module Cat.Univalent.Rezk where
The Rezk completionπ
In the same way that we can freely complete a proset into a poset, it is possible to, in a universal way, replace any precategory by a category , such that there is a weak equivalence (a fully faithful, essentially surjective functor) , such that any map from to a univalent category factors uniquely through .
The construction is essentially piecing together a handful of pre-existing results: The univalence principle for -types implies that Sets is a univalent category, and functor categories with univalent codomain are univalent; The Yoneda lemma implies that any precategory admits a full inclusion into , and full subcategories of univalent categories are univalent β so, like Grothendieck cracking the nut, the sea of theory has risen to the point where our result is trivial:
module Rezk-large (A : Precategory o h) where Rezk-completion : Precategory (o β lsuc h) (o β h) Rezk-completion = Full-inclusionβFull-subcat {F = γ A} (γ-is-fully-faithful A) Rezk-completion-is-category : is-category Rezk-completion Rezk-completion-is-category = Restrict-is-category _ (Ξ» _ β squash) (Functor-is-category Sets-is-category) Complete : Functor A Rezk-completion Complete = Ff-domainβFull-subcat {F = γ A} (γ-is-fully-faithful A) Complete-is-ff : is-fully-faithful Complete Complete-is-ff = is-fully-faithful-domainβFull-subcat {F = γ _} (γ-is-fully-faithful _) Complete-is-eso : is-eso Complete Complete-is-eso = is-eso-domainβFull-subcat {F = γ _} (γ-is-fully-faithful _)
However, this construction is a bit disappointing, because weβve had to pass to a larger universe than the one we started with. If originally we had with objects living in a universe and homs in , we now have with objects living in .
Itβs unavoidable that the objects in will live in an universe satisfying , since we want their identity type to be equivalent to something living in , but going up a level is unfortunate. However, itβs also avoidable!
Since is a category, isomorphism is an identity system on its objects, which lives at the level of its morphisms β β rather than of its objects, . Therefore, using the construction of small images, we may take to be a -type!
module _ (A : Precategory o h) where private PSh[A] = PSh h A module PSh[A] = Precategory PSh[A] PSh[A]-is-cat : is-category PSh[A] PSh[A]-is-cat = Functor-is-category Sets-is-category module γim = Replacement PSh[A]-is-cat (γβ A) Rezk-completion : Precategory (o β h) (o β h) Rezk-completion .Ob = γim.Image Rezk-completion .Hom x y = γim.embed x => γim.embed y Rezk-completion .Hom-set _ _ = PSh[A].Hom-set _ _ Rezk-completion .id = PSh[A].id Rezk-completion ._β_ = PSh[A]._β_ Rezk-completion .idr = PSh[A].idr Rezk-completion .idl = PSh[A].idl Rezk-completion .assoc = PSh[A].assoc
Our resized Rezk completion factors the Yoneda embedding as a functor
where the first functor is a weak equivalence, and the second functor is fully faithful. Letβs first define the functors:
complete : Functor A Rezk-completion complete .Fβ = γim.inc complete .Fβ = γ A .Fβ complete .F-id = γ A .F-id complete .F-β = γ A .F-β RezkβͺPSh : Functor Rezk-completion (PSh h A) RezkβͺPSh .Fβ = γim.embed RezkβͺPSh .Fβ f = f RezkβͺPSh .F-id = refl RezkβͺPSh .F-β _ _ = refl
From the existence of the second functor, we can piece together pre-existing lemmas about the image and about identity systems in general to show that this resized Rezk completion is also a category: We can pull back the identity system on to one on , since we know of a (type-theoretic) embedding between their types of objects.
That gives us an identity system which is slightly off, that of β-isomorphisms on the image of the functor β, but since we know that this functor is fully faithful, thatβs equivalent to what we want.
private module RezkβͺPSh = Ffr RezkβͺPSh id-equiv abstract Rezk-completion-is-category : is-category Rezk-completion Rezk-completion-is-category = transfer-identity-system (pullback-identity-system PSh[A]-is-cat (_ , γim.embed-is-embedding)) (Ξ» _ _ β RezkβͺPSh.iso-equiv eβ»ΒΉ) Ξ» x β Cr.β -pathp Rezk-completion refl refl refl
It remains to show that the functor is a weak equivalence. Itβs fully faithful because the Yoneda embedding is, and itβs essentially surjective because itβs literally surjective-on-objects.
complete-is-ff : is-fully-faithful complete complete-is-ff = γ-is-fully-faithful A complete-is-eso : is-eso complete complete-is-eso x = do t β γim.inc-is-surjective x pure (t .fst , pathβiso (t .snd))