English
There is an equivalence between monoid homomorphisms M → G and GrothendieckGroup M → G, i.e., a lifting universal property.
Русский
Существует эквивалентность между моноид-гомоморфизмами M → G и GrothendieckGroup M → G, то есть каноническое свойство траекторий.
LaTeX
$$$\\text{lift}: (M \\to_* G) \\simeq (\\mathrm{GrothendieckGroup}(M) \\to_* G)$$$
Lean4
/-- A monoid homomorphism from a monoid `M` to a group `G` lifts to a group homomorphism from the
Grothendieck group of `M` to `G`. -/
@[to_additive (attr := simps symm_apply) /--
A monoid homomorphism from a monoid `M` to a group `G` lifts to a group homomorphism from the
Grothendieck group of `M` to `G`. -/
]
noncomputable def lift : (M →* G) ≃ (GrothendieckGroup M →* G)
where
toFun f := (monoidOf ⊤).lift (g := f) fun _ ↦ Group.isUnit _
invFun f := f.comp of
left_inv f := by ext; simp
right_inv f := by ext; simp
