forget_preservesFilteredColimits — Mathlib

English

The forgetful functor forget CommRingCat : CommRingCat ⥤ Type preserves filtered colimits; i.e., limits and colimits of diagrams of commutative rings exist and are computed by forgetting to sets, and the colimit in Type agrees with the underlying colimit in the algebraic setting.

Русский

Функтор forgetting CommRingCat сохраняет фильтрованные колимиты; то есть колимит в CommRingCat совместим с колимитом при забывании до множеств.

LaTeX

$$$ \\mathrm{forget CommRingCat\\to Type}\\big( \\operatorname{Colim}_J F \\big) \\cong \\operatorname{Colim}_J (\\mathrm{forget}\\circ F)$$$

Lean4

instance forget_preservesFilteredColimits : PreservesFilteredColimits (forget CommRingCat.{u}) :=
  Limits.comp_preservesFilteredColimits (forget₂ CommRingCat RingCat) (forget RingCat.{u})

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