category of pairs of sets

 set theory
Notation Set×Set\Set \times \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 Set\Set, Set→\Set^{\rightarrow}, [(N,≤),Set][(\IN,\leq),\Set], Sh(X)\Sh(X), SetI\Set^I, Quiv\Quiv, (Set×Set)∅,fin(\Set \times \Set)_{\varnothing,\fin}

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

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: (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

Indistinguishable categories

These categories in the database currently have exactly the same properties as the category of pairs of sets. This indicates that the data may be incomplete or that a distinguishing property may be missing from the database.

Functors

The database stores 3 functors based on the category of pairs of sets.