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`(IsInitialCategory zero) (HI : P zero) : IsInitialObject (sub_pre_cat P HF) (zero; HI). Proof. typeclasses eauto. Defined. | Lemma | Require Import Category.Objects InitialTerminalCategory.Core InitialTerminalCategory.Functors Functor.Core Category.Strict Functor.Paths. Require Import Functor.Identity Functor.Composition.Core Functor.Composition.Laws. | Categories\Cat\Core.v | is_initial_object__is_initial_category | 2,000 |
s' d' u : @transport _ (fun m => morphism _ (Fm m s') d') _ _ p u = u o components_of (Category.Morphisms.idtoiso (_ -> _) (ap Fm p) : morphism _ _ _)^-1 s'. Proof. case p; clear p; simpl; autorewrite with morphism; reflexivity. Qed. | Lemma | Require Import Category.Core Functor.Core FunctorCategory.Core NaturalTransformation.Core. Require Import Category.Morphisms. | Categories\Cat\Morphisms.v | transport_Fc_to_idtoiso | 2,001 |
s' d' u : @transport _ (fun m => morphism _ s' (Fm m d')) _ _ p u = components_of (Category.Morphisms.idtoiso (_ -> _) (ap Fm p) : morphism _ _ _) d' o u. Proof. case p; clear p; simpl; autorewrite with morphism; reflexivity. Qed. | Lemma | Require Import Category.Core Functor.Core FunctorCategory.Core NaturalTransformation.Core. Require Import Category.Morphisms. | Categories\Cat\Morphisms.v | transport_cF_to_idtoiso | 2,002 |
Record | null | Categories\Category\Core.v | PreCategory | 2,003 |
|
object morphism identity compose associativity left_identity right_identity := @' object morphism identity compose associativity (fun _ _ _ _ _ _ _ => symmetry _ _ (associativity _ _ _ _ _ _ _)) left_identity right_identity (fun _ => left_identity _ _ _). | Definition | null | Categories\Category\Core.v | Build_PreCategory | 2,004 |
(id0 id1 : forall x, morphism C x x) (id1_left : forall s d (m : morphism C s d), id1 _ o m = m) (id0_right : forall s d (m : morphism C s d), m o id0 _ = m) : id0 == id1. Proof. intro. etransitivity; [ symmetry; apply id1_left | apply id0_right ]. Qed. | Lemma | null | Categories\Category\Core.v | identity_unique | 2,005 |
s d (m : morphism C s d) i : i = 1 -> i o m = m := fun H => (ap10 (ap _ H) _ @ left_identity _ _ _ m)%path. | Definition | null | Categories\Category\Core.v | concat_left_identity | 2,006 |
s d (m : morphism C s d) i : i = 1 -> m o i = m := fun H => (ap _ H @ right_identity _ _ _ m)%path. | Definition | null | Categories\Category\Core.v | concat_right_identity | 2,007 |
(C : PreCategory) : PreCategory := @Build_PreCategory' C (fun s d => morphism C d s) (identity (C := C)) (fun _ _ _ m1 m2 => m2 o m1) (fun _ _ _ _ _ _ _ => @associativity_sym _ _ _ _ _ _ _ _) (fun _ _ _ _ _ _ _ => @associativity _ _ _ _ _ _ _ _) (fun _ _ => @right_identity _ _ _) (fun _ _ => @left_identity _ _ _) (@identity_identity C) _. | Definition | Require Import Category.Core Category.Objects. | Categories\Category\Dual.v | opposite | 2,008 |
C : (C^op)^op = C := idpath. | Definition | Require Import Category.Core Category.Objects. | Categories\Category\Dual.v | opposite_involutive | 2,009 |
(x : C) `(H : IsTerminalObject C x) : IsInitialObject (C^op) x := fun x' => H x'. | Definition | Require Import Category.Core Category.Objects. | Categories\Category\Dual.v | terminal_opposite_initial | 2,010 |
(x : C) `(H : IsInitialObject C x) : IsTerminalObject (C^op) x := fun x' => H x'. | Definition | Require Import Category.Core Category.Objects. | Categories\Category\Dual.v | initial_opposite_terminal | 2,011 |
(m_inv0 m_inv1 : morphism C d s) (left_inverse_0 : m_inv0 o m = identity _) (right_inverse_1 : m o m_inv1 = identity _) : m_inv0 = m_inv1. Proof. transitivity (m_inv0 o m o m_inv1); try solve [ by (rewrite -> ?associativity; rewrite_hyp; autorewrite with morphism) | by (rewrite <- ?associativity; rewrite_hyp; autorewrite with morphism) ]. Qed. | Lemma | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | inverse_unique | 2,012 |
IsIsomorphism_sig_T m <~> IsIsomorphism m. Proof. issig. Defined. | Lemma | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | issig_isisomorphism | 2,013 |
Isomorphic_sig_T <~> Isomorphic s d. Proof. issig. Defined. | Lemma | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | issig_isomorphic | 2,014 |
