module Cat.Functor.FullSubcategory {o h} {C : Precategory o h} where
Full subcategoriesπ
A full subcategory of some larger category is the category generated by some predicate on the objects of of : You keep only those objects for which holds, and all the morphisms between them. An example is the category of abelian groups, as a full subcategory of groups: being abelian is a proposition (thereβs βat most one way for a group to be abelianβ).
We can interpret full subcategories, by analogy, as being the βinduced subgraphsβ of the categorical world: Keep only some of the vertices (objects), but all of the arrows (arrows) between them.
record Restrict-ob (P : C.Ob β Type β) : Type (o β β) where no-eta-equality constructor restrict field object : C.Ob witness : P object open Restrict-ob public Restrict : (P : C.Ob β Type β) β Precategory (o β β) h Restrict P .Ob = Restrict-ob P Restrict P .Hom A B = C.Hom (A .object) (B .object) Restrict P .Hom-set _ _ = C.Hom-set _ _ Restrict P .id = C.id Restrict P ._β_ = C._β_ Restrict P .idr = C.idr Restrict P .idl = C.idl Restrict P .assoc = C.assoc
Restrict-ob-path : β {P : C.Ob β Type β} β {x y : Restrict-ob P} β (p : x .object β‘ y .object) β PathP (Ξ» i β P (p i)) (x .witness) (y .witness) β x β‘ y Restrict-ob-path p q i .object = p i Restrict-ob-path p q i .witness = q i
A very important property of full subcategories (Restrict
ions) is that any
full subcategory of a univalent category is
univalent. The argument is roughly as follows: Since
is univalent, an isomorphism
gives us a path
,
so in particular if we know
and
,
then we have
.
But, since the morphisms in the full subcategory coincide with those of
,
any iso in the subcategory is an iso in
,
thus a path!
module _ (P : C.Ob β Type β) where import Cat.Reasoning (Restrict P) as R
We begin by translating between isomorphisms in the subcategory (called here) and in , which can be done by destructuring and reassembling:
sub-isoβsuper-iso : β {A B : Restrict-ob P} β (A R.β B) β (A .object C.β B .object) sub-isoβsuper-iso x = C.make-iso x.to x.from x.invl x.invr where module x = R._β _ x super-isoβsub-iso : β {A B : Restrict-ob P} β (A .object C.β B .object) β (A R.β B) super-isoβsub-iso y = R.make-iso y.to y.from y.invl y.invr where module y = C._β _ y
module _ (P : C.Ob β Type β) (pprop : β x β is-prop (P x)) where import Cat.Reasoning (Restrict P) as R
We then prove that object-isomorphism pairs in the subcategory (i.e. inhabitants of ) coincide with those in the supercategory; Hence, since is by assumption univalent, so is .
Restrict-is-category : is-category C β is-category (Restrict P) Restrict-is-category cids = Ξ» where .to-path im i .object β Univalent.isoβpath cids (sub-isoβsuper-iso P im) i .to-path {a = a} {b = b} im i .witness β is-propβpathp (Ξ» i β pprop (cids .to-path (sub-isoβsuper-iso P im) i)) (a .witness) (b .witness) i .to-path-over p β R.β -pathp _ _ Ξ» i β cids .to-path-over (sub-isoβsuper-iso P p) i .C.to
From full inclusionsπ
There is another way of representing full subcategories: By giving a full inclusion, i.e.Β a fully faithful functor . Each full inclusion canonically determines a full subcategory of , namely that consisting of the objects in merely in the image of .
module _ {o' h'} {D : Precategory o' h'} {F : Functor D C} (ff : is-fully-faithful F) where open Functor F Full-inclusionβFull-subcat : Precategory _ _ Full-inclusionβFull-subcat = Restrict (Ξ» x β β[ d β Ob D ] (Fβ d C.β x))
This canonical full subcategory is weakly equivalent to , meaning that it admits a fully faithful, essentially surjective functor from . This functor is actually just again:
Ff-domainβFull-subcat : Functor D Full-inclusionβFull-subcat Ff-domainβFull-subcat .Functor.Fβ x = restrict (Fβ x) (inc (x , C.id-iso)) Ff-domainβFull-subcat .Functor.Fβ = Fβ Ff-domainβFull-subcat .Functor.F-id = F-id Ff-domainβFull-subcat .Functor.F-β = F-β is-fully-faithful-domainβFull-subcat : is-fully-faithful Ff-domainβFull-subcat is-fully-faithful-domainβFull-subcat = ff is-eso-domainβFull-subcat : is-eso Ff-domainβFull-subcat is-eso-domainβFull-subcat yo = β₯-β₯-map (Ξ» (preimg , isom) β preimg , super-isoβsub-iso _ isom) (yo .witness)
Up to weak equivalence, admitting a full inclusion is equivalent to being a full subcategory: Every full subcategory admits a full inclusion, given on objects by projecting the first component and on morphisms by the identity function.
module _ {P : C.Ob β Type β} where Forget-full-subcat : Functor (Restrict P) C Forget-full-subcat .Functor.Fβ = object Forget-full-subcat .Functor.Fβ f = f Forget-full-subcat .Functor.F-id = refl Forget-full-subcat .Functor.F-β f g i = f C.β g is-fully-faithful-Forget-full-subcat : is-fully-faithful Forget-full-subcat is-fully-faithful-Forget-full-subcat = id-equiv