module Cat.Functor.WideSubcategory where

Wide subcategoriesπŸ”—

A wide subcategory is specified by a predicate on the morphisms of rather than one on the objects. Since is nontrivial, we must take care that the result actually form a category: must contain the identities and be closed under composition.

To start, we package up all the data required to define a wide subcategory up into a record.

  record Wide-subcat (β„“' : Level) : Type (o βŠ” β„“ βŠ” lsuc β„“') where
    no-eta-equality
    field
      P      : βˆ€ {x y} β†’ C.Hom x y β†’ Type β„“'
      P-prop : βˆ€ {x y} (f : C.Hom x y) β†’ is-prop (P f)

      P-id : βˆ€ {x} β†’ P (C.id {x})
      P-∘  : βˆ€ {x y z} {f : C.Hom y z} {g : C.Hom x y}
          β†’ P f β†’ P g β†’ P (f C.∘ g)

  open Wide-subcat

Morphisms of wide subcategories are defined as morphisms in where holds.

  record Wide-hom {β„“'} (sub : Wide-subcat C β„“') (x y : C.Ob) : Type (h βŠ” β„“') where
    no-eta-equality
    constructor wide
    field
      hom     : C.Hom x y
      witness : hom ∈ sub

We can then use this data to construct a category.

  Wide : Wide-subcat C β„“ β†’ Precategory o (h βŠ” β„“)
  Wide sub .Ob = C.Ob
  Wide sub .Hom         = Wide-hom sub
  Wide sub .Hom-set _ _ = Wide-hom-is-set

  Wide sub .id .hom     = C.id
  Wide sub .id .witness = sub .P-id

  Wide sub ._∘_ f g .hom     = f .hom C.∘ g .hom
  Wide sub ._∘_ f g .witness = sub .P-∘ (f .witness) (g .witness)

  Wide sub .idr _ = ext $ C.idr _
  Wide sub .idl _ = ext $ C.idl _
  Wide sub .assoc _ _ _ = ext $ C.assoc _ _ _

From split essentially surjective inclusionsπŸ”—

There is another way of representing wide subcategories: By giving a pseudomonic split essential surjection

We construct the wide subcategory by restricting to the morphisms in that lie in the image of Since is a faithful functor, this is indeed a proposition.

    Split-eso-inclusion→Wide-subcat : Precategory _ _
    Split-eso-inclusion→Wide-subcat = Wide sub where
      sub : Wide-subcat C (h βŠ” h')
      sub .P {x = x} {y = y} f =
        Σ[ g ∈ D.Hom (eso x .fst) (eso y .fst) ]
        (eso.to y C.∘ F₁ g C.∘ eso.from x ≑ f)
      sub .P-prop {x} {y} f (g , p) (g' , q) =
        Ξ£-prop-path!                          $
        is-pseudomonic.faithful pseudomonic   $
        C.isoβ†’epic (eso x .snd C.Iso⁻¹) _ _   $
        C.isoβ†’monic (eso y .snd) _ _ (p βˆ™ sym q)
      sub .P-id {x} =
        (D.id , apβ‚‚ C._∘_ refl (C.eliml F-id) βˆ™ C.invl (eso x .snd))
      sub .P-∘ {x} {y} {z} {f} {g} (f' , p) (g' , q) =
        f' D.∘ g' , (
          eso.to z C.∘ F₁ (f' D.∘ g') C.∘ eso.from x                                    β‰‘βŸ¨ C.push-inner (F-∘ f' g') βŸ©β‰‘
          (eso.to z C.∘ F₁ f') C.∘ (F₁ g' C.∘ eso.from x)                               β‰‘βŸ¨ C.insert-inner (eso.invr y) βŸ©β‰‘
          ((eso.to z C.∘ F₁ f') C.∘ eso.from y) C.∘ (eso.to y C.∘ F₁ g' C.∘ eso.from x) β‰‘βŸ¨ apβ‚‚ C._∘_ (sym (C.assoc _ _ _) βˆ™ p) q βŸ©β‰‘
          f C.∘ g ∎)

This canonical wide subcategory is equivalent to

    Wide-subcat→Split-eso-domain : Functor (Split-eso-inclusion→Wide-subcat) D
    Wide-subcat→Split-eso-domain .Functor.F₀ x = eso x .fst
    Wide-subcatβ†’Split-eso-domain .Functor.F₁ f = f .witness .fst
    Wide-subcat→Split-eso-domain .Functor.F-id = refl
    Wide-subcatβ†’Split-eso-domain .Functor.F-∘ _ _ = refl

    is-fully-faithful-Wide-subcat→domain : is-fully-faithful Wide-subcat→Split-eso-domain
    is-fully-faithful-Wide-subcat→domain = is-iso→is-equiv $ iso
      (Ξ» f β†’ wide (eso.to _ C.∘ F₁ f C.∘ eso.from _) (f , refl))
      (Ξ» _ β†’ refl)
      (Ξ» f β†’ ext (f .witness .snd))

    is-eso-Wide-subcat→domain : is-split-eso Wide-subcat→Split-eso-domain
    is-eso-Wide-subcat→domain x =
      F₀ x , pseudomonic→essentially-injective pseudomonic (eso (F₀ x) .snd)

We did cheat a bit earlier when defining wide subcategories: our predicate isn’t required to respect isomorphisms! This means that we could form a β€œsubcategory” that kills off all the isomorphisms, which allows us to distinguish between isomorphic objects. Therefore, wide subcategories are not invariant under equivalence of categories.

This in turn means that the forgetful functor from a wide subcategory is not pseudomonic! To ensure that it is, we need to require that the predicate holds on all isomorphisms. Arguably this is something that should be part of the definition of a wide subcategory, but it isn’t strictly required, so we opt to leave it as a side condition.

    Forget-wide-subcat : Functor (Wide sub) C
    Forget-wide-subcat .Fβ‚€ x = x
    Forget-wide-subcat .F₁ f = f .hom
    Forget-wide-subcat .F-id = refl
    Forget-wide-subcat .F-∘ _ _ = refl

    is-faithful-Forget-wide-subcat : is-faithful Forget-wide-subcat
    is-faithful-Forget-wide-subcat = Wide-hom-path

    is-pseudomonic-Forget-wide-subcat
      : (P-invert : βˆ€ {x y} {f : C.Hom x y} β†’ C.is-invertible f β†’ f ∈ sub)
      β†’ is-pseudomonic Forget-wide-subcat
    is-pseudomonic-Forget-wide-subcat P-invert .is-pseudomonic.faithful =
      is-faithful-Forget-wide-subcat
    is-pseudomonic-Forget-wide-subcat P-invert .is-pseudomonic.isos-full f =
      pure $
        Wide.make-iso
          (wide f.to   (P-invert (C.iso→invertible f)))
          (wide f.from (P-invert (C.isoβ†’invertible (f C.Iso⁻¹))))
          (ext f.invl)
          (ext f.invr) ,
        trivial!
      where module f = C._β‰…_ f

    is-split-eso-Forget-wide-subcat : is-split-eso Forget-wide-subcat
    is-split-eso-Forget-wide-subcat y = y , C.id-iso