open import Cat.Instances.Shape.Terminal
open import Cat.Instances.Shape.Join
open import Cat.Diagram.Limit.Base
open import Cat.Instances.Discrete
open import Cat.Instances.Functor
open import Cat.Diagram.Pullback
open import Cat.Diagram.Terminal
open import Cat.Diagram.Product
open import Cat.Functor.Base
open import Cat.Univalent
open import Cat.Prelude

open import Data.Sum

import Cat.Reasoning

module Cat.Instances.Slice where

Slice categories🔗

When working in $\sets$, there is an evident notion of family indexed by a set: a family of sets $(F_i)_{i \in I}$ is equivalently a functor $[\id{Disc}(I), \sets]$, where we have equipped the set $I$ with the discrete category structure. This works essentially because of the discrete category-global sections adjunction, but in general this can not be applied to other categories, like $\id{Groups}$. How, then, should we work with “indexed families” in general categories?

The answer is to consider, rather than families themselves, the projections from their total spaces as the primitive objects. A family indexed by $I$, then, would consist of an object $A$ and a morphism $t : A \to I$, where $A$ is considered as the “total space” object and $t$ assigns gives the “tag” of each object. By analysing how $t$ pulls back along maps $B \to I$, we recover a notion of “fibres”: the collection with index $i$ can be recovered as the pullback $t^*i$.

Note that, since the discussion in the preceding paragraph made no mention of the category of sets, it applies in any category! More generally, for any category $\ca{C}$ and object $c : \ca{C}$, we have a category of objects indexed by $c$, the slice category $\ca{C}/c$. An object of “the slice over $c$” is given by an object $d : \ca{C}$ to serve as the domain, and a map $f : d \to c$.

  record /-Obj (c : C.Ob) : Type (o ⊔ ℓ) where
    no-eta-equality
    constructor cut
    field
      {domain} : C.Ob
      map      : C.Hom domain c

A map between $f : a \to c$ and $g : b \to c$ is given by a map $h : a \to b$ such that the triangle below commutes. Since we’re thinking of $f$ and $g$ as families indexed by $c$, commutativity of the triangle says that the map $h$ “respects reindexing”, or less obliquely “preserves fibres”.

  record /-Hom (a b : /-Obj c) : Type ℓ where
    no-eta-equality
    private
      module a = /-Obj a
      module b = /-Obj b
    field
      map      : C.Hom a.domain b.domain
      commutes : b.map C.∘ map ≡ a.map

The slice category $\ca{C}/c$ is given by the /-Obj and /-Homs.

Slice : (C : Precategory o ℓ) → Precategory.Ob C → Precategory _ _
Slice C c = precat where
  import Cat.Reasoning C as C
  open Precategory
  open /-Hom
  open /-Obj

  precat : Precategory _ _
  precat .Ob = /-Obj {C = C} c
  precat .Hom = /-Hom
  precat .Hom-set x y = /-Hom-is-set
  precat .id .map      = C.id
  precat .id .commutes = C.idr _

For composition in the slice over $c$, note that if the triangle (the commutativity condition for $f$) and the rhombus (the commutativity condition for $g$) both commute, then so does the larger triangle (the commutativity for $g \circ f$).

  precat ._∘_ {x} {y} {z} f g = fog where
    module f = /-Hom f
    module g = /-Hom g
    fog : /-Hom _ _
    fog .map = f.map C.∘ g.map
    fog .commutes =
      z .map C.∘ f.map C.∘ g.map ≡⟨ C.pulll f.commutes ⟩≡
      y .map C.∘ g.map           ≡⟨ g.commutes ⟩≡
      x .map                     ∎
  precat .idr f = /-Hom-path (C.idr _)
  precat .idl f = /-Hom-path (C.idl _)
  precat .assoc f g h = /-Hom-path (C.assoc _ _ _)

Limits🔗

We discuss some limits in the slice of $\ca{C}$ over $c$. First, every slice category has a terminal object, given by the identity map $\id{id} : c \to c$.

module _ {o ℓ} {C : Precategory o ℓ} {c : Precategory.Ob C} where
  import Cat.Reasoning C as C
  import Cat.Reasoning (Slice C c) as C/c
  open /-Hom
  open /-Obj

  Slice-terminal-object : is-terminal (Slice C c) (cut C.id)
  Slice-terminal-object obj .centre .map = obj .map
  Slice-terminal-object obj .centre .commutes = C.idl _
  Slice-terminal-object obj .paths other =
    /-Hom-path (sym (other .commutes) ∙ C.idl _)

Products in a slice category are slightly more complicated, but recall that another word for pullback is “fibred product”. Indeed, in the pullback page we noted that the pullback of $X \to Z$ and $Y \to Z$ is exactly the product of those maps in the slice over $Z$.

