# The Unapologetic Mathematician

## Diagram categories

A light post today, as I finish up my packing and get ready to head out tomorrow.

Diagram categories are one of those things that at first blush seem almost trivial, but they turn out to be very useful. In general, we start with some small category $\mathcal{D}$ that describes the form of a diagram, and then we take the category of functors $\mathcal{C}^\mathcal{D}$ into the category we’re interested in studying.

An easy example is a set — a category with nothing but identity arrows. A functor from a set to $\mathcal{C}$ just picks out one object of $\mathcal{C}$ for each element of the set. A little more interesting is the category $\bullet\rightrightarrows\bullet$. This has two objects and two (nontrivial) morphisms. A functor from this category picks out two objects from $\mathcal{C}$ and two (in general different) parallel arrows from one to the other in $\mathcal{C}$.

Even more useful is the category
$\begin{matrix}\bullet&\rightarrow&\bullet\\\downarrow&&\downarrow\\\bullet&\rightarrow&\bullet\end{matrix}$
Of course we can compose the arrows around the upper right or those around the lower left to get diagonal arrows, but we insist here that those two diagonal arrows are the same. A functor from this category picks out four objects and four morphisms that describe a commuting square in $\mathcal{C}$.

As a bit of a teaser, notice that the setup for finding a binary equalizer in $\mathcal{C}$ is exactly a functor from the category $\bullet\rightrightarrows\bullet$ to $\mathcal{C}$. Similarly, the setup for a binary product is a functor from the category $\bullet\qquad\bullet$. What categories give rise to diagram categories for the setups for multiple (co)products? (multiple) (co)equalizers? pushouts? pullbacks? How are the setups for pushouts and pullbacks different from the setups for equalizers and coequalizers, and why?