The pullback of sets and whose morphisms are and is the intersection of and :
This diagram can only commute if and at the same time, i.e. .
Pullback of functions
Given and , then their pullback will be — alongside projection morphisms and — the fiber product of these functions: the elements in the preimage of the intersection of their images, or rather, the cartesian product of these elements’ fibers over and . That is,
Note that, for any , we’ll have that
which means that, since is in the intersection of their images, .
This example can also be seen as the Equalizer of the morphisms and (where is their Product) — I assume the equalizer’s object is the fiber product itself, and its morphism to is just an inclusion map? to-be-elaborated Note that, since it’s an equalizer, it ensures that , i.e. that (equals) in a “common domain” (given by , embedded in via ).
Function’s graph
Given a function , its graph can be seen as ‘s pullback with .
For all ,
Preimage of a function
Given a function and a subset , then the pullback of and the inclusion map is the preimage , alongside morphisms to and the restriction to :
Note that, for some ,
Do note that it also works when , since is an Initial Object in : it’s guaranteed to exist morphisms from it to and . In any case, it’d be a pretty boring diagram, going from nowhere to nowhere.
