module Cat.Functor.Hom.Representable {o ΞΊ} {C : Precategory o ΞΊ} where
private module C = Cat.Reasoning C module C^ = Cat.Reasoning Cat[ C ^op , Sets ΞΊ ] module [C,Sets] = Cat.Reasoning Cat[ C , Sets ΞΊ ] module Sets = Cat.Reasoning (Sets ΞΊ) open Element-hom open Functor open Element open _=>_
Representable functorsπ
A functor (from a locally -small category) is said to be representable when it is naturally isomorphic to for an object (called the representing object) β that is, the functor classifies the maps into . Note that we can evidently dualise the definition, to get what is called a corepresentable functor, one of the form , but we refer informally to both of these situations as βrepresentablesβ and βrepresenting objectsβ.
record Representation (F : Functor (C ^op) (Sets ΞΊ)) : Type (o β ΞΊ) where no-eta-equality field rep : C.Ob represents : natural-iso F (Hom-into C rep) module rep = natural-iso represents equiv : β {a} β C.Hom a rep β β£ F .Fβ a β£ equiv = IsoβEquiv Ξ» where .fst β rep.from .Ξ· _ .snd .is-iso.inv β rep.to .Ξ· _ .snd .is-iso.rinv x β rep.invr Ξ·β _ $β x .snd .is-iso.linv x β rep.invl Ξ·β _ $β x module Rep {a} = Equiv (equiv {a}) open Representation open Representation using (module Rep) public
This definition is deceptively simple: the idea of representable functor (and of representing object) is key to understanding the idea of universal property, which could be called the most important concept in category theory. Most constructions in category theory specified in terms of the existence of certain maps are really instances of representing objects for functors: limits, colimits, coends, adjoint functors, Kan extensions, etc.
The first thing we will observe is an immediate consequence of the Yoneda lemma: representing objects are unique. Intuitively this is because β is a representation of β determines how reacts to being mapped into, and since the only thing we can probe objects in an arbitrary category by are morphisms, two objects which react to morphisms in the same way must be isomorphic.
representation-unique : {F : Functor (C ^op) (Sets ΞΊ)} (X Y : Representation F) β X .rep C.β Y .rep representation-unique X Y = is-ffβessentially-injective {F = γ C} (γ-is-fully-faithful C) γXβ γY where γXβ γY : γβ C (X .rep) C^.β γβ C (Y .rep) γXβ γY = (X .represents C^.Isoβ»ΒΉ) C^.βIso Y .represents
Therefore, if is a univalent category, then the type of representations for a functor is a proposition. This does not follow immediately from the lemma above: we also need to show that the isomorphism computed by the full-faithfulness of the Yoneda embedding commutes with the specified representation isomorphism. This follows by construction, but the proof needs to commute
applications of functors and paths-from-isos, which is never pretty:
Representation-is-prop : β {F} β is-category C β is-prop (Representation F) Representation-is-prop {F = F} c-cat x y = path where module X = Representation x module Y = Representation y objs : X.rep β‘ Y.rep objs = c-cat .to-path (representation-unique x y) path : x β‘ y path i .rep = objs i path i .represents = C^.β -pathp refl (ap (γβ C) objs) {f = X.represents} {g = Y.represents} (Nat-pathp _ _ Ξ» a β Hom-pathp-reflr (Sets _) {A = F .Fβ a} {q = Ξ» i β el! (C.Hom a (objs i))} (funext Ξ» x β ap (Ξ» e β e .Sets.to) (ap-Fβ-iso c-cat (Hom-from C a) _) $β _ Β·Β· sym (Y.rep.to .is-natural _ _ _) $β _ Β·Β· ap Y.Rep.from (sym (X.rep.from .is-natural _ _ _ $β _) Β·Β· ap X.Rep.to (C.idl _) Β·Β· X.Rep.Ξ΅ _))) i
As terminal objectsπ
We begin to connect the idea of representing objects to other universal constructions by proving this alternative characterisation of representations: A functor is representable if, and only if, its category of elements has a terminal object.
