewrite (efmap_comp0 x y z (f, g)).
- destruct X; simpl in *.
construct.
+ now apply fobj.
+ cons... | Theorem | Functor_is_Enriched_over_Set | Construction | Construction/Enriched.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Structure.Monoidal",
"Category.Instance.Sets"
] | [
"Category",
"Category_is_Enriched_over_Set",
"EnrichedFunctor",
"Sets",
"construct",
"fmap",
"fobj",
"proper",
"split",
"srewrite"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Free@{fo fh fp} : Category@{fo fh fp} := {|
obj := C;
hom := tlist hom;
homset := fun _ _ => {| equiv := eq |};
id := fun _ => tnil;
compose := fun _ _ _ f g => g +++ f
|}. | Next Obligation. equivalence; congruence. Qed. | Definition | Free | Construction | Construction/Free.v | [
"Category.Lib",
"Category.Lib.TList",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Adjunction",
"Category.Instance.Sets"
] | [
"Category",
"equivalence",
"obj",
"tlist"
] | Wikipedia: "In mathematics, the free category or path category generated by
a directed graph or quiver is the category that results from freely
concatenating arrows together, whenever the target of one arrow is the
source of the next."
"More precisely, the objects of the category are the vertices of the
... | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb |
composition {x y : C} : tlist hom x y → x ~{C}~> y. | Proof.
intros.
induction X.
- exact id.
- exact (compose IHX b).
Defined. | Definition | composition | Construction | Construction/Free.v | [
"Category.Lib",
"Category.Lib.TList",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Adjunction",
"Category.Instance.Sets"
] | [
"tlist"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
composition_tnil {x : C} : composition tnil ≈ id[x]. | Proof. now cat. Qed. | Definition | composition_tnil | Construction | Construction/Free.v | [
"Category.Lib",
"Category.Lib.TList",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Adjunction",
"Category.Instance.Sets"
] | [
"cat",
"composition"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
composition_tapp {x y z : C}
(g : tlist hom y z) (f : tlist hom x y) :
composition (f +++ g) ≈ composition g ∘ composition f. | Proof.
induction f; simpl.
- rewrite tlist_app_tnil_l.
now cat.
- rewrite <- tlist_app_comm_cons.
simpl.
rewrite IHf.
now cat.
Qed. | Definition | composition_tapp | Construction | Construction/Free.v | [
"Category.Lib",
"Category.Lib.TList",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Adjunction",
"Category.Instance.Sets"
] | [
"cat",
"composition",
"tlist",
"tlist_app_comm_cons",
"tlist_app_tnil_l"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
FreeFunctor : Free ⟶ C | := {|
fobj := fun x => x;
fmap := fun _ _ f => composition f;
fmap_id := fun _ => composition_tnil;
fmap_comp := fun _ _ _ => composition_tapp
|}. | Definition | FreeFunctor | Construction | Construction/Free.v | [
"Category.Lib",
"Category.Lib.TList",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Adjunction",
"Category.Instance.Sets"
] | [
"Free",
"composition",
"composition_tapp",
"composition_tnil",
"fmap",
"fobj"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Mor : C → C → Type | :=
| Ident {x} : Mor x x
| Morph {x y} (f : x ~> y) : Mor x y
| Comp {x y z} (f : Mor y z) (g : Mor x y) : Mor x z. | Inductive | Mor | Construction | Construction/Free.v | [
"Category.Lib",
"Category.Lib.TList",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Adjunction",
"Category.Instance.Sets"
] | [
"Comp"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
morD `(t : Mor x y) : x ~> y | :=
match t with
| Ident => id
| Morph f => f
| Comp f g => morD f ∘ morD g
end. | Fixpoint | morD | Construction | Construction/Free.v | [
"Category.Lib",
"Category.Lib.TList",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Adjunction",
"Category.Instance.Sets"
] | [
"Comp",
"Mor"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
morDA `(t : Mor x y) : tlist hom x y | :=
match t with
| Ident => tnil
| Morph f => tcons _ f tnil
| Comp f g => morDA g +++ morDA f
end. | Fixpoint | morDA | Construction | Construction/Free.v | [
"Category.Lib",
"Category.Lib.TList",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Adjunction",
"Category.Instance.Sets"
] | [
"Comp",
"Mor",
"tlist"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Mor_Setoid {x y} : Setoid (Mor x y) | := {
equiv f g := morDA f = morDA g
}. | Instance | Mor_Setoid | Construction | Construction/Free.v | [
"Category.Lib",
"Category.Lib.TList",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Adjunction",
"Category.Instance.Sets"
] | [
"Mor",
"Setoid",
"morDA"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
morD_sound `{t : Mor x y} :
morD t ≈ composition (morDA t). | Proof.
