English
There is a canonical isomorphism between the diagram functor and its unit-counit composites via the cone equivalence.
Русский
Существует канонический изоморфизм между диаграммой и единично-коэффициентной композицией через эквивалентность конов.
LaTeX
$$$\mathrm{diagram}(F,U) \simeq \mathrm{cone}(F,U)$$$
Lean4
/-- Isomorphic presheaves have isomorphic sheaf condition diagrams. -/
def isoOfIso (α : F ≅ G) : diagram F U ≅ diagram.{v'} G U :=
NatIso.ofComponents
(by
rintro ⟨⟩
· exact piOpens.isoOfIso U α
· exact piInters.isoOfIso U α)
(by
rintro ⟨⟩ ⟨⟩ ⟨⟩
· simp
· dsimp
ext
simp only [leftRes, limit.lift_map, limit.lift_π, Cones.postcompose_obj_π, NatTrans.comp_app, Fan.mk_π_app,
Discrete.natIso_hom_app, Iso.app_hom, Category.assoc, NatTrans.naturality, limMap_π_assoc]
· dsimp [diagram]
ext
simp only [rightRes, limit.lift_map, limit.lift_π, Cones.postcompose_obj_π, NatTrans.comp_app, Fan.mk_π_app,
Discrete.natIso_hom_app, Iso.app_hom, Category.assoc, NatTrans.naturality, limMap_π_assoc]
· simp)
