The notion of a relation is a decategorification of that of a profunctor:
- 1.
A profunctor from a category
to a category is a functor - 2.
A relation on sets
and is a function
Here we notice that:
- The opposite
of a set is itself, as restricts to the identity endofunctor on . - The values that profunctors and relations take are analogous:
- A category is enriched over the category
of sets, with profunctors taking values on it.
- A set is enriched over the set
of classical truth values, with relations taking values on it.
- A category is enriched over the category