category of sets and relations

  • notation: Rel\mathbf{Rel}
  • objects: sets
  • morphisms: A morphism from AA to BB is a relation, i.e. a subset of A×BA \times B.
  • nLab Link
  • Related categories: Set\mathbf{Set}

This category is self-dual as it can be: There is an isomorphism Rel≅Relop\mathbf{Rel} \cong \mathbf{Rel}^{\mathrm{op}} that is the identity on objects and maps a relation to its opposite relation. It is the prototype of a dagger-category.

Properties

Properties from the database

Deduced properties

Non-Properties

Non-Properties from the database

Deduced Non-Properties*

*This also uses the deduced properties.

Unknown properties

For these properties the database currently doesn't have an answer if they are satisfied or not. Please help to complete the data!

Special morphisms

  • Isomorphisms: bijective functions
  • Monomorphisms: A relation R:A→BR : A \to B is a monomorphism iff the map R∗:P(A)→P(B)R_* : P(A) \to P(B) defined by T↦{b∈B:∃ a∈T:(a,b)∈R}T \mapsto \{b \in B : \exists \, a \in T: (a,b) \in R \} is injective.
  • Epimorphisms: A relation R:A→BR : A \to B is an epimorphism iff the map R∗:P(B)→P(A)R^* : P(B) \to P(A) defined by S↦{a∈A:∃ b∈S:(a,b)∈R}S \mapsto \{a \in A : \exists \, b \in S: (a,b) \in R \} is injective.

Comments

  • Lots of properties are unknown here. Please help to fill in the gaps!