Isomorphic_full_sig_T <~> Isomorphic s d := (issig_isomorphic oE equiv_functor_sigma_id issig_isisomorphism). Global Instance trunc_Isomorphic : IsHSet (Isomorphic s d). Proof. eapply istrunc_equiv_istrunc; [ exact issig_isomorphic | ]. typeclasses eauto. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | issig_full_isomorphic | 2,015 |
(i j : Isomorphic s d) : @morphism_isomorphic _ _ _ i = @morphism_isomorphic _ _ _ j -> i = j. Proof. destruct i, j; simpl. intro; path_induction. f_ap. exact (center _). Defined. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | path_isomorphic | 2,016 |
(i j : Isomorphic s d) p : ap (@morphism_isomorphic _ _ _) (path_isomorphic i j p) = p. Proof. unfold path_isomorphic. destruct i, j. path_induction_hammer. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | ap_morphism_isomorphic_path_isomorphic | 2,017 |
(i j : Isomorphic s d) p q : ap (fun e : Isomorphic s d => e^-1)%morphism (path_isomorphic i j p) = q. Proof. apply path_ishprop. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | ap_morphism_inverse_path_isomorphic | 2,018 |
`(@IsIsomorphism C x y m) : IsIsomorphism m^-1 := {| morphism_inverse := m; left_inverse := right_inverse; right_inverse := left_inverse |}. Local Ltac iso_comp_t inv_lemma := etransitivity; [ | apply inv_lemma ]; first [ rewrite -> ?associativity; apply ap | rewrite <- ?associativity; apply ap ]; first [ rewrite -> ?associativity; rewrite inv_lemma | rewrite <- ?associativity; rewrite inv_lemma ]; auto with morphism. Global Instance isisomorphism_compose `(@IsIsomorphism C y z m0) `(@IsIsomorphism C x y m1) : IsIsomorphism (m0 o m1). Proof. exists (m1^-1 o m0^-1); [ abstract iso_comp_t @left_inverse | abstract iso_comp_t @right_inverse ]. Defined. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | isisomorphism_inverse | 2,019 |
(x y : C) (H : x = y) : Isomorphic x y := match H in (_ = y0) return (x <~=~> y0) with | 1%path => reflexivity x end. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | idtoiso | 2,020 |
{C} x y := | Record | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | Epimorphism | 2,021 |
{C} x y := | Record | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | Monomorphism | 2,022 |
(C D : PreCategory) s d (m1 m2 : morphism C s d) (p : m1 = m2) (s' d' : morphism C s d -> D) u : @transport _ (fun m => morphism D (s' m) (d' m)) _ _ p u = idtoiso _ (ap d' p) o u o (idtoiso _ (ap s' p))^-1. Proof. idtoiso_t. Qed. | Lemma | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | idtoiso_of_transport | 2,023 |
(C : PreCategory) (s d : C) (p : s = d) : (idtoiso _ p)^-1 = idtoiso _ (p^)%path. Proof. path_induction; reflexivity. Defined. | Lemma | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | idtoiso_inv | 2,024 |
(C : PreCategory) (s d d' : C) (m1 : d = d') (m2 : s = d) : idtoiso _ m1 o idtoiso _ m2 = idtoiso _ (m2 @ m1)%path. Proof. idtoiso_t. Qed. | Lemma | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | idtoiso_comp | 2,025 |
(C : PreCategory) (s d d' d'' : C) (m0 : d' = d'') (m1 : d = d') (m2 : s = d) : idtoiso _ m0 o (idtoiso _ m1 o idtoiso _ m2) = idtoiso _ ((m2 @ m1) @ m0)%path. Proof. idtoiso_t. Qed. | Lemma | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | idtoiso_comp3 | 2,026 |
idtoiso_comp3' (C : PreCategory) (s d d' d'' : C) (m0 : d' = d'') (m1 : d = d') (m2 : s = d) : (idtoiso _ m0 o idtoiso _ m1) o idtoiso _ m2 = idtoiso _ (m2 @ (m1 @ m0))%path. Proof. idtoiso_t. Qed. | Lemma | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | idtoiso_comp3' | 2,027 |
(C : PreCategory) (s d d' d'' d''' : C) (m0 : d'' = d''') (m1 : d' = d'') (m2 : d = d') (m3 : s = d) : idtoiso _ m0 o (idtoiso _ m1 o (idtoiso _ m2 o idtoiso _ m3)) = idtoiso _ (((m3 @ m2) @ m1) @ m0)%path. Proof. idtoiso_t. Qed. | Lemma | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | idtoiso_comp4 | 2,028 |
idtoiso_comp4' (C : PreCategory) (s d d' d'' d''' : C) (m0 : d'' = d''') (m1 : d' = d'') (m2 : d = d') (m3 : s = d) : ((idtoiso _ m0 o idtoiso _ m1) o idtoiso _ m2) o idtoiso _ m3 = idtoiso _ (m3 @ (m2 @ (m1 @ m0)))%path. Proof. idtoiso_t. Qed. | Lemma | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | idtoiso_comp4' | 2,029 |
