module Cat.Diagram.Duals where
Dualities of diagrams🔗
Following Hu and Carette, we’ve opted to have different concrete definitions for diagrams and their opposites. In particular, we have the following pairs of “redundant” definitions:
- Products and coproducts
- Pullbacks and pushouts
- Equalisers and coequalisers
- Terminal objects and initial objects
For all four of the above pairs, we have that one in is the other in We prove these correspondences here:
Co/products🔗
is-co-product→is-coproduct : ∀ {A B P} {i1 : C.Hom A P} {i2 : C.Hom B P} → is-product (C ^op) i1 i2 → is-coproduct C i1 i2 is-co-product→is-coproduct isp = record { [_,_] = isp.⟨_,_⟩ ; in₀∘factor = isp.π₁∘factor ; in₁∘factor = isp.π₂∘factor ; unique = isp.unique } where module isp = is-product isp is-coproduct→is-co-product : ∀ {A B P} {i1 : C.Hom A P} {i2 : C.Hom B P} → is-coproduct C i1 i2 → is-product (C ^op) i1 i2 is-coproduct→is-co-product iscop = record { ⟨_,_⟩ = iscop.[_,_] ; π₁∘factor = iscop.in₀∘factor ; π₂∘factor = iscop.in₁∘factor ; unique = iscop.unique } where module iscop = is-coproduct iscop
Co/equalisers🔗
is-co-equaliser→is-coequaliser : ∀ {A B E} {f g : C.Hom A B} {coeq : C.Hom B E} → is-equaliser (C ^op) f g coeq → is-coequaliser C f g coeq is-co-equaliser→is-coequaliser eq = record { coequal = eq.equal ; universal = eq.universal ; factors = eq.factors ; unique = eq.unique } where module eq = is-equaliser eq is-coequaliser→is-co-equaliser : ∀ {A B E} {f g : C.Hom A B} {coeq : C.Hom B E} → is-coequaliser C f g coeq → is-equaliser (C ^op) f g coeq is-coequaliser→is-co-equaliser coeq = record { equal = coeq.coequal ; universal = coeq.universal ; factors = coeq.factors ; unique = coeq.unique } where module coeq = is-coequaliser coeq
Initial/terminal🔗
open Terminal open Initial is-coterminal→is-initial : ∀ {A} → is-terminal (C ^op) A → is-initial C A is-coterminal→is-initial x = x is-initial→is-coterminal : ∀ {A} → is-initial C A → is-terminal (C ^op) A is-initial→is-coterminal x = x Coterminal→Initial : Terminal (C ^op) → Initial C Coterminal→Initial term .bot = term .top Coterminal→Initial term .has⊥ = is-coterminal→is-initial (term .has⊤) Initial→Coterminal : Initial C → Terminal (C ^op) Initial→Coterminal init .top = init .bot Initial→Coterminal init .has⊤ = is-initial→is-coterminal (init .has⊥)
Pullback/pushout🔗
is-co-pullback→is-pushout : ∀ {P X Y Z} {p1 : C.Hom X P} {f : C.Hom Z X} {p2 : C.Hom Y P} {g : C.Hom Z Y} → is-pullback (C ^op) p1 f p2 g → is-pushout C f p1 g p2 is-co-pullback→is-pushout pb = record { square = pb.square ; universal = pb.universal ; i₁∘universal = pb.p₁∘universal ; i₂∘universal = pb.p₂∘universal ; unique = pb.unique } where module pb = is-pullback pb is-pushout→is-co-pullback : ∀ {P X Y Z} {p1 : C.Hom X P} {f : C.Hom Z X} {p2 : C.Hom Y P} {g : C.Hom Z Y} → is-pushout C f p1 g p2 → is-pullback (C ^op) p1 f p2 g is-pushout→is-co-pullback po = record { square = po.square ; universal = po.universal ; p₁∘universal = po.i₁∘universal ; p₂∘universal = po.i₂∘universal ; unique = po.unique } where module po = is-pushout po
Co/cones🔗
module _ {o ℓ} {J : Precategory o ℓ} {F : Functor J C} where open Functor F renaming (op to F^op) open Cocone-hom open Cone-hom open Cocone open Cone Co-cone→Cocone : Cone F^op → Cocone F Co-cone→Cocone cone .coapex = cone .apex Co-cone→Cocone cone .ψ = cone .ψ Co-cone→Cocone cone .commutes = cone .commutes Cocone→Co-cone : Cocone F → Cone F^op Cocone→Co-cone cone .apex = cone .coapex Cocone→Co-cone cone .ψ = cone .ψ Cocone→Co-cone cone .commutes = cone .commutes Cocone→Co-cone→Cocone : ∀ K → Co-cone→Cocone (Cocone→Co-cone K) ≡ K Cocone→Co-cone→Cocone K i .coapex = K .coapex Cocone→Co-cone→Cocone K i .ψ = K .ψ Cocone→Co-cone→Cocone K i .commutes = K .commutes Co-cone→Cocone→Co-cone : ∀ K → Cocone→Co-cone (Co-cone→Cocone K) ≡ K Co-cone→Cocone→Co-cone K i .apex = K .apex Co-cone→Cocone→Co-cone K i .ψ = K .ψ Co-cone→Cocone→Co-cone K i .commutes = K .commutes Co-cone-hom→Cocone-hom : ∀ {x y} → Cone-hom F^op y x → Cocone-hom F (Co-cone→Cocone x) (Co-cone→Cocone y) Co-cone-hom→Cocone-hom ch .hom = ch .hom Co-cone-hom→Cocone-hom ch .commutes o = ch .commutes o Cocone-hom→Co-cone-hom : ∀ {x y} → Cocone-hom F x y → Cone-hom F^op (Cocone→Co-cone y) (Cocone→Co-cone x) Cocone-hom→Co-cone-hom ch .hom = ch .hom Cocone-hom→Co-cone-hom ch .commutes = ch .commutes
Co/limits🔗
Limits and colimits are defined via Kan extensions,
so it’s reasonable to expect that duality of Kan extensions would
apply to (co)limits. Unfortunately, proof assistants: (co)limits are
extensions of !F, but duality of
Kan extensions inserts an extra Functor.op. We could work
around this, but it’s easier to just do the proofs by hand.
Colimit→Co-limit : Colimit F → Limit F^op Colimit→Co-limit colim = to-limit (to-is-limit ml) where module colim = Colimit colim open make-is-limit ml : make-is-limit F^op colim.coapex ml .ψ = colim.ψ ml .commutes = colim.commutes ml .universal eta p = colim.universal eta p ml .factors eta p = colim.factors eta p ml .unique eta p other q = colim.unique eta p other q Co-limit→Colimit : Limit F^op → Colimit F Co-limit→Colimit lim = to-colimit (to-is-colimit mc) where module lim = Limit lim open make-is-colimit mc : make-is-colimit F lim.apex mc .ψ = lim.ψ mc .commutes = lim.commutes mc .universal eta p = lim.universal eta p mc .factors eta p = lim.factors eta p mc .unique eta p other q = lim.unique eta p other q