English
In a directed system of additive monoids G with a direct limit, the natural and nonnegative rational casts embed into the direct limit as predictable classes; for each natural n and index i, n is sent to the class of (i,n). Similarly for NNRat and Int cast, the corresponding casts land inside the direct limit as the class of (i,n).
Русский
В направленной системе абелевых монойдов прямой предел вводит естественные отображения натуральных чисел и округленных дробей как классов элементов пары (i, n) в DirectLimit.
LaTeX
$$$ (n : DirectLimit G f) = \overline{\langle i, n \rangle}, \quad (q : NNRat) (i) : (q : DirectLimit G f) = \overline{\langle i, q \rangle}, \quad (q : DirectLimit G f) (\text{Int}) (i) : (q : DirectLimit G f) = \overline{\langle i, q \rangle} $$$
Lean4
/-- `Under.pushout f` preserves finite limits if `f` is flat. -/
theorem preservesFiniteLimits_of_flat (hf : RingHom.Flat f.hom) : PreservesFiniteLimits (Under.pushout f) where
preservesFiniteLimits
_ :=
letI : Algebra R S := f.hom.toAlgebra
haveI : Module.Flat R S := hf
preservesLimitsOfShape_of_natIso (tensorProdIsoPushout R S)