Suppose we have a pullback diagram like the one below, i.e., a limit of the diagram $a \xrightarrow{f} c \xleftarrow{g} b$, in the category $\ca{C}$. We’ll show that it’s also a limit of the (discrete) diagram consisting of $f$ and $g$, but now in the slice category $\ca{C}/c$.

For starters, note that we have seemingly “two” distinct choices for maps $a \times_c b \to c$, but since the square above commutes, either one will do. For definiteness, we go with the composite $f \circ \pi_1$.

  module
    _ {f g : /-Obj c} {Pb : C.Ob} {π₁ : C.Hom Pb (f .domain)}
                                  {π₂ : C.Hom Pb (g .domain)}
      (pb : is-pullback C {X = f .domain} {Z = c} {Y = g .domain} {P = Pb}
        π₁ (map {_} {_} {C} {c} f) π₂ (map {_} {_} {C} {c} g))
    where
    private module pb = is-pullback pb

    is-pullback→product-over : C/c.Ob
    is-pullback→product-over = cut (f .map C.∘ π₁)

Now, by commutativity of the square, the maps $\pi_1$ and $\pi_2$ in the diagram above extend to maps $(f \circ \pi_1) \to f$ and $(f \circ \pi_1) \to g$ in $\ca{C}/c$. Indeed, note that by scribbling a red line across the diagonal of the diagram, we get the two needed triangles as the induced subdivisions.

    is-pullback→π₁ : C/c.Hom is-pullback→product-over f
    is-pullback→π₁ .map      = π₁
    is-pullback→π₁ .commutes i = f .map C.∘ π₁

    is-pullback→π₂ : C/c.Hom is-pullback→product-over g
    is-pullback→π₂ .map        = π₂
    is-pullback→π₂ .commutes i = pb.square (~ i)

    open is-product

Unfolding what it means for a diagram to be a universal cone over the discrete diagram consisting of $f$ and $g$ in the category $\ca{C}/c$, we see that it is exactly the data of the pullback of $f$ and $g$ in $\ca{C}$, as below:

    is-pullback→is-fibre-product
      : is-product (Slice C c) is-pullback→π₁ is-pullback→π₂
    is-pullback→is-fibre-product .⟨_,_⟩ {Q} /f /g = factor
      where
        module f = /-Hom /f
        module g = /-Hom /g

        factor : C/c.Hom Q _
        factor .map = pb.limiting (f.commutes ∙ sym g.commutes)
        factor .commutes =
          (f .map C.∘ π₁) C.∘ pb.limiting _ ≡⟨ C.pullr pb.p₁∘limiting ⟩≡
          f .map C.∘ f.map                  ≡⟨ f.commutes ⟩≡
          Q .map                            ∎

    is-pullback→is-fibre-product .π₁∘factor = /-Hom-path pb.p₁∘limiting
    is-pullback→is-fibre-product .π₂∘factor = /-Hom-path pb.p₂∘limiting
    is-pullback→is-fibre-product .unique other p q =
      /-Hom-path (pb.unique (ap map p) (ap map q))

  Pullback→Fibre-product
    : ∀ {f g : /-Obj c}
    → Pullback C (f .map) (g .map) → Product (Slice C c) f g
  Pullback→Fibre-product pb .Product.apex = _
  Pullback→Fibre-product pb .Product.π₁ = _
  Pullback→Fibre-product pb .Product.π₂ = _
  Pullback→Fibre-product pb .Product.has-is-product =
    is-pullback→is-fibre-product (pb .Pullback.has-is-pb)

While products and terminal objects in $\ca{C}/X$ do not correspond to those in $\ca{C}$, pullbacks (and equalisers) are precisely equivalent. A square is a pullback in $\ca{C}/X$ iff. the square consisting of their underlying objects and maps is a square in $\ca{C}$.