(C : PreCategory) (s d d' d'' d''' d'''' : C) (m0 : d''' = d'''') (m1 : d'' = d''') (m2 : d' = d'') (m3 : d = d') (m4 : s = d) : idtoiso _ m0 o (idtoiso _ m1 o (idtoiso _ m2 o (idtoiso _ m3 o idtoiso _ m4))) = idtoiso _ ((((m4 @ m3) @ m2) @ m1) @ m0)%path. Proof. idtoiso_t. Qed. | Lemma | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | idtoiso_comp5 | 2,030 |
idtoiso_comp5' (C : PreCategory) (s d d' d'' d''' d'''' : C) (m0 : d''' = d'''') (m1 : d'' = d''') (m2 : d' = d'') (m3 : d = d') (m4 : s = d) : (((idtoiso _ m0 o idtoiso _ m1) o idtoiso _ m2) o idtoiso _ m3) o idtoiso _ m4 = idtoiso _ (m4 @ (m3 @ (m2 @ (m1 @ m0))))%path. Proof. idtoiso_t. Qed. | Lemma | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | idtoiso_comp5' | 2,031 |
(C D : PreCategory) (s d : C) (m : s = d) (F : Functor C D) : F _1 (idtoiso _ m) = idtoiso _ (ap (object_of F) m). Proof. path_induction; simpl; apply identity_of. Defined. | Lemma | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | idtoiso_functor | 2,032 |
`(@IsIsomorphism C x y p) : p o p^-1 = identity _ := right_inverse. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_compose_pV | 2,033 |
`(@IsIsomorphism C x y p) : p^-1 o p = identity _ := left_inverse. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_compose_Vp | 2,034 |
`(@IsIsomorphism C y z p) `(q : morphism C x y) : p^-1 o (p o q) = q. Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_compose_V_pp | 2,035 |
`(@IsIsomorphism C x z p) `(q : morphism C y z) : p o (p^-1 o q) = q. Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_compose_p_Vp | 2,036 |
`(p : morphism C y z) `(@IsIsomorphism C x y q) : (p o q) o q^-1 = p. Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_compose_pp_V | 2,037 |
`(p : morphism C x z) `(@IsIsomorphism C x y q) : (p o q^-1) o q = p. Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_compose_pV_p | 2,038 |
`(@IsIsomorphism C y z p) `(@IsIsomorphism C x y q) : (p o q)^-1 = q^-1 o p^-1. Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_inv_pp | 2,039 |
`(@IsIsomorphism C y z p) `(@IsIsomorphism C x z q) : (p^-1 o q)^-1 = q^-1 o p. Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_inv_Vp | 2,040 |
`(@IsIsomorphism C x y p) `(@IsIsomorphism C x z q) : (p o q^-1)^-1 = q o p^-1. Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_inv_pV | 2,041 |
`(@IsIsomorphism C x y p) `(@IsIsomorphism C y z q) : (p^-1 o q^-1)^-1 = q o p. Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_inv_VV | 2,042 |
`(p : morphism C x y) `(q : morphism C x z) `(@IsIsomorphism C y z r) : p = (r^-1 o q) -> (r o p) = q. Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_moveR_Mp | 2,043 |
`(@IsIsomorphism C x y p) `(q : morphism C x z) `(r : morphism C y z) : r = (q o p^-1) -> (r o p) = q. Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_moveR_pM | 2,044 |
`(p : morphism C x y) `(q : morphism C x z) `(@IsIsomorphism C z y r) : p = (r o q) -> (r^-1 o p) = q. Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_moveR_Vp | 2,045 |
`(@IsIsomorphism C x y p) `(q : morphism C y z) `(r : morphism C x z) : r = (q o p) -> (r o p^-1) = q. Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_moveR_pV | 2,046 |
`(p : morphism C x y) `(q : morphism C x z) `(@IsIsomorphism C y z r) : (r^-1 o q) = p -> q = (r o p). Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_moveL_Mp | 2,047 |
`(@IsIsomorphism C x y p) `(q : morphism C x z) `(r : morphism C y z) : (q o p^-1) = r -> q = (r o p). Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_moveL_pM | 2,048 |
`(p : morphism C x y) `(q : morphism C x z) `(@IsIsomorphism C _ _ r) : (r o q) = p -> q = (r^-1 o p). Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_moveL_Vp | 2,049 |
`(@IsIsomorphism C x y p) `(q : morphism C y z) r : (q o p) = r -> q = (r o p^-1). Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_moveL_pV | 2,050 |
`(p : morphism C x y) `(@IsIsomorphism C x y q) : p o q^-1 = identity _ -> p = q. Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_moveL_1M | 2,051 |
`(p : morphism C x y) `(@IsIsomorphism C x y q) : q^-1 o p = identity _ -> p = q. Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_moveL_M1 | 2,052 |
`(p : morphism C x y) `(@IsIsomorphism C y x q) : p o q = identity _ -> p = q^-1. Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_moveL_1V | 2,053 |
`(p : morphism C x y) `(@IsIsomorphism C y x q) : q o p = identity _ -> p = q^-1. Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_moveL_V1 | 2,054 |
