llp_rlp_of_hasSmallObjectArgument

English

Equivalently, I.rlp.llp equals the retracts of transfinite compositions of pushouts of coproducts of arrows in C, indexed by κ.

Русский

Эквивалентно, I.rlp.llp — это retracts трансфинитных композицигпх пушаутов копроподктов стрел в C, индексированных κ.

LaTeX

$$$ I.rlp.llp = (\text{transfiniteCompositions}(\text{coproducts}(I).pushouts).retracts) $$$

Lean4

/-- If `I : MorphismProperty C` permits the small object argument,
then the class of morphism that have the left lifting property with respect to
the maps that have the right lifting property with respect to `I` are
exactly the retracts of transfinite compositions
of pushouts of coproducts of morphisms in `C`. -/
theorem llp_rlp_of_hasSmallObjectArgument :
    I.rlp.llp = (transfiniteCompositions.{w} (coproducts.{w} I).pushouts).retracts :=
  llp_rlp_of_isCardinalForSmallObjectArgument I I.smallObjectκ

Похожие утверждения