The “if” direction (what is pullback-above→pullback-below) in the code is essentially an immediate consequence of the fact that equality of morphisms in $\ca{C}/X$ may be tested in $\ca{C}$, but we do have to take some care when extending the “limiting” morphism back down to the slice category (see the calculation marked {- * -}).

module _ where
  open /-Obj
  open /-Hom

  pullback-above→pullback-below
    : ∀ {o ℓ} {C : Precategory o ℓ} {X : Precategory.Ob C}
    → ∀ {P A B c} {p1 f p2 g}
    → is-pullback C (p1 .map) (f .map) (p2 .map) (g .map)
    → is-pullback (Slice C X) {P} {A} {B} {c} p1 f p2 g
  pullback-above→pullback-below {C = C} {P = P} {a} {b} {c} {p1} {f} {p2} {g} pb
    = pb′ where
      open is-pullback
      open Cat.Reasoning C

      pb′ : is-pullback (Slice _ _) _ _ _ _
      pb′ .square = /-Hom-path (pb .square)
      pb′ .limiting p .map = pb .limiting (ap map p)
      pb′ .limiting {P'} {p₁' = p₁'} p .commutes =
        (c .map ∘ pb .limiting (ap map p))           ≡˘⟨ (pulll (p1 .commutes)) ⟩≡˘
        (P .map ∘ p1 .map ∘ pb .limiting (ap map p)) ≡⟨ ap (_ ∘_) (pb .p₁∘limiting) ⟩≡
        (P .map ∘ p₁' .map)                          ≡⟨ p₁' .commutes ⟩≡
        P' .map                                      ∎ {- * -}
      pb′ .p₁∘limiting = /-Hom-path (pb .p₁∘limiting)
      pb′ .p₂∘limiting = /-Hom-path (pb .p₂∘limiting)
      pb′ .unique p q = /-Hom-path (pb .unique (ap map p) (ap map q))

  pullback-below→pullback-above
    : ∀ {o ℓ} {C : Precategory o ℓ} {X : Precategory.Ob C}
    → ∀ {P A B c} {p1 f p2 g}
    → is-pullback (Slice C X) {P} {A} {B} {c} p1 f p2 g
    → is-pullback C (p1 .map) (f .map) (p2 .map) (g .map)
  pullback-below→pullback-above {C = C} {P = P} {p1 = p1} {f} {p2} {g} pb = pb′ where
    open Cat.Reasoning C
    open is-pullback
    pb′ : is-pullback _ _ _ _ _
    pb′ .square = ap map (pb .square)
    pb′ .limiting {P′ = P'} {p₁'} {p₂'} p =
      pb .limiting {P′ = cut (P .map ∘ p₁')}
        {p₁' = record { map = p₁' ; commutes = refl }}
        {p₂' = record { map = p₂' ; commutes = sym (pulll (g .commutes))
                                             ∙ sym (ap (_ ∘_) p)
                                             ∙ pulll (f .commutes)
                      }}
        (/-Hom-path p)
        .map
    pb′ .p₁∘limiting = ap map $ pb .p₁∘limiting
    pb′ .p₂∘limiting = ap map $ pb .p₂∘limiting
    pb′ .unique p q = ap map $ pb .unique
      {lim' = record { map = _ ; commutes = sym (pulll (p1 .commutes)) ∙ ap (_ ∘_) p }}
      (/-Hom-path p) (/-Hom-path q)

Slices of Sets🔗

We now prove the correspondence between slices of $\sets$ and functor categories into $\sets$, i.e. the corresponding between indexing and slicing mentioned in the first paragraph.

module _ {I : Set ℓ} where
  open /-Obj
  open /-Hom

We shall prove that the functor Total-space, defined below, is an equivalence of categories, i.e. that it is fully faithful and essentially surjective. But first, we must define the functor! Like its name implies, it maps the functor $F : I → \sets$ to the first projection map $\id{fst} : \sum F \to I$.

  Total-space : Functor Cat[ Disc′ I , Sets ℓ ] (Slice (Sets ℓ) I)
  Total-space .F₀ F .domain = Σ (∣_∣ ⊙ F₀ F)
                            , Σ-is-hlevel  (I .is-tr) (is-tr ⊙ F₀ F)
  Total-space .F₀ F .map = fst

  Total-space .F₁ nt .map (i , x) = i , nt .η _ x
  Total-space .F₁ nt .commutes    = refl

  Total-space .F-id    = /-Hom-path refl
  Total-space .F-∘ _ _ = /-Hom-path refl

To prove that the Total-space functor is fully faithful, we will exhibit a quasi-inverse to its action on morphisms. Given a fibre-preserving map between $\id{fst} : \sum F \to I$ and $\id{fst} : \sum G \to I$, we recover a natural transformation between $F$ and $G$. The hardest part is showing naturality, where we use path induction.

