Category Theory, Pushout and Pullback
Pushout
The pushout is a special case of the colimit,
which was described in the previous section.
Start with a morphism f from Z into X,
and another morphism g from Z into Y.
The pushout, if it exists, is an object P,
and morphisms from X into P and from Y into P,
such that the entire square commutes.
In other words, you can go from Z to X to P, or from Z to Y to P,
and the result is the same.
Furthermore, P is an initial object having this property.
If Q also builds a commutative square, a morphism from P to Q makes the entire diagram,
{Z,X,Y,P,Q}, commute.
The simplest example comes from sets and arbitrary functions on sets.
Let Z be a subset of both X and Y, so that the morphisms into X and Y are inclusion.
Let P be X union Y, with inclusion from X / Y into P as the pushout morphisms.
If Q is some other extension of X and Y,
map c ∈ X or Y into P,
giving d,
and into Q, giving e.
Then map d to e.
This is the unique morphism from P into Q that makes everything commute,
hence P is initial, and is the unique (up to isomorphism) pushout of X and Y.
If X and Y are abelian groups, or modules,
the pushout P is the direct product of X and Y,
mod the relations f(c)-g(c), for each c in Z.
Both X and Y embed in the direct product, and map into the quotient module.
These are the pushout morphisms.
Verify that P is an initial object.
Finally, X and Y could be topological spaces,
whence P becomes the adjunction space.
It is basically the disjoint union of X and Y, glued together at Z.
Pullback
The pullback is just like the pushout, but the arrows are reversed.
Let X and Y map into Z, and the pullback P maps into X and Y,
such that the square commutes, and P is terminal.
In other words, P is the limit of our 3 object diagram.
These are not as common as pushouts,
though they are sometimes used in connection with fiber bundles.