restrictFullyFaithful_homEquiv_apply

English

Let L' ⊣ R' be an adjunction between D' and C' and suppose iC, iD embed categories C and D faithfully and fully. Given the commutative data comm1, comm2, the hom-set bijection obtained from the restricted, fully faithful adjunction is given by precomposing with the relevant unit and the images of commutations, i.e., the hom-equivalence on L.obj X → Y is computed by hiC.preimage of a specific composite built from adjunction units and the natural isomorphisms comm1, comm2.

Русский

Пусть дано окольная пара L' ⊣ R' между категориями D' и C' и вложения iC, iD соответствуют полному и полному инЪективному отражению объектов. При задании данных comm1, comm2 биекция гом-множеств для ограниченного, полнофункторного соседствия задаётся через предобразие по iC от конкретного композиции единиц и изображений из коммутаторов comm1, comm2.

LaTeX

$$$\big(\text{adj.restrictFullyFaithful}(hiC, hiD, comm1, comm2)\big) .\mathrm{homEquiv}_{X,Y}(f) = hiC.\mathrm{preimage}\Big(\mathrm{adj}.\mathrm{unit}}_{iC X} \circ R'\big( comm1.hom.app X \big) \circ R'\big( iD.map f \big) \circ comm2.hom.app Y\Big).$$$

Lean4

theorem restrictFullyFaithful_homEquiv_apply {X : C} {Y : D} (f : L.obj X ⟶ Y) :
    (adj.restrictFullyFaithful hiC hiD comm1 comm2).homEquiv X Y f =
      hiC.preimage (adj.unit.app (iC.obj X) ≫ R'.map (comm1.hom.app X) ≫ R'.map (iD.map f) ≫ comm2.hom.app Y) :=
  by
  -- This proof was just `simp [restrictFullyFaithful]` before https://github.com/leanprover-community/mathlib4/pull/16317

  apply hiC.map_injective
  simp only [homEquiv_apply, Functor.comp_obj, Functor.map_comp, map_restrictFullyFaithful_unit_app, Functor.id_obj,
    assoc, Functor.FullyFaithful.map_preimage]
  congr 2
  exact (comm2.hom.naturality _).symm

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