English
Cocycle relation tying three fst maps in a threefold pullback; expresses associativity constraints.
Русский
Кокосовое соотношение, связывающее три fst в трехкратном обратном, задающее ассоциативность.
LaTeX
$$$ \\text{cocycle}_{fstfstfst}(i,j,k) : t'_{\\mathcal{U}} f g i j k \\;\\gg\\; t'_{\\mathcal{U}} f g j k i \\;\\gg\\; t'_{\\mathcal{U}} f g k i j = \\mathbf{1} $$
Lean4
/-- (Implementation)
The canonical map `(s.X ×[X] Uᵢ) ×[s.X] (s.X ×[X] Uⱼ) ⟶ (Uᵢ ×[Z] Y) ×[X] Uⱼ`
This is used in `gluedLift`. -/
def gluedLiftPullbackMap (i j : 𝒰.I₀) :
pullback ((𝒰.pullback₁ s.fst).f i) ((𝒰.pullback₁ s.fst).f j) ⟶ (gluing 𝒰 f g).V ⟨i, j⟩ :=
by
refine (pullbackRightPullbackFstIso _ _ _).hom ≫ ?_
refine pullback.map _ _ _ _ ?_ (𝟙 _) (𝟙 _) ?_ ?_
· exact (pullbackSymmetry _ _).hom ≫ pullback.map _ _ _ _ (𝟙 _) s.snd f (Category.id_comp _).symm s.condition
· simpa using pullback.condition
· simp only [Category.comp_id, Category.id_comp]