`(@IsIsomorphism C x y p) q : identity _ = p^-1 o q -> p = q. Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_moveR_M1 | 2,055 |
`(@IsIsomorphism C x y p) q : identity _ = q o p^-1 -> p = q. Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_moveR_1M | 2,056 |
`(@IsIsomorphism C x y p) q : identity _ = q o p -> p^-1 = q. Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_moveR_1V | 2,057 |
`(@IsIsomorphism C x y p) q : identity _ = p o q -> p^-1 = q. Proof. iso_concat_t. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | iso_moveR_V1 | 2,058 |
(a : morphism C x3 x4) (b : morphism C x2 x3) (c : morphism C x1 x2) (d : morphism C x0 x1) : a o b o c o d = (a o ((b o c) o d)). Proof. rewrite !associativity; reflexivity. Qed. | Lemma | Require Import Category.Core Functor.Core. Require Import HoTT.Tactics Basics.Trunc Basics.Tactics Types.Sigma Equivalences. | Categories\Category\Morphisms.v | compose4associativity_helper | 2,059 |
(C : PreCategory) (P : C -> Type) := forall x (_ : P x) x' (_ : P x'), { c : Contr (morphism C x x') | IsIsomorphism (center (morphism C x x')) }. | Definition | Require Import Category.Core Category.Morphisms. | Categories\Category\Objects.v | unique_up_to_unique_isomorphism | 2,060 |
(C : PreCategory) := | Record | Require Import Category.Core Category.Morphisms. | Categories\Category\Objects.v | TerminalObject | 2,061 |
(C : PreCategory) := | Record | Require Import Category.Core Category.Morphisms. | Categories\Category\Objects.v | InitialObject | 2,062 |
unique_up_to_unique_isomorphism (fun x => IsTerminalObject C x). Proof. unique. Qed. | Theorem | Require Import Category.Core Category.Morphisms. | Categories\Category\Objects.v | terminal_object_unique | 2,063 |
unique_up_to_unique_isomorphism (fun x => IsInitialObject C x). Proof. unique. Qed. | Theorem | Require Import Category.Core Category.Morphisms. | Categories\Category\Objects.v | initial_object_unique | 2,064 |
`{Funext} (C D : PreCategory) (Heq : path_precategory'_T C D) : transport _ Heq.2.1 (transportD (fun obj => obj -> obj -> Type) (fun obj mor => forall x, mor x x) Heq.1 (morphism C) (@identity C)) = @identity D. Proof. destruct Heq as [? [? ?]]; cbn in *. repeat (intro || apply path_forall). apply identity_unique; cbn in *; auto with morphism. destruct C, D; cbn in *. path_induction; cbn in *. auto. Qed. | Lemma | Require Import Category.Core. Require Import HoTT.Basics.Equivalences HoTT.Basics.PathGroupoids HoTT.Basics.Trunc HoTT.Basics.Tactics. Require Import HoTT.Types.Sigma HoTT.Types.Arrow HoTT.Types.Forall. Require Import HoTT.Tactics. | Categories\Category\Paths.v | path_precategory_uncurried__identity_helper | 2,065 |
`{Funext} C D : path_precategory'_T C D -> path_precategory''_T C D := fun H => (H. | Definition | Require Import Category.Core. Require Import HoTT.Basics.Equivalences HoTT.Basics.PathGroupoids HoTT.Basics.Trunc HoTT.Basics.Tactics. Require Import HoTT.Types.Sigma HoTT.Types.Arrow HoTT.Types.Forall. Require Import HoTT.Tactics. | Categories\Category\Paths.v | path_precategory''_T__of__path_precategory'_T | 2,066 |
A B P Q `{forall a b, IsHProp (Q a b)} (x : { a : A & { b : B a & P a b /\ Q a b }}) q' : (x.1; (x.2.1; (fst x.2.2, q'))) = x. Proof. destruct x as [? [? [? ?]]]; cbn in *. repeat f_ap; apply path_ishprop. Defined. | Lemma | Require Import Category.Core. Require Import HoTT.Basics.Equivalences HoTT.Basics.PathGroupoids HoTT.Basics.Trunc HoTT.Basics.Tactics. Require Import HoTT.Types.Sigma HoTT.Types.Arrow HoTT.Types.Forall. Require Import HoTT.Tactics. | Categories\Category\Paths.v | eta2_sigma_helper | 2,067 |
path_precategory_uncurried' `{fs : Funext} (C D : PreCategory) : path_precategory''_T C D -> C = D. Proof. intros [? [? [? ?]]]. destruct C, D; cbn in *. path_induction; cbn in *. f_ap; eapply @center; abstract exact _. Defined. | Definition | Require Import Category.Core. Require Import HoTT.Basics.Equivalences HoTT.Basics.PathGroupoids HoTT.Basics.Trunc HoTT.Basics.Tactics. Require Import HoTT.Types.Sigma HoTT.Types.Arrow HoTT.Types.Forall. Require Import HoTT.Tactics. | Categories\Category\Paths.v | path_precategory_uncurried' | 2,068 |