induction t; simpl; cat.
rewrite IHt1, IHt2; simpl.
now rewrite composition_tapp.
Qed. | Lemma | morD_sound | Construction | Construction/Free.v | [
"Category.Lib",
"Category.Lib.TList",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Adjunction",
"Category.Instance.Sets"
] | [
"Mor",
"cat",
"composition",
"composition_tapp",
"morD",
"morDA"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
tlistDA `(t : tlist hom x y) : Mor x y | :=
match t with
| tnil => Ident
| tcons _ f fs => Comp (tlistDA fs) (Morph f)
end. | Fixpoint | tlistDA | Construction | Construction/Free.v | [
"Category.Lib",
"Category.Lib.TList",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Adjunction",
"Category.Instance.Sets"
] | [
"Comp",
"Mor",
"tlist"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
morDA_tlistDA `{f : tlist hom x y} :
morDA (tlistDA f) = f. | Proof.
induction f; simpl; auto.
rewrite <- tlist_app_cons.
now rewrite IHf.
Qed. | Lemma | morDA_tlistDA | Construction | Construction/Free.v | [
"Category.Lib",
"Category.Lib.TList",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Adjunction",
"Category.Instance.Sets"
] | [
"morDA",
"tlist",
"tlistDA",
"tlist_app_cons"
] | Note that this yields an equality. | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb |
tlistDA_morDA `{f : Mor x y} :
tlistDA (morDA f) ≈ f. | Proof.
induction f; simpl; auto.
rewrite <- IHf1, <- IHf2.
now rewrite morDA_tlistDA.
Qed. | Lemma | tlistDA_morDA | Construction | Construction/Free.v | [
"Category.Lib",
"Category.Lib.TList",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Adjunction",
"Category.Instance.Sets"
] | [
"Mor",
"morDA",
"morDA_tlistDA",
"tlistDA"
] | While this is merely an equivalence. Such is the essence of adjointness
between pseudocategories. | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb |
FreeSyntax : Category := {|
obj := C;
hom := Mor;
homset := @Mor_Setoid;
id := fun _ => Ident;
compose := fun _ _ _ => Comp
|}. | Next Obligation. now apply tlist_app_tnil_r. Qed. | Definition | FreeSyntax | Construction | Construction/Free.v | [
"Category.Lib",
"Category.Lib.TList",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Adjunction",
"Category.Instance.Sets"
] | [
"Category",
"Comp",
"Mor",
"Mor_Setoid",
"obj",
"tlist_app_tnil_r"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
ForgetSyntax : FreeSyntax ⟶ Free | := {
fobj := λ x, x;
fmap := λ x y f, morDA f;
}. | Instance | ForgetSyntax | Construction | Construction/Free.v | [
"Category.Lib",
"Category.Lib.TList",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Adjunction",
"Category.Instance.Sets"
] | [
"Free",
"FreeSyntax",
"fmap",
"fobj",
"morDA"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
CanonicalMap : Free ⟶ FreeSyntax := {
fobj := λ x, x;
fmap := λ x y f, tlistDA f;
}. | Next Obligation.
generalize dependent f.
generalize dependent z.
induction g; simpl; intros.
- now rewrite !tlist_app_tnil_l.
