The notion of a relation is a decategorification of that of a profunctor:

  1. 1. A profunctor from a category C to a category D is a functor
    p:Dop×CSets.
  2. 2. A relation on sets A and B is a function
    R:A×B{true,false}.

Here we notice that:

  • The opposite Xop of a set X is itself, as ()op:CatsCats restricts to the identity endofunctor on Sets.
  • The values that profunctors and relations take are analogous:
    • A category is enriched over the category

      Sets=defCats0

      of sets, with profunctors taking values on it.

    • A set is enriched over the set

      {true,false}=defCats1

      of classical truth values, with relations taking values on it.


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