`{Funext} C D HO HM HC HI : ap object (@path_precategory_uncurried' _ C D (HO; (HM; (HC, HI)))) = HO. Proof. destruct C, D; cbn in *. path_induction_hammer. Qed. | Lemma | Require Import Category.Core. Require Import HoTT.Basics.Equivalences HoTT.Basics.PathGroupoids HoTT.Basics.Trunc HoTT.Basics.Tactics. Require Import HoTT.Types.Sigma HoTT.Types.Arrow HoTT.Types.Forall. Require Import HoTT.Tactics. | Categories\Category\Paths.v | path_precategory_uncurried'_fst | 2,069 |
`{Funext} C : @path_precategory_uncurried' _ C C (idpath; (idpath; (idpath, idpath))) = idpath. Proof. destruct C; cbn in *. rewrite !(contr idpath). reflexivity. Qed. | Lemma | Require Import Category.Core. Require Import HoTT.Basics.Equivalences HoTT.Basics.PathGroupoids HoTT.Basics.Trunc HoTT.Basics.Tactics. Require Import HoTT.Types.Sigma HoTT.Types.Arrow HoTT.Types.Forall. Require Import HoTT.Tactics. | Categories\Category\Paths.v | path_precategory_uncurried'_idpath | 2,070 |
(C D : PreCategory) : C = D -> path_precategory''_T C D. Proof. intro H'. exists (ap object H'). exists ((transport_compose _ object _ _) ^ @ apD (@morphism) H'). split. - refine (_ @ apD (@compose) H'); cbn. refine (transport_pp _ _ _ _ @ _). refine ((ap _ (transportD_compose (fun obj => obj -> obj -> Type) (fun obj mor => forall s d d' : obj, mor d d' -> mor s d -> mor s d') object H' (morphism C) (@compose C))^) @ (transport_apD_transportD _ morphism (fun x mor => forall s d d' : x, mor d d' -> mor s d -> mor s d') H' (@compose C))). - refine (_ @ apD (@identity) H'); cbn. refine (transport_pp _ _ _ _ @ _). refine ((ap _ (transportD_compose (fun obj => obj -> obj -> Type) (fun obj mor => forall x : obj, mor x x) object H' (morphism C) (@identity C))^) @ (transport_apD_transportD _ morphism (fun x mor => forall s : x, mor s s) H' (@identity C))). Defined. | Definition | Require Import Category.Core. Require Import HoTT.Basics.Equivalences HoTT.Basics.PathGroupoids HoTT.Basics.Trunc HoTT.Basics.Tactics. Require Import HoTT.Types.Sigma HoTT.Types.Arrow HoTT.Types.Forall. Require Import HoTT.Tactics. | Categories\Category\Paths.v | path_precategory_uncurried'_inv | 2,071 |
`{Funext} (C D : PreCategory) : forall x : path_precategory''_T C D, path_precategory_uncurried'_inv (path_precategory_uncurried' C D x) = x. Proof. destruct C, D; cbn in *. intros [H0' [H1' [H2' H3']]]. path_induction. cbn. repeat (edestruct (center (_ = _)); try reflexivity). Qed. | Lemma | Require Import Category.Core. Require Import HoTT.Basics.Equivalences HoTT.Basics.PathGroupoids HoTT.Basics.Trunc HoTT.Basics.Tactics. Require Import HoTT.Types.Sigma HoTT.Types.Arrow HoTT.Types.Forall. Require Import HoTT.Tactics. | Categories\Category\Paths.v | equiv_path_precategory_uncurried'__eissect | 2,072 |
equiv_path_precategory_uncurried' `{Funext} (C D : PreCategory) : path_precategory''_T C D <~> C = D. Proof. apply (equiv_adjointify (@path_precategory_uncurried' _ C D) (@path_precategory_uncurried'_inv C D)). - hnf. intros []. apply path_precategory_uncurried'_idpath. - hnf. apply __eissect. Defined. | Lemma | Require Import Category.Core. Require Import HoTT.Basics.Equivalences HoTT.Basics.PathGroupoids HoTT.Basics.Trunc HoTT.Basics.Tactics. Require Import HoTT.Types.Sigma HoTT.Types.Arrow HoTT.Types.Forall. Require Import HoTT.Tactics. | Categories\Category\Paths.v | equiv_path_precategory_uncurried' | 2,073 |
`{Funext} (C D : PreCategory) : path_precategory'_T C D <~> C = D := ((' C D) oE (Build_Equiv _ _ _ (isequiv__path_precategory''_T__of__path_precategory'_T C D))). | Definition | Require Import Category.Core. Require Import HoTT.Basics.Equivalences HoTT.Basics.PathGroupoids HoTT.Basics.Trunc HoTT.Basics.Tactics. Require Import HoTT.Types.Sigma HoTT.Types.Arrow HoTT.Types.Forall. Require Import HoTT.Tactics. | Categories\Category\Paths.v | equiv_path_precategory_uncurried | 2,074 |
`{Funext} C D : _ -> _ := equiv_path_precategory_uncurried C D. | Definition | Require Import Category.Core. Require Import HoTT.Basics.Equivalences HoTT.Basics.PathGroupoids HoTT.Basics.Trunc HoTT.Basics.Tactics. Require Import HoTT.Types.Sigma HoTT.Types.Arrow HoTT.Types.Forall. Require Import HoTT.Tactics. | Categories\Category\Paths.v | path_precategory_uncurried | 2,075 |
