Theorems · Theorem · category theory

CategoryTheory.MorphismProperty.hasOfPostcompProperty_iff_le_diagonal

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasPullbacks C]
  {P : CategoryTheory.MorphismProperty C} [P.IsStableUnderBaseChange] [P.IsMultiplicative]
  {Q : CategoryTheory.MorphismProperty C} [Q.IsStableUnderBaseChange], P.HasOfPostcompProperty Q ↔ Q ≤ P.diagonal

If P is multiplicative and stable under base change, having the of-postcomp property w.r.t. Q is equivalent to Q implying P on the diagonal.

Defined in: Mathlib.CategoryTheory.MorphismProperty.Limits
Cited by: 0 results in Mathlib
Foundations: Depth 62 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Limits.HasPullbacks CategoryTheory.MorphismProperty.IsStableUnderBaseChange CategoryTheory.MorphismProperty.IsMultiplicative CategoryTheory.MorphismProperty.IsStableUnderBaseChange

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CategoryTheory.Categorystatement and proof · cited by 32,673
Quiver.Homproof · cited by 32,603
CategoryTheory.CategoryStruct.compproof · cited by 17,999
CategoryTheory.CategoryStruct.idproof · cited by 6,235
CategoryTheory.MorphismPropertystatement and proof · cited by 2,179
CategoryTheory.Category.id_compproof · cited by 1,998
CategoryTheory.Limits.pullbackproof · cited by 864
CategoryTheory.Limits.pullback.fstproof · cited by 639
CategoryTheory.Limits.pullback.sndproof · cited by 637
CategoryTheory.Limits.HasPullbacksstatement and proof · cited by 439
CategoryTheory.MorphismProperty.IsMultiplicativestatement and proof · cited by 332
CategoryTheory.Limits.limit.lift_πproof · cited by 266

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