of_colim_preservesFiniteProducts — Mathlib

English

If the colimit functor preserves limits of shape Discrete Fin 0, then C is nonempty.

Русский

Если колим сохраняет пределы дискретного типа Fin 0, то C непуста.

LaTeX

$$[PreservesLimitsOfShape (Discrete Fin 0) colim] ⇒ Nonempty C$$

Lean4

/-- If the `colim` functor `(C ⥤ Type) ⥤ Type` preserves finite products, then `C` is sifted. -/
theorem of_colim_preservesFiniteProducts [h : PreservesFiniteProducts (colim : (C ⥤ Type u) ⥤ Type u)] : IsSifted C :=
  by
  have := isSiftedOrEmpty_of_colim_preservesFiniteProducts C
  have := nonempty_of_colim_preservesLimitsOfShapeFinZero C
  constructor

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