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feat(hott/algebra) finish functor category up to that missing sigma characterization

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Jakob von Raumer 2014-12-31 19:36:07 -05:00 committed by Leonardo de Moura
parent 31d1076bd7
commit 4af0a911b3
1 changed files with 24 additions and 16 deletions
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40
hott/algebra/precategory/constructions.hlean
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@ -6,13 +6,13 @@
import .natural_transformation
import types.prod types.sigma
import types.prod types.sigma types.pi
open eq prod eq eq.ops
open eq prod eq eq.ops equiv truncation
namespace precategory
namespace opposite
section
definition opposite {ob : Type} (C : precategory ob) : precategory ob :=
mk (λ a b, hom b a)
(λ b a, !homH)
@ -34,25 +34,33 @@ namespace precategory
-- take the trick they use in Coq-HoTT, and introduce a further
-- axiom in the definition of precategories that provides thee
-- symmetric associativity proof.
theorem op_op' {ob : Type} (C : precategory ob) : opposite (opposite C) = C :=
universe variables l k
definition op_op' {ob : Type} (C : precategory.{l k} ob) : opposite (opposite C) = C :=
sorry
/-begin
apply (rec_on C), intros (hom', homH', comp', ID', assoc', id_left', id_right'),
apply (ap (λassoc'', precategory.mk hom' @homH' comp' ID' assoc'' id_left' id_right')),
apply (@funext.path_pi _ _ _ _ assoc'), intro a,
apply (@funext.path_pi _ _ _ _ (@assoc' a)), intro b,
apply (@funext.path_pi _ _ _ _ (@assoc' a b)), intro c,
apply (@funext.path_pi _ _ _ _ (@assoc' a b c)), intro d,
apply (@funext.path_pi _ _ _ _ (@assoc' a b c d)), intro f,
apply (@funext.path_pi _ _ _ _ (@assoc' a b c d f)), intro g,
apply (@funext.path_pi _ _ _ _ (@assoc' a b c d f g)), intro h,
apply ap,
show @assoc ob (@opposite ob (@precategory.mk ob hom' @homH' comp' ID' assoc' id_left' id_right')) d c b a h
g
f ⁻¹ = @assoc' a b c d f g h,
begin
apply is_hset.elim,
end
end-/
theorem op_op : Opposite (Opposite C) = C :=
(ap (Precategory.mk C) (op_op' C)) ⬝ !Precategory.equal
end
end opposite
/-definition type_category : precategory Type :=
mk (λa b, a → b)
(λ a b c, function.compose)
(λ a, function.id)
(λ a b c d h g f, symm (function.compose_assoc h g f))
(λ a b f, function.compose_id_left f)
(λ a b f, function.compose_id_right f)
definition Type_category : Category := Mk type_category-/
-- Note: Discrete precategory doesn't really make sense in HoTT,
-- We'll define a discrete _category_ later.
/-section
@ -155,7 +163,7 @@ namespace precategory
variables (C D : Precategory)
definition functor_category [fx : funext] : precategory (functor C D) :=
mk (λa b, natural_transformation a b)
sorry --TODO: Prove that the nat trafos between two functors are an hset
(λ a b, @natural_transformation.to_hset C D a b)
(λ a b c g f, natural_transformation.compose g f)
(λ a, natural_transformation.id)
(λ a b c d h g f, !natural_transformation.assoc)