  Total-space-is-ff : is-fully-faithful Total-space
  Total-space-is-ff {f1} {f2} = is-iso→is-equiv
    (iso from linv (λ x → Nat-path (λ x → funext (λ _ → transport-refl _))))
    where
      from : /-Hom (Total-space .F₀ f1) (Total-space .F₀ f2) → f1 => f2
      from mp = nt where
        eta : ∀ i → ∣ F₀ f1 i ∣ → ∣ F₀ f2 i ∣
        eta i j =
          subst (∣_∣ ⊙ F₀ f2) (happly (mp .commutes) _) (mp .map (i , j) .snd)

        nt : f1 => f2
        nt .η = eta
        nt .is-natural _ _ f =
          J (λ _ p → eta _ ⊙ F₁ f1 p ≡ F₁ f2 p ⊙ eta _)
            (ap (eta _ ⊙_) (F-id f1) ∙ sym (ap (_⊙ eta _) (F-id f2)))
            f

For essential surjectivity, given a map $f : X \to I$, we recover a family of sets $(f^*i)_{i \in I}$ by taking the fibre of $f$ over each point, which cleanly extends to a functor. To show that the Total-space of this functor is isomorphic to the map we started with, we use one of the auxilliary lemmas used in the construction of an object classifier: Total-equiv. This is cleaner than exhibiting an isomorphism directly, though it does involve an appeal to univalence.

  Total-space-is-eso : is-split-eso Total-space
  Total-space-is-eso fam = functor , path→iso _ path
    where
      functor : Functor _ _
      functor .F₀ i = fibre (fam .map) i
                    , Σ-is-hlevel  (fam .domain .is-tr)
                                    λ _ → is-prop→is-set (I .is-tr _ _)
      functor .F₁ p = subst (fibre (fam .map)) p
      functor .F-id = funext transport-refl
      functor .F-∘ f g = funext (subst-∙ (fibre (fam .map)) _ _)

      path : F₀ Total-space functor ≡ fam
      path = /-Obj-path (n-ua (Total-equiv _  e⁻¹)) (ua→ λ a → sym (a .snd .snd))

Slices preserve univalence🔗

An important property of slice categories is that they preserve univalence. This can be seen intuitively: If $\ca{C}$ is a univalent category, then let $a, b, c$ be some objects, with the pairs $(a, f)$ and $(b, g)$ objects in the slice $\ca{C}/c$. An isomorphism $h : (a, f) \cong (b, g)$ induces an identification $a \equiv b$, which extends to an identification $(a, f) \equiv (b, g)$ since $h \circ g = f$.

module _ {C : Precategory o ℓ} {o : Precategory.Ob C} (isc : is-category C) where
  private
    module C   = Cat.Reasoning C
    module C/o = Cat.Reasoning (Slice C o)
    module Cu  = Cat.Univalent C

    open /-Obj
    open /-Hom
    open C/o._≅_
    open C._≅_

  slice-is-category : is-category (Slice C o)
  slice-is-category A .centre = A , C/o.id-iso
  slice-is-category A .paths (B , isom) = Σ-pathp A≡B eql where
    Ad≡Bd : A .domain ≡ B .domain
    Ad≡Bd = Cu.iso→path isc
      (C.make-iso (isom .to .map) (isom .from .map)
        (ap map (C/o._≅_.invl isom)) (ap map (C/o._≅_.invr isom)))

    Af≡Bf : PathP (λ i → C.Hom (Ad≡Bd i) o) (A .map) (B .map)
    Af≡Bf = Cu.Hom-pathp-refll-iso isc (isom .from .commutes)

    A≡B = /-Obj-path Ad≡Bd Af≡Bf

    eql : PathP (λ i → A C/o.≅ A≡B i) C/o.id-iso isom
    eql = C/o.≅-pathp refl A≡B
      (/-Hom-pathp _ _ (Cu.Hom-pathp-reflr-iso isc (C.idr _)))

Arbitrary limits in slices🔗

Suppose we have some really weird diagram $F : \ca{J} \to \ca{C}/c$, like the one below. Well, alright, it’s not that weird, but it’s not a pullback or a terminal object, so we don’t a priori know how to compute it.

