category of large families of sets

 set theory
Notation SetI\Set^I Objects families of sets (Xi)i∈I(X_i)_{i \in I} indexed by a collection II that is not essentially small Morphisms families of maps Related Set\Set, Set×Set\Set \times \Set, Set0(I)\Set^{(I)}_0, Set1(I)\Set^{(I)}_1, VectK(I)\Vect^{(I)}_K, AbI\Ab^I, [CRing,Set][\CRing, \Set]

This is the product category SetI=∏i∈ISet\Set^I = \prod_{i \in I} \Set, or equivalently, the functor category [Idisc⁡,Set][I_{\disc},\Set]. For some background on why this product category is well-defined even though II is a collection, see Foundations. It is a larger variant of Set×Set\Set \times \Set, and most of its properties are inherited from Set\Set, but it is not locally essentially small.

Satisfied Properties

Assigned properties

Deduced properties

Unsatisfied Properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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Special objects

  • terminal object: family of singleton sets
  • initial object: family of empty sets
  • products: pointwise defined direct products
  • coproducts: pointwise defined disjoint unions

Special morphisms

  • isomorphisms: families of bijective maps
  • monomorphisms: families of injective maps
  • epimorphisms: families of surjective maps
  • regular monomorphisms: same as monomorphisms
  • regular epimorphisms: same as epimorphisms