# The Unapologetic Mathematician

## Arrow Categories

One very useful example of a category is the category of arrows of a given category $\mathcal{C}$.

We start with any category $\mathcal{C}$ with objects ${\rm Ob}(\mathcal{C})$ and morphisms ${\rm Mor}(\mathcal{C})$. From this we build a new category called $\mathcal{C}^\mathbf{2}$, for reasons that I’ll explain later. The objects of $\mathcal{C}^\mathbf{2}$ are just the morphisms of $\mathcal{C}$. The morphisms of this new category are where things start getting interesting.

Let’s take two objects of $\mathcal{C}^\mathbf{2}$ — that is, two morphisms of $\mathcal{C}$ — and lay them side-by-side:
$\begin{matrix}A&&C\\\downarrow^f&&\downarrow^g\\B&&D\end{matrix}$
Now we want something that transforms one into the other. What we’ll do is connect each of the objects on the left to the corresponding object on the right by an arrow:
$\begin{matrix}A&\rightarrow^h&C\\\downarrow^f&&\downarrow^g\\B&\rightarrow^k&D\end{matrix}$
and require that the resulting square commute: $g\circ h=k\circ f$ as morphisms in $\mathcal{C}$. This is a morphism from $f$ to $g$. Sometimes we’ll write $(h,k):f\rightarrow g$, and sometimes we’ll name the square and write $\alpha:f\rightarrow g$.

If we have three morphisms $f$, $g$, and $h$ in $\mathcal{C}$, and commuting squares $(k_1,k_2):f\rightarrow g$ and $(k_3,k_4):g\rightarrow h$ then we can get a commuting square $(k_3\circ k_1,k_4\circ k_2):f\rightarrow h$. We check that this square commutes: $h\circ k_3\circ k_1=k_4\circ g\circ k_1=k_4\circ k_2\circ f$. This gives a composition of commuting squares. It’s easily checked that this is associative.

Given any morphism $f:A\rightarrow B$ in $\mathcal{C}$ we can just apply the identity arrows to each of $A$ and $B$ to get a commuting square $(1_A,1_B)$ between $f$ and itself. It is clear that this square serves as the identity arrow on the object $f$ in $\mathcal{C}^\mathbf{2}$, completing our proof that arrows and commuting squares in $\mathcal{C}$ do form a category.

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Posted by John Armstrong | Category theory

## 3 Comments	»

1. Thanks for this — nice crisp explanation. Would be even more helpful if it had a couple of simple, finite examples. Posets maybe?

Comment by Brian Rotman | January 28, 2010 | Reply

2. Well, these sorts of things show up in all sorts of other categorical constructions I’ve discussed, although I haven’t explicitly linked back here. Look at categories of natural transformations, or of representations, or (coming eventually) bundles.

Comment by | January 28, 2010 | Reply

3. How do you show that \$hom(f,g)\$ and \$hom(f’,g’)\$ are disjoint?

Comment by Tony | December 12, 2012 | Reply

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

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