Prologue~
Said the Duck to the Kangaroo,
'Good gracious! how you hop!
Over the fields and the water too,
As if you never would stop!
My life is a bore in this nasty pond,
And I long to go out in the world beyond...'
— Edward Lear, The Duck and the Kangaroo (1871)
As a continuation to the previous post, let us look at pushouts. However, we take a different approach this time, we first examine the general architecture of a colimit, and then see how the pushout emerges as a specific, beautiful instance of it.
Verse I: The Roads~
A diagram of type $J$ in a category $\mathscr{C}$ is a covariant functor: $ D : J \to \mathscr{C}$, where category $J$ is called the index category and the functor $D$ is called a $J-$shaped diagram.
Think of a diagram as the categorical version of an indexed family in Set theory. While a family of sets is just a collection, a diagram includes the relationships (morphisms) between those objects. It is a network of roads (morphisms) connecting cities (objects).
Here on, we will look at the functorial aspect of the diagram $D$, so for the sake of better understanding, we refer to $D$ as $F$.
Verse II: At the Crossroads~
If the diagram is the network of roads, the colimit is where they all converge.
We define a co-cone of a diagram $F: J \to \mathscr{C}$ as an object $N \in Ob(\mathscr{C})$ together with a family of morphisms $\psi_X : F(X) \to N$ for all $X \in Ob(J)$ such that for every morphism $f: X \to Y$ in $J$, we have $$\psi_Y \circ F(f) = \psi_X$$.
Some examples of colimits are disjoint unions, direct sums, coproducts, pushouts, or direct limits. Note that pushouts are a form of colimits, and similarly, pullbacks are a form of limits.
In other words, $L$ is the closest or best meeting point, any other meeting point $N$ is just a shadow of $L$.
Verse III: Squaring the Circle~
Ideally, we want to glue two objects together.
Now, given a diagram consisting of two morphisms $f: Z \to X$ and $g: Z \to Y$ with a common domain. This $Z$ acts as the seam or the overlap between $X$ and $Y$. We define a pushout consisting of an object $P$ slong with two morphisms $p_1 : X \to P$ and $p_2 : Y \to P$ such that the square commutes.
$(P,p_1, p_2)$ is universal with respect to the diagram, that is to say that for any other such triple $(Q, q_1, q_2)$, the following diagram commutes such that there exists a unique $u: P \to Q$, also making the diagram commute.
Pushouts are also called fibered coproducts, fibered sums, amalagamated sums, or cocartesian squares.
Similar to before, a pushout, if exists, is unique up to a unique isomorphism. As with all universal properties, a pushout (if it exists) is unique up to a unique isomorphism.
Verse IV: Through the Looking Glass~
If we step through the looking glass, the arrows reverse, and colimits become limits.
A cone to $F$ is an object $N \in \mathscr{C}$ together with a family of morphisms $\psi_X : N \to F(X)$ indexed by the objects $X \in J$, such that for every morphism $f : X \to Y$ in $J$, we have $$F(f) \circ \psi_X = \psi_Y$$
A limit of the diagram $F: J \to \mathscr{C}$ is a cone $(L, \phi)$ to $F$ such that for every cone $(N, \psi)$ to $F$, there exists a unique morphism $u: N \to L$ such that for all $X \in Ob (J)$, we have $$\phi_X \circ u = \psi_X$$
Epilogue~
So away they went with a hop and a bound,
And they hopped the whole world three times round;
And who so happy,—O who,
As the Duck and the Kangaroo?
— Edward Lear, The Duck and the Kangaroo (1871)