# The Unapologetic Mathematician

## Universal arrows and universal elements

Remember that we defined a colimit as an initial object in the comma category $(F\downarrow\Delta)$, and a limit as a terminal object in the comma category $(\Delta\downarrow F)$. It turns out that initial and terminal objects in more general comma categories are also useful.

So, say we have a categories $\mathcal{C}$ and $\mathcal{D}$, an object $C\in\mathcal{C}$, and a functor $F:\mathcal{D}\rightarrow\mathcal{C}$, so we can set up the category $(C\downarrow F)$. Recall that an object of this category consists of an object $D\in\mathcal{D}$ and an arrow $f:C\rightarrow F(D)$, and a morphism from $f_1:C\rightarrow F(D_1)$ to $f_2:C\rightarrow F(D_2)$ is an arrow $d:D_1\rightarrow D_2$ such that $f_2=F(d)\circ f_1$.

Now an initial object in this category is an object $U$ of $\mathcal{D}$ and an arrow $u:C\rightarrow F(U)$ so that for any other arrow $f:C\rightarrow F(D)$ there is a unique arrow $d:U\rightarrow D$ satisfying $f=F(d)\circ u$. We call such an object, when it exists, a “universal arrow from $C$ to $F$“. For example, a colimit of a functor $F$ is a universal arrow (in $\mathcal{C}^\mathcal{J}$) from $F$ to $\Delta$.

Dually, a terminal object in $(F \downarrow C)$ consists of an object $U\in\mathcal{D}$ and an arrow $u:F(U)\rightarrow C$ so that for any other arrow $f:F(D)\rightarrow C$ there exists a unique arrow $d:D\rightarrow U$ satisfying $f=u\circ F(d)$. We call this a “couniversal arrow from $F$ to $C$“. A limit of a functor $F$ is a couniversal arrow from $\Delta$ to $F$.

Here’s another example of a universal arrow we’ve seen before: let’s say we have an integral domain $D$ and the forgetful functor $U:\mathbf{Field}\rightarrow\mathbf{Dom}$ from fields to integral domains. That is, $U$ takes a field and “forgets” that it’s a field, remembering only that it’s a ring with no zerodivisors. An object of $(D\rightarrow U)$ consists of a field $F$ and a homomorphism of integral domains $D\rightarrow U(F)$. The universal such arrow is the field of fractions of $D$.

As a more theoretical example, we can consider a functor $F:\mathcal{C}\rightarrow\mathbf{Set}$ and the comma category $(*\downarrow F)$. Since an arrow $x:*\rightarrow X$ in $\mathbf{Set}$ is an element of the set $X$, we call a universal arrow from $*$ to $F$ a “universal element” of $F$. It consists of an object $U\in\mathcal{C}$ and an element $u\in F(U)$ so that for any other element $c\in F(C)$ we have a unique arrow $f:U\rightarrow C$ with $\left[F(f)\right](u)=c$.

As an example, let’s take $N$ to be a normal subgroup of a group $G$. For any group $H$ we can consider the set of group homomorphisms $f:G\rightarrow H$ that send every element of $N$ to the identity element of $H$. Verify for yourself that this gives a functor $F:\mathbf{Grp}\rightarrow\mathbf{Set}$. Any such homomorphism factors through the projection $\pi_{(G,N)}:G\rightarrow G/N$, so the group $G/N$ and the projection $\pi_{(G,N)}$ constitute a universal element of $F$. We used cosets of $N$ to show that such a universal element exists, but everything after that follows from the universal property.

Another example: let $M_1$ be a right module over a ring $R$, and $M_2$ be a left $R$-module. Now for any abelian group $A$ we can consider the set of all $R$-middle-linear functions from $M_1\times M_2$ to $A$ — functions $f$ which satisfy $f(m_1+m_1',m_2)=f(m_1,m_2)+f(m_1',m_2)$, $f(m_1,m_2+m_2')=f(m_1,m_2)+f(m_1,m_2')$, and $f(m_1r,m_2)=f(m_1,rm_2)$. This gives a functor from $\mathbf{Ab}$ to $\mathbf{Set}$, and the universal element is the function $M_1\times M_2\rightarrow M_1\otimes_RM_2$ to the tensor product.

The concept of a universal element is a special case of a universal arrow. Very interestingly, though, when $\mathcal{C}$ is locally small (as it usually is for us)\$ the reverse is true. Indeed, if we’re considering an object $C\in\mathcal{C}$ and a functor $F:\mathcal{D}\rightarrow\mathcal{C}$ then we could instead consider the set $*$ and the functor $\hom_\mathcal{C}(C,F(\underline{\hphantom{X}})):\mathcal{D}\rightarrow\mathbf{Set}$. Universal arrows for the former setup are equivalent to universal elements in the latter.