path_precategory' `{fs : Funext} (C D : PreCategory) : forall (Hobj : object C = object D) (Hmor : transport (fun obj => obj -> obj -> Type) Hobj (morphism C) = morphism D), transport _ Hmor (transportD (fun obj => obj -> obj -> Type) (fun obj mor => forall s d d', mor d d' -> mor s d -> mor s d') Hobj (morphism C) (@compose C)) = @compose D -> C = D. Proof. intros. apply path_precategory_uncurried. repeat esplit; eassumption. Defined. | Lemma | Require Import Category.Core. Require Import HoTT.Basics.Equivalences HoTT.Basics.PathGroupoids HoTT.Basics.Trunc HoTT.Basics.Tactics. Require Import HoTT.Types.Sigma HoTT.Types.Arrow HoTT.Types.Forall. Require Import HoTT.Tactics. | Categories\Category\Paths.v | path_precategory' | 2,076 |
`{fs : Funext} (C D : PreCategory) : forall (Hobj : object C = object D) (Hmor : forall s d, morphism C (transport idmap Hobj^ s) (transport idmap Hobj^ d) = morphism D s d), (forall s d d' m m', transport idmap (Hmor _ _) (@compose C _ _ _ (transport idmap (Hmor _ _)^ m) (transport idmap (Hmor _ _)^ m')) = @compose D s d d' m m') -> C = D. Proof. intros Hobj Hmor Hcomp. pose (path_forall _ _ (fun s => path_forall _ _ (fun d => (ap10 (@transport_arrow Type idmap (fun x => x -> Type) _ _ Hobj (@morphism C) _) _) @ (@transport_arrow Type idmap _ _ _ Hobj (@morphism C _) _) @ (transport_const _ _) @ Hmor s d))) as Hmor'. eapply (' C D Hobj Hmor'). repeat (apply path_forall; intro). refine (_ @ Hcomp _ _ _ _ _); clear Hcomp. subst Hmor'. cbn. abstract ( destruct C, D; cbn in *; destruct Hobj; cbn in *; repeat match goal with | _ => reflexivity | _ => rewrite !concat_1p | _ => rewrite !transport_forall_constant, !transport_arrow | _ => progress transport_path_forall_hammer | [ |- transport ?P ?p^ ?u = ?v ] => (apply (@moveR_transport_V _ P _ _ p u v); progress transport_path_forall_hammer) | [ |- ?u = transport ?P ?p^ ?v ] => (apply (@moveL_transport_V _ P _ _ p u v); progress transport_path_forall_hammer) | [ |- context[?H ?x ?y] ] => (destruct (H x y); clear H) | _ => progress f_ap end ). Defined. | Lemma | Require Import Category.Core. Require Import HoTT.Basics.Equivalences HoTT.Basics.PathGroupoids HoTT.Basics.Trunc HoTT.Basics.Tactics. Require Import HoTT.Types.Sigma HoTT.Types.Arrow HoTT.Types.Forall. Require Import HoTT.Tactics. | Categories\Category\Paths.v | path_precategory | 2,077 |
PreCategory. refine (@Build_PreCategory (forall a : A, P a) (fun s d => forall a : A, morphism (P a) (s a) (d a)) (fun x => fun a => identity (x a)) (fun s d d' m2 m1 => fun a => m2 a o m1 a) _ _ _ _); abstract ( repeat (intro || apply path_forall); auto with morphism ). Defined. | Definition | Require Import Category.Strict. Require Import Basics.Trunc. | Categories\Category\Pi.v | pi | 2,078 |
PreCategory. refine (@Build_PreCategory (C * D) (fun s d => morphism C (fst s) (fst d) * morphism D (snd s) (snd d)) (fun x => (identity (fst x), identity (snd x))) (fun s d d' m2 m1 => (fst m2 o fst m1, snd m2 o snd m1)) _ _ _ _); abstract ( repeat (simpl || intros [] || intro); try f_ap; auto with morphism ). Defined. | Definition | Require Import Basics.Tactics. Require Import Category.Strict. Require Import Types.Prod. | Categories\Category\Prod.v | prod | 2,079 |
Record | null | Categories\Category\Strict.v | StrictCategory | 2,080 |
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(s d : C + D) : Type := match s, d with | inl s, inl d => morphism C s d | inr s, inr d => morphism D s d | _, _ => Empty end. | Definition | null | Categories\Category\Sum.v | sum_morphism | 2,081 |
(x : C + D) : sum_morphism x x := match x with | inl x => identity x | inr x => identity x end. | Definition | null | Categories\Category\Sum.v | sum_identity | 2,082 |
(s d d' : C + D) (m1 : sum_morphism d d') (m2 : sum_morphism s d) : sum_morphism s d'. Proof. case s, d, d'; simpl in *; solve [ case m1 | case m2 | eapply compose; eassumption ]. Defined. | Definition | null | Categories\Category\Sum.v | sum_compose | 2,083 |
(C D : PreCategory) : PreCategory. Proof. refine (@Build_PreCategory (C + D) (sum_morphism C D) (sum_identity C D) (sum_compose C D) _ _ _ _); abstract ( repeat (simpl || apply istrunc_S || intros [] || intro); auto with morphism; typeclasses eauto ). Defined. | Definition | null | Categories\Category\Sum.v | sum | 2,084 |
