English
A variant of lift after reindexing symmetry holds with the symmetric pullback map.
Русский
Вариант подъема после симметричной переиндексации сохраняется через симметричную вытяжку.
LaTeX
$$$\\mathrm{lift} \\phi \\circ (\\operatorname{reindex}_{R,s,e})^{\\mathrm{symm}} = \\mathrm{lift}(\\operatorname{domDomCongrLinearEquiv'}_{R,R,s,e}^{\\}\\mathrm{symm} \\phi)$$$
Lean4
/-- The tensor product over an empty index type `ι` is isomorphic to the base ring. -/
@[simps symm_apply]
def isEmptyEquiv [IsEmpty ι] : (⨂[R] i : ι, s i) ≃ₗ[R] R
where
toFun := lift (constOfIsEmpty R _ 1)
invFun r := r • tprod R (@isEmptyElim _ _ _)
left_inv
x := by
refine x.induction_on ?_ ?_
· intro x y
simp only [map_smulₛₗ, RingHom.id_apply, lift.tprod, constOfIsEmpty_apply, const_apply, smul_eq_mul, mul_one]
congr
aesop
· simp only
intro x y hx hy
rw [map_add, add_smul, hx, hy]
right_inv t := by simp
map_add' := LinearMap.map_add _
map_smul' := fun r x => by exact LinearMap.map_smul _ r x