- rewrite <- tlist_app_cons.
rewrite <- !tlist_app_comm_cons.
simpl.
rewrite <- tlist_app_cons.
now rewrite <- IHg.
Qed. | Instance | CanonicalMap | Construction | Construction/Free.v | [
"Category.Lib",
"Category.Lib.TList",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Adjunction",
"Category.Instance.Sets"
] | [
"Free",
"FreeSyntax",
"fmap",
"fobj",
"tlistDA",
"tlist_app_comm_cons",
"tlist_app_cons",
"tlist_app_tnil_l"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
ForgetSyntax_CanonicalMap :
ForgetSyntax ⊣ CanonicalMap := {
adj := λ x y,
{| to := {| morphism := tlistDA |}
; from := {| morphism := morDA |} |}
}. | Next Obligation.
simpl; intros;
now rewrite tlistDA_morDA.
Qed. | Instance | ForgetSyntax_CanonicalMap | Construction | Construction/Free.v | [
"Category.Lib",
"Category.Lib.TList",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Theory.Adjunction",
"Category.Instance.Sets"
] | [
"CanonicalMap",
"ForgetSyntax",
"from",
"morDA",
"morphism",
"tlistDA",
"tlistDA_morDA",
"to"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Groupoid (C : Category) : Category | := {|
obj := @obj C;
hom := @Isomorphism C;
homset := @iso_setoid C;
id := @iso_id C;
compose := @iso_compose C
|}. | Definition | Groupoid | Construction | Construction/Groupoid.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism"
] | [
"Category",
"Isomorphism",
"iso_compose",
"iso_id",
"iso_setoid",
"obj"
] | A Groupoid is a category where all morphisms are isomorphisms, and morphism
equivalence is equivalence of isomorphisms. | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb |
Opposite `(C : Category) : Category | := {|
obj := @obj C;
hom := fun x y => @hom C y x;
homset := fun x y => @homset C y x;
id := @id C;
compose := fun _ _ _ f g => g ∘ f;
compose_respects := fun x y z f g fg h i hi =>
@compose_respects C z y x h i hi f g fg;
id_left := fun x y f => @id_right C y x f;
id_right := fun x... | Definition | Opposite | Construction | Construction/Opposite.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism"
] | [
"Category",
"obj"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
"C ^op" | := (@Opposite C)
(at level 7, format "C ^op", left associativity) : category_scope. | Notation | C ^op | Construction | Construction/Opposite.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism"
] | [
"Opposite",
"left"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
op_invol {C : Category} : (C^op)^op = C. | Proof.
unfold Opposite; simpl.
destruct C; simpl.
f_equal.
Qed. | Lemma | op_invol | Construction | Construction/Opposite.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism"
] | [
"Category",
"Opposite",
"op"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
op {C : Category} {x y} (f : y ~{C}~> x) : x ~{C^op}~> y | := f. | Definition | op | Construction | Construction/Opposite.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism"
] | [
"Category"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
unop {C : Category} {x y} (f : x ~{C^op}~> y) : y ~{C}~> x | := f. | Definition | unop | Construction | Construction/Opposite.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism"
] | [
"Category",
"op"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Isomorphism_Opposite {C : Category} {x y : C}
(iso : @Isomorphism C x y) :
@Isomorphism (C^op) x y | := {
to := from iso;
from := to iso;
iso_to_from := iso_to_from iso;
iso_from_to := iso_from_to iso
}. | Instance | Isomorphism_Opposite | Construction | Construction/Opposite.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism"
] | [
"Category",
"Isomorphism",
"from",
"op",
"to"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Product (C D : Category) : Category | := {|
obj := C * D;
hom := fun x y => (fst x ~> fst y) * (snd x ~> snd y);
homset := fun x y =>
let setoid_C := @homset C (fst x) (fst y) in
let setoid_D := @homset D (snd x) (snd y) in
{| equiv := fun f g =>
(@equiv _ setoid_C (fst f) (fst g) *