The observation that will let us compute a limit for this diagram is inspecting the computation of products in slice categories, above. To compute the product of $(a, f)$ and $(b, g)$, we had to pass to a pullback of $a \xto{f} c \xot{b}$ in $\ca{C}$ — which we had assumed exists. But! Take a look at what that diagram looks like:

We “exploded” a diagram of shape $\bullet \quad \bullet$ to one of shape $\bullet \to \bullet \ot \bullet$. This process can be described in a way easier to generalise: We “exploded” our diagram $F : \{*,*\} \to \ca{C}/c$ to one indexed by a category which contains $\{*,*\}$, contains an extra point, and has a unique map between each object of $\{*,*\}$ — the join of these categories.

Generically, if we have a diagram $F : J \to \ca{C}/c$, we can “explode” this into a diagram $F' : (J \star \{*\}) \to \ca{C}$, compute the limit in $\ca{C}$, then pass back to the slice category.

    F′ : Functor (J ⋆ ⊤Cat) C
    F′ .F₀ (inl x) = F.₀ x .domain
    F′ .F₀ (inr x) = o
    F′ .F₁ {inl x} {inl y} (lift f) = F.₁ f .map
    F′ .F₁ {inl x} {inr y} _ = F.₀ x .map
    F′ .F₁ {inr x} {inr y} (lift h) = C.id
    F′ .F-id {inl x} = ap map F.F-id
    F′ .F-id {inr x} = refl
    F′ .F-∘ {inl x} {inl y} {inl z} (lift f) (lift g) = ap map (F.F-∘ f g)
    F′ .F-∘ {inl x} {inl y} {inr z} (lift f) (lift g) = sym (F.F₁ g .commutes)
    F′ .F-∘ {inl x} {inr y} {inr z} (lift f) (lift g) = C.introl refl
    F′ .F-∘ {inr x} {inr y} {inr z} (lift f) (lift g) = C.introl refl

  limit-above→limit-in-slice : Limit F′ → Limit F
  limit-above→limit-in-slice lims = lim where
    module lim = Terminal lims
    module limob = Cone lim.top

    nadir : Cone F
    nadir .apex .domain = limob.apex
    nadir .apex .map = limob.ψ (inr tt)
    nadir .ψ x .map = limob.ψ (inl x)
    nadir .ψ x .commutes = limob.commutes (lift tt)
    nadir .commutes f = /-Hom-path (limob.commutes (lift f))

    lim : Limit F
    lim .top = nadir
    lim .has⊤ other = contr ch cont where
      other′ : Cone F′
      other′ .apex = other .apex .domain
      other′ .ψ (inl x) = other .ψ x .map
      other′ .ψ (inr tt) = other .apex .map
      other′ .commutes {inl x} {inl y} (lift f) = ap map (other .commutes f)
      other′ .commutes {inl x} {inr y} (lift f) = other .ψ x .commutes
      other′ .commutes {inr x} {inr y} (lift f) = C.idl _

      module cont = is-contr (lim.has⊤ other′)
      ch : Cone-hom F other nadir
      ch .hom .map = cont.centre .hom
      ch .hom .commutes = cont.centre .commutes (inr tt)
      ch .commutes o = /-Hom-path (cont.centre .commutes (inl o))

      cont : ∀ c → ch ≡ c
      cont c = Cone-hom-path _ (/-Hom-path (ap hom uniq)) where
        c′ : Cone-hom F′ other′ lim.top
        c′ .hom = c .hom .map
        c′ .commutes (inl x) = ap map (c .commutes x)
        c′ .commutes (inr tt) = c .hom .commutes

        uniq = cont.paths c′

In particular, if a category $\ca{C}$ is complete, then so are its slices:

is-complete→slice-is-complete
  : ∀ {ℓ o ℓ′} {C : Precategory o ℓ} {c : Precategory.Ob C}
  → is-complete o′ ℓ′ C
  → is-complete o′ ℓ′ (Slice C c)
is-complete→slice-is-complete lims F = limit-above→limit-in-slice F (lims _)