category of pairs of sets

  • notation: Set×Set\mathbf{Set} \times \mathbf{Set}
  • objects: pairs (A,B)(A,B) of sets AA and BB
  • morphisms: A morphism (A,B)→(C,D)(A,B) \to (C,D) consists of a map A→CA \to C and a map B→DB \to D.
  • Related categories: Set\mathbf{Set}, Sh(X)\mathrm{Sh}(X)

This is an example of the product of categories. It inherits most (but not all) properties from Set\mathbf{Set}. It can also be seen as the category Sh(1+1)\mathbf{Sh}(1+1) of sheaves on a discrete space with two points, and also as the slice category Set/(1+1)\mathbf{Set}/(1+1).

Satisfied Properties

Properties from the database

Deduced properties

Unsatisfied Properties

Properties from the database

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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Special objects

  • terminal object: (1,1)(1,1)
  • initial object: (0,0)(0,0)
  • products: component-wise direct product
  • coproducts: component-wise disjoint union

Special morphisms

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