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Mathlib.CategoryTheory.Limits.Sifted

Sifted categories #

A category C is sifted if C is nonempty and the diagonal functor C ⥤ C × C is final. Sifted categories can be characterized as those such that the colimit functor (C ⥤ Type) ⥤ Type preserves finite products.

Main results #

References #

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@[reducible, inline]
abbrev CategoryTheory.IsSiftedOrEmpty (C : Type u) [CategoryTheory.Category.{v, u} C] :
Prop

A category C IsSiftedOrEmpty if the diagonal functor C ⥤ C × C is final.

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Instances For
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    class CategoryTheory.IsSifted (C : Type u) [CategoryTheory.Category.{v, u} C] extends CategoryTheory.Functor.Final :
    Prop

    A category C IsSfited if

    1. the diagonal functor C ⥤ C × C is final.
    2. there exists some object.
    Instances
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      theorem CategoryTheory.IsSifted.nonempty {C : Type u} :
      ∀ {inst : CategoryTheory.Category.{v, u} C} [self : CategoryTheory.IsSifted C], Nonempty C
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      theorem CategoryTheory.IsSifted.isSifted_of_equiv {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsSifted C] {D : Type u₁} [CategoryTheory.Category.{v₁, u₁} D] (e : D ≌ C) :
      CategoryTheory.IsSifted D

      Being sifted is preserved by equivalences of categories

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      theorem CategoryTheory.IsSifted.isSifted_iff_asSmallIsSifted {C : Type u} [CategoryTheory.Category.{v, u} C] :
      CategoryTheory.IsSifted C ↔ CategoryTheory.IsSifted (CategoryTheory.AsSmall C)

      In particular a category is sifted iff and only if it is so when viewed as a small category

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      instance CategoryTheory.IsSifted.instIsConnected {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsSifted C] :
      CategoryTheory.IsConnected C

      A sifted category is connected.

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      • ⋯ = ⋯
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      instance CategoryTheory.IsSifted.instIsSiftedOrEmptyOfHasBinaryCoproducts {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] :
      CategoryTheory.IsSiftedOrEmpty C

      A category with binary coproducts is sifted or empty.

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      • ⋯ = ⋯
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      instance CategoryTheory.IsSifted.isSifted_of_hasBinaryCoproducts_and_nonempty {C : Type u} [CategoryTheory.Category.{v, u} C] [Nonempty C] [CategoryTheory.Limits.HasBinaryCoproducts C] :
      CategoryTheory.IsSifted C

      A nonempty category with binary coproducts is sifted.

      Equations
      • ⋯ = ⋯