@equiv _ setoid_D (snd f) (s... | Definition | Product | Construction | Construction/Product.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Functor",
"Category.Construction.Opposite"
] | [
"Category",
"obj",
"setoid_equiv"
] | A product of two categories forms a category. All of the methods are
spelled out here to ease simplification. | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb |
"C ∏ D" | := (@Product C D) (at level 90) : category_scope. | Notation | C ∏ D | Construction | Construction/Product.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Functor",
"Category.Construction.Opposite"
] | [
"Product"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Fst {C D : Category} : C ∏ D ⟶ C | := {
fobj := fst;
fmap := fun _ _ => fst
}. | Instance | Fst | Construction | Construction/Product.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Functor",
"Category.Construction.Opposite"
] | [
"Category",
"fmap",
"fobj"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Snd {C D : Category} : C ∏ D ⟶ D | := {
fobj := snd;
fmap := fun _ _ => snd
}. | Instance | Snd | Construction | Construction/Product.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Functor",
"Category.Construction.Opposite"
] | [
"Category",
"fmap",
"fobj"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Swap
{C : Category} {D : Category} : (C ∏ D) ⟶ (D ∏ C) | := {|
fobj := fun x => (snd x, fst x);
fmap := fun _ _ f => (snd f, fst f);
|}. | Definition | Swap | Construction | Construction/Product.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Functor",
"Category.Construction.Opposite"
] | [
"Category",
"fmap",
"fobj"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
fst_comp {C : Category} {D : Category} x y z
(f : y ~{C ∏ D}~> z) (g : x ~{C ∏ D}~> y) :
fst f ∘ fst g ≈ fst (f ∘ g). | Proof. reflexivity. Qed. | Corollary | fst_comp | Construction | Construction/Product.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Functor",
"Category.Construction.Opposite"
] | [
"Category"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
snd_comp {C : Category} {D : Category} x y z
(f : y ~{C ∏ D}~> z) (g : x ~{C ∏ D}~> y) :
snd f ∘ snd g ≈ snd (f ∘ g). | Proof. reflexivity. Qed. | Corollary | snd_comp | Construction | Construction/Product.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Functor",
"Category.Construction.Opposite"
] | [
"Category"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
Product_Opposite {C D : Category} : (C ∏ D) ^op = (C^op ∏ D^op). | Proof. reflexivity. Qed. | Corollary | Product_Opposite | Construction | Construction/Product.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Functor",
"Category.Construction.Opposite"
] | [
"Category",
"op"
] | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb | |
PROP : Type | := {
prop_cat : Category;
prop_strict : @StrictMonoidal prop_cat;
prop_symmetric : @SymmetricMonoidal prop_cat;
(** Coherence between the two [Monoidal] paths through a PROP.
[prop_strict] supplies a [Monoidal] via [strict_is_monoidal];
[prop_symmetric] supplies one via
[braided_is_monoidal ... | Class | PROP | Construction | Construction/PROP.v | [
"Category.Lib",
"Category.Theory.Category",
"Category.Theory.Isomorphism",
"Category.Theory.Functor",
"Category.Functor.Bifunctor",
"Category.Construction.Product",
"Category.Structure.Monoidal",
"Category.Structure.Monoidal.Braided",
"Category.Structure.Monoidal.Symmetric",
"Category.Structure.Mo... | [
"Category",
"StrictMonoidal",
"SymmetricMonoidal",
"braided_is_monoidal",
"obj",
"object",
"prop_cat",
"strict_is_monoidal",
"symmetric_is_braided"
] | ** The PROP class
Bundles a category, its strict symmetric monoidal structure, and the
object correspondence with [nat]. The object correspondence is given
by [prop_of_nat : nat -> obj], with [prop_unit_zero] / [prop_tensor_plus]
pinning down [I] and [(⨂)] on objects.
The full "objects are exactl... | https://github.com/jwiegley/category-theory | e4f9c6c4db80ed1c93ac52a27233ae45b022fbcb |
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