Theorems · Theorem · category theory

CategoryTheory.AB4.of_AB5

∀ (C : Type u) [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
  [inst_2 : CategoryTheory.Limits.HasFiniteBiproducts C] [CategoryTheory.Limits.HasFiniteLimits C]
  [inst_4 : CategoryTheory.Limits.HasFilteredColimitsOfSize.{w, w, v, u} C] [CategoryTheory.AB5OfSize.{w, w, v, u} C],
  CategoryTheory.AB4OfSize.{w, v, u} C

A category with finite biproducts and finite limits is AB4 if it is AB5.

Defined in: Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Basic
Cited by: 0 results in Mathlib
Foundations: Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Limits.HasZeroMorphisms CategoryTheory.Limits.HasFiniteBiproducts CategoryTheory.Limits.HasFiniteLimits CategoryTheory.Limits.HasFilteredColimitsOfSize CategoryTheory.AB5OfSize

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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

CategoryTheory.Categorystatement and proof · cited by 32,673
CategoryTheory.Limits.HasZeroMorphismsstatement and proof · cited by 3,275
CategoryTheory.Limits.HasFiniteBiproductsstatement and proof · cited by 106
CategoryTheory.Limits.HasFiniteLimitsstatement and proof · cited by 36
CategoryTheory.Limits.HasFilteredColimitsOfSizestatement and proof · cited by 13
CategoryTheory.AB5OfSizestatement and proof · cited by 5
CategoryTheory.Limits.hasCoproducts_of_finite_and_filteredstatement · cited by 3
CategoryTheory.hasExactColimitsOfShape_discrete_of_hasExactColimitsOfShape_finset_discreteproof · cited by 2
CategoryTheory.AB4OfSizestatement · cited by 2

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