Record | Require Import Category.Core Category.Morphisms. Require Import HoTT.Basics HoTT.Tactics. | Categories\Category\Univalent.v | Category | 2,085 |
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sig' : PreCategory. Proof. refine (@Build_PreCategory obj (fun s d => mor s d) (fun x => identity x) (fun s d d' m1 m2 => compose m1 m2) _ _ _ _); assumption. Defined. | Definition | Require Import Category.Core Functor.Core. Require Import Basics.Trunc Types.Sigma. | Categories\Category\Sigma\Core.v | sig' | 2,086 |
pr1' : Functor sig' A := Build_Functor sig' A (@pr1 _ _) (fun _ _ => @pr1 _ _) (fun _ _ _ _ _ => idpath) (fun _ => idpath). End sig_obj_mor. Arguments pr1' {A Pobj Pmor HPmor Pidentity Pcompose P_associativity P_left_identity P_right_identity}. Section sig_obj_mor_hProp. Variable A : PreCategory. Variable Pobj : A -> Type. Local Notation obj := (sig_type Pobj). Variable Pmor : forall s d : obj, morphism A s.1 d.1 -> Type. Local Notation mor s d := (sig_type (Pmor s d)). Context `(HPmor : forall s d m, IsHProp (Pmor s d m)). Variable Pidentity : forall x, @Pmor x x (@identity A _). Variable Pcompose : forall s d d' m1 m2, @Pmor d d' m1 -> @Pmor s d m2 -> @Pmor s d' (m1 o m2). Local Notation identity x := (@identity A x.1; @Pidentity x). Local Notation compose m1 m2 := (m1.1 o m2.1; @Pcompose _ _ _ m1.1 m2.1 m1.2 m2.2)%morphism. Local Ltac t ex_tac := intros; simpl; apply path_sigma_uncurried; simpl; ex_tac; apply path_ishprop. Let P_associativity : forall x1 x2 x3 x4 (m1 : mor x1 x2) (m2 : mor x2 x3) (m3 : mor x3 x4), compose (compose m3 m2) m1 = compose m3 (compose m2 m1). Proof. abstract t ltac:(exists (associativity _ _ _ _ _ _ _ _)) using P_associativity_core_subproof. Defined. | Definition | Require Import Category.Core Functor.Core. Require Import Basics.Trunc Types.Sigma. | Categories\Category\Sigma\Core.v | pr1' | 2,087 |
PreCategory := Eval cbv delta [P_associativity P_left_identity P_right_identity] in @' A Pobj Pmor _ Pidentity Pcompose P_associativity P_left_identity P_right_identity. | Definition | Require Import Category.Core Functor.Core. Require Import Basics.Trunc Types.Sigma. | Categories\Category\Sigma\Core.v | sig | 2,088 |
Functor sig A := pr1'. | Definition | Require Import Category.Core Functor.Core. Require Import Basics.Trunc Types.Sigma. | Categories\Category\Sigma\Core.v | proj1_sig | 2,089 |
sig_mor' : PreCategory. Proof. refine (@Build_PreCategory (object A) (fun s d => mor s d) (fun x => identity x) (fun s d d' m1 m2 => compose m1 m2) _ _ _ _); assumption. Defined. | Definition | Require Import HoTT.Tactics Types.Forall Types.Sigma Basics.Trunc. Require Import Category.Core Functor.Core Category.Sigma.Core. Require Import Functor.Paths. Import Functor.Identity.FunctorIdentityNotations. Import Functor.Composition.Core.FunctorCompositionCoreNotations. | Categories\Category\Sigma\OnMorphisms.v | sig_mor' | 2,090 |
Functor sig_mor' A := Build_Functor sig_mor' A idmap (fun _ _ => @pr1_type _ _) (fun _ _ _ _ _ => idpath) (fun _ => idpath). | Definition | Require Import HoTT.Tactics Types.Forall Types.Sigma Basics.Trunc. Require Import Category.Core Functor.Core Category.Sigma.Core. Require Import Functor.Paths. Import Functor.Identity.FunctorIdentityNotations. Import Functor.Composition.Core.FunctorCompositionCoreNotations. | Categories\Category\Sigma\OnMorphisms.v | pr1_mor | 2,091 |
PreCategory. Proof. refine (@sig' A (fun _ => Unit) (fun s d => @Pmor (pr1_type s) (pr1_type d)) _ (fun _ => Pidentity _) (fun _ _ _ _ _ m1 m2 => Pcompose m1 m2) _ _ _); intros; trivial. Defined. | Definition | Require Import HoTT.Tactics Types.Forall Types.Sigma Basics.Trunc. Require Import Category.Core Functor.Core Category.Sigma.Core. Require Import Functor.Paths. Import Functor.Identity.FunctorIdentityNotations. Import Functor.Composition.Core.FunctorCompositionCoreNotations. | Categories\Category\Sigma\OnMorphisms.v | sig_mor_as_sig | 2,092 |
Functor sig_mor_as_sig sig_mor' := Build_Functor sig_mor_as_sig sig_mor' (@pr1_type _ _) (fun _ _ => idmap) (fun _ _ _ _ _ => idpath) (fun _ => idpath). | Definition | Require Import HoTT.Tactics Types.Forall Types.Sigma Basics.Trunc. Require Import Category.Core Functor.Core Category.Sigma.Core. Require Import Functor.Paths. Import Functor.Identity.FunctorIdentityNotations. Import Functor.Composition.Core.FunctorCompositionCoreNotations. | Categories\Category\Sigma\OnMorphisms.v | sig_functor_mor | 2,093 |
