Theorems · Theorem · category theory
CategoryTheory.preservesFiniteLimits_iff_lan_preservesFiniteLimits
∀ {C D : Type u₁} [inst : CategoryTheory.SmallCategory C] [inst_1 : CategoryTheory.SmallCategory D]
[CategoryTheory.Limits.HasFiniteLimits C] (F : CategoryTheory.Functor C D),
CategoryTheory.Limits.PreservesFiniteLimits F ↔ CategoryTheory.Limits.PreservesFiniteLimits F.op.lanIf C is finitely complete, then F : C ⥤ D preserves finite limits iff
Lan F.op : (Cᵒᵖ ⥤ Type*) ⥤ (Dᵒᵖ ⥤ Type*) preserves finite limits.
- Defined in
- Mathlib.CategoryTheory.Functor.Flat
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CategoryTheory.Functorstatement and proof · cited by 16,252
- Oppositestatement · cited by 8,081
- CategoryTheory.Functor.compstatement · cited by 6,529
- CategoryTheory.Functor.opstatement and proof · cited by 997
- CategoryTheory.CostructuredArrowstatement · cited by 536
- CategoryTheory.SmallCategorystatement and proof · cited by 480
- CategoryTheory.CostructuredArrow.projstatement · cited by 122
- CategoryTheory.Limits.PreservesFiniteLimitsstatement and proof · cited by 121
- CategoryTheory.FinCategoryproof · cited by 107
- CategoryTheory.Limits.HasFiniteLimitsstatement and proof · cited by 36
- CategoryTheory.Functor.lanstatement and proof · cited by 22
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