Let
- 1.
As a Coequaliser. We have an isomorphism of sets
where
is the equivalence relation generated by . - 2.
As a Pushout. We have an isomorphism of sets1 where
is the equivalence relation generated by . - 3.
The First Isomorphism Theorem for Sets. We have an isomorphism of sets23
- 4.
Descending Functions to Quotient Sets, I. Let
be an equivalence relation on . The following conditions are equivalent:- (a)
There exists a map
making the diagram
commute.
- (b)
We have
. - (c)
For each
, if , then .
- (a)
There exists a map
- 5.
Descending Functions to Quotient Sets, II. Let
be an equivalence relation on . If the conditions of Item 4 hold, then is the unique map making the diagramcommute.
- 6.
Descending Functions to Quotient Sets, III. Let
be an equivalence relation on . We have a bijectionnatural in
, given by the assignment of Item 4 and Item 5, where is the set defined by - 7.
Descending Functions to Quotient Sets, IV. Let
be an equivalence relation on . If the conditions of Item 4 hold, then the following conditions are equivalent:- (a)
The map
is an injection. - (b)
We have
. - (c)
For each
, we have iff .
- (a)
The map
- 8.
Descending Functions to Quotient Sets, V. Let
be an equivalence relation on . If the conditions of Item 4 hold, then the following conditions are equivalent:- (a)
The map
is surjective. - (b)
The map
is surjective.
- (a)
The map
- 9.
Descending Functions to Quotient Sets, VI. Let
be a relation on and let be the equivalence relation associated to . The following conditions are equivalent:
1Dually, we also have an isomorphism of sets
2Further Terminology: The set is often called the coimage of , and denoted by .
3In a sense this is a result relating the monad in induced by with the comonad in induced by , as the kernel and image
of Chapter 7: Constructions With Relations, Item 2 of Proposition 7.3.1.1.2.
of