terminal-elementβrepresentation : {F : Functor (C ^op) (Sets ΞΊ)} β Terminal (β« C F) β Representation F terminal-elementβrepresentation {F} term = f-rep where module F = Functor F open Terminal term
From the terminal object in 1, we obtain a natural transformation , given componentwise by interpreting each pair as an object of , then taking the terminating morphism , which satisfies (by definition) . This natural transformation is componentwise invertible, as the calculation below shows, so it constitutes a natural isomorphism.
nat : F => γβ C (top .ob) nat .Ξ· ob section = hasβ€ (elem ob section) .centre .hom nat .is-natural x y f = funext Ξ» sect β ap hom $ hasβ€ _ .paths $ elem-hom _ $ F.β (hasβ€ _ .centre .hom C.β f) (top .section) β‘β¨ happly (F.F-β _ _) _ β©β‘ F.β f (F.β (hasβ€ _ .centre .hom) (top .section)) β‘β¨ ap (F.β f) (hasβ€ _ .centre .commute) β©β‘ F.β f sect β inv : β x β Sets.is-invertible (nat .Ξ· x) inv x = Sets.make-invertible (Ξ» f β F.β f (top .section)) (funext Ξ» x β ap hom $ hasβ€ _ .paths (elem-hom x refl)) (funext Ξ» x β hasβ€ _ .centre .commute) f-rep : Representation F f-rep .rep = top .ob f-rep .represents = C^.invertibleβiso nat $ componentwise-invertibleβinvertible nat inv
Universal constructionsπ
We now show a partial converse to the calculation above: That terminal objects are representing objects for a particular functor. Consider, to be more specific, the constant functor which sends everything to the terminal set. When is representable?
Well, unfolding the definition, itβs when we have an object with a natural isomorphism . Unfolding that, itβs an object for which, given any other object , we have an isomorphism of sets 2. Hence, a representing object for the βconstantly β functor is precisely a terminal object. It turns out the
representable-unitβterminal : Representation (Const (el (Lift _ β€) (hlevel 2))) β Terminal C representable-unitβterminal repr .Terminal.top = repr .rep representable-unitβterminal repr .Terminal.hasβ€ ob = retractβis-contr (Rep.from repr) (Ξ» _ β lift tt) (Rep.Ξ· repr) (hlevel 0)
Corepresentable functorsπ
As noted earlier, we can dualise the definition of a representable functor to the covariant setting to get corepresentable functors.
record Corepresentation (F : Functor C (Sets ΞΊ)) : Type (o β ΞΊ) where no-eta-equality field corep : C.Ob corepresents : natural-iso F (Hom-from C corep) module corep = natural-iso corepresents coequiv : β {a} β C.Hom corep a β β£ F .Fβ a β£ coequiv = IsoβEquiv Ξ» where .fst β corep.from .Ξ· _ .snd .is-iso.inv β corep.to .Ξ· _ .snd .is-iso.rinv x β corep.invr Ξ·β _ $β x .snd .is-iso.linv x β corep.invl Ξ·β _ $β x module Corep {a} = Equiv (coequiv {a}) open Corepresentation open Corepresentation using (module Corep) public
Much like their contravariant cousins, corepresenting objects are unique up to isomorphism.
corepresentation-unique : {F : Functor C (Sets ΞΊ)} (X Y : Corepresentation F) β X .corep C.β Y .corep
We omit the proof, as it is identical to the representable case.
corepresentation-unique X Y = is-ffβessentially-injective {F = Functor.op (γcov C)} (γcov-is-fully-faithful C) (isoβco-iso (Cat[ C , Sets ΞΊ ]) ni) where ni : natural-iso (Hom-from C (Y .corep)) (Hom-from C (X .corep)) ni = (Y .corepresents niβ»ΒΉ) niβ X .corepresents
This implies that the type of corepresentations is a proposition when is univalent.
Corepresentation-is-prop : β {F} β is-category C β is-prop (Corepresentation F)
We opt to not show the proof, as it is even nastier than the proof for representables due to the fact that the yoneda embedding for covariant functors is itself contravariant.