Functor sig_mor' sig_mor_as_sig := Build_Functor sig_mor' sig_mor_as_sig (fun x => exist _ x tt) (fun _ _ => idmap) (fun _ _ _ _ _ => idpath) (fun _ => idpath). | Definition | Require Import HoTT.Tactics Types.Forall Types.Sigma Basics.Trunc. Require Import Category.Core Functor.Core Category.Sigma.Core. Require Import Functor.Paths. Import Functor.Identity.FunctorIdentityNotations. Import Functor.Composition.Core.FunctorCompositionCoreNotations. | Categories\Category\Sigma\OnMorphisms.v | sig_functor_mor_inv | 2,094 |
`{Funext} : sig_functor_mor o sig_functor_mor_inv = 1 /\ sig_functor_mor_inv o sig_functor_mor = 1. Proof. split; path_functor; simpl; trivial. refine (exist _ (path_forall _ _ (fun x => match x as x return (x.1; tt) = x with | (_; tt) => idpath end)) _). repeat (apply path_forall; intro). destruct_head @sig_type. destruct_head Unit. rewrite !transport_forall_constant. transport_path_forall_hammer. reflexivity. Qed. | Lemma | Require Import HoTT.Tactics Types.Forall Types.Sigma Basics.Trunc. Require Import Category.Core Functor.Core Category.Sigma.Core. Require Import Functor.Paths. Import Functor.Identity.FunctorIdentityNotations. Import Functor.Composition.Core.FunctorCompositionCoreNotations. | Categories\Category\Sigma\OnMorphisms.v | sig_mor_eq | 2,095 |
pr1_mor o sig_functor_mor = pr1' := idpath. End sig_mor. Arguments pr1_mor {A Pmor _ Pidentity Pcompose P_associativity P_left_identity P_right_identity}. Section sig_mor_hProp. Variable A : PreCategory. Variable Pmor : forall s d, morphism A s d -> Type. Local Notation mor s d := (sig_type (Pmor s d)). Context `(HPmor : forall s d m, IsHProp (Pmor s d m)). Variable Pidentity : forall x, @Pmor x x (@identity A _). Variable Pcompose : forall s d d' m1 m2, @Pmor d d' m1 -> @Pmor s d m2 -> @Pmor s d' (m1 o m2). Local Notation identity x := (@identity A x; @Pidentity x). Local Notation compose m1 m2 := (m1.1 o m2.1; @Pcompose _ _ _ m1.1 m2.1 m1.2 m2.2)%morphism. Local Ltac t ex_tac := intros; simpl; apply path_sigma_uncurried; simpl; ex_tac; apply path_ishprop. Let P_associativity : forall x1 x2 x3 x4 (m1 : mor x1 x2) (m2 : mor x2 x3) (m3 : mor x3 x4), compose (compose m3 m2) m1 = compose m3 (compose m2 m1). Proof. abstract t ltac:(exists (associativity _ _ _ _ _ _ _ _)) using P_associativity_on_morphisms_subproof. Defined. | Definition | Require Import HoTT.Tactics Types.Forall Types.Sigma Basics.Trunc. Require Import Category.Core Functor.Core Category.Sigma.Core. Require Import Functor.Paths. Import Functor.Identity.FunctorIdentityNotations. Import Functor.Composition.Core.FunctorCompositionCoreNotations. | Categories\Category\Sigma\OnMorphisms.v | sig_mor_compat | 2,096 |
PreCategory := Eval cbv delta [P_associativity P_left_identity P_right_identity] in @' A Pmor _ Pidentity Pcompose P_associativity P_left_identity P_right_identity. | Definition | Require Import HoTT.Tactics Types.Forall Types.Sigma Basics.Trunc. Require Import Category.Core Functor.Core Category.Sigma.Core. Require Import Functor.Paths. Import Functor.Identity.FunctorIdentityNotations. Import Functor.Composition.Core.FunctorCompositionCoreNotations. | Categories\Category\Sigma\OnMorphisms.v | sig_mor | 2,097 |
Functor sig_mor A := pr1_mor. | Definition | Require Import HoTT.Tactics Types.Forall Types.Sigma Basics.Trunc. Require Import Category.Core Functor.Core Category.Sigma.Core. Require Import Functor.Paths. Import Functor.Identity.FunctorIdentityNotations. Import Functor.Composition.Core.FunctorCompositionCoreNotations. | Categories\Category\Sigma\OnMorphisms.v | proj1_sig_mor | 2,098 |
PreCategory := @Build_PreCategory (sig_type Pobj) (fun s d => morphism A (pr1_type s) (pr1_type d)) (fun x => @identity A (pr1_type x)) (fun s d d' m1 m2 => m1 o m2)%morphism (fun _ _ _ _ => associativity A _ _ _ _) (fun _ _ => left_identity A _ _) (fun _ _ => right_identity A _ _) _. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core Category.Sigma.Core. Require Import Functor.Paths. Import Functor.Identity.FunctorIdentityNotations. Import Functor.Composition.Core.FunctorCompositionCoreNotations. | Categories\Category\Sigma\OnObjects.v | sig_obj | 2,099 |