English
Conjugation formula for the costar under symmetrification: φ.symmetrify.costar u equals the conjugation of Sum.map by the symmetrify equivalence.
Русский
Формула конъюгации для костара при симметризации: φ.symmetrify.costar u равен конъюгации Sum.map симметризационным эквивалентом.
LaTeX
$$φ.symmetrify.costar u = (Quiver.symmetrifyCostar (φ.obj u)).symm ∘ Sum.map (φ.costar u) (φ.star u) ∘ Quiver.symmetrifyCostar u$$
Lean4
/-- In a quiver with involutive inverses, the star and costar at every vertex are equivalent.
This map is induced by `Quiver.reverse`. -/
@[simps]
def starEquivCostar (u : U) : Quiver.Star u ≃ Quiver.Costar u
where
toFun e := ⟨e.1, reverse e.2⟩
invFun e := ⟨e.1, reverse e.2⟩
left_inv e := by simp
right_inv e := by simp