Corepresentation-is-prop {F = F} c-cat X Y = path where objs : X .corep β‘ Y .corep objs = c-cat .to-path (corepresentation-unique X Y) path : X β‘ Y path i .corep = objs i path i .corepresents = [C,Sets].β -pathp refl (ap (Hom-from C) objs) {f = X .corepresents} {g = Y .corepresents} (Nat-pathp _ _ Ξ» a β Hom-pathp-reflr (Sets _) {A = F .Fβ a} {q = Ξ» i β el! (C.Hom (objs i) a)} (funext Ξ» x β ap (Ξ» e β e .Sets.to) (ap-Fβ-iso (opposite-is-category c-cat) (Hom-into C a) _) $β _ Β·Β· sym (corep.to Y .is-natural _ _ _ $β _) Β·Β· ap (Corep.from Y) (sym (corep.from X .is-natural _ _ _ $β _) Β·Β· ap (Corep.to X) (C.idr _) Β·Β· Corep.Ξ΅ X _))) i
Corepresentable functors preserve limitsπ
A useful fact about corepresentable functors is that they preserve all limits. To show this, we first need to show that the covariant hom functor preserves limits.
To get an intuition for why this is true, consider how the functor
behaves on products. The set of morphisms
is equivalent to the set
of pairs of morphisms (See product-repr
for a proof of this equivalence).
Hom-from-preserves-limits : β {oβ² ΞΊβ²} β (c : C.Ob) β is-continuous oβ² ΞΊβ² (Hom-from C c) Hom-from-preserves-limits c {Diagram = Dia} {K} {eps} lim = to-is-limitp ml (funext Ξ» _ β refl) where open make-is-limit module lim = is-limit lim ml : make-is-limit _ _ ml .Ο j f = lim.Ο j C.β f ml .commutes f = funext Ξ» g β C.pulll (sym (eps .is-natural _ _ _)) β (C.elimr (K .F-id) C.β©ββ¨refl) ml .universal eta p x = lim.universal (Ξ» j β eta j x) (Ξ» f β p f $β x) ml .factors _ _ = funext Ξ» _ β lim.factors _ _ ml .unique eps p other q = funext Ξ» x β lim.unique _ _ _ Ξ» j β q j $β x
Preservation of limits by corepresentable functors then follows from a general fact about functors: if preserves limits, and is naturally isomorphic to , then must also preserve limits.
corepresentable-preserves-limits : β {oβ² ΞΊβ²} {F} β Corepresentation F β is-continuous oβ² ΞΊβ² F corepresentable-preserves-limits F-corep lim = natural-isoβpreserves-limits (F-corep .corepresents niβ»ΒΉ) (Hom-from-preserves-limits (F-corep .corep)) lim
We can show a similar fact for representable functors, but with a twist: they reverse colimits! This is due to the fact that a representable functor is contravariant. Specifically, will take limits in to limits in , but limits in are colimits, so will take colimits in to limits in .
A less formal perspective on this is that the collection of maps out of a colimit is still defined as a limit in . For instance, to give a out of a coproduct, we are required to give a pair of maps and .
γ-reverses-colimits : β {oβ² ΞΊβ²} β (c : C.Ob) β is-cocontinuous oβ² ΞΊβ² (Functor.op (γβ C c)) γ-reverses-colimits c {Diagram = Dia} {K} {eta} colim = to-is-colimitp mc (funext Ξ» _ β refl) where open make-is-colimit module colim = is-colimit colim mc : make-is-colimit _ _ mc .Ο j f = f C.β colim.Ο j mc .commutes f = funext Ξ» g β C.pullr (eta .is-natural _ _ _) β (C.reflβ©ββ¨ C.eliml (K .F-id)) mc .universal eps p x = colim.universal (Ξ» j β eps j x) (Ξ» f β p f $β x) mc .factors eps p = funext Ξ» _ β colim.factors _ _ mc .unique eps p other q = funext Ξ» x β colim.unique _ _ _ Ξ» j β q j $β x representable-reverses-colimits : β {oβ² ΞΊβ²} {F} β Representation F β is-cocontinuous oβ² ΞΊβ² (Functor.op F) representable-reverses-colimits F-rep colim = natural-isoβpreserves-colimits ((F-rep .represents ni^op) niβ»ΒΉ) (γ-reverses-colimits (F-rep .rep)) colim