# The Unapologetic Mathematician

## Initial and Terminal Objects

Since the next subject I’ll move into starts with a few simple definitions, I’ll kick it off today. Light fare for a Sunday.

An initial object $I$ in a category $\mathcal{C}$ is one so that $\hom_\mathcal{C}(I,X)$ has exactly one element for each object $X\in\mathcal{C}$. Many of the categories we’ve considered have initial objects. For example:

• The empty set is the initial object of $\mathbf{Set}$, since there’s exactly one function from the empty set to any other set.
• The trivial group is the initial object of $\mathbf{Grp}$, since (again) there’s only one group homomorphism from the trivial group to any other group.
• The initial object of the category $\mathbf{Gpd}$ of groupoids is the empty groupoid with no objects and no morphisms.
• The initial object of an ordinal number considered as a category is the least element of the ordinal.
• More generally, the initial object of any preorder is a minimal element, which is below-or-equal-to every other element.

Not all categories have an initial object. A discrete category, for instance, has none. Neither does a preorder with no minimal element (consider the integers as an ordered set). However, if a category has an initial object it’s unique up to isomorphism. Let’s say that $I_1$ and $I_2$ are both initial. Then there is exactly one arrow $I_1\rightarrow I_2$, and exactly one $I_2\rightarrow I_1$. They compose to give arrows $I_1\rightarrow I_1$ and $I_2\rightarrow I_2$, but again there is only one such arrow in each case: the identity arrows. So these compositions must be the respective identities, and we have an isomorphism.

The dual notion is that of a terminal object $T$ so that every $\hom_\mathcal{C}(X,T)$ has exactly one element. Again, terminal objects are unique up to isomorphism. And again, many of our favorite categories have terminal objects, but not all. Some examples:

• Any set with a single element is a terminal object in $\mathbf{Set}$, since there’s only one function from any other set into it: send all elements of $X$ to the same point.
• The trivial group is also the terminal object of $\mathbf{Grp}$, since there’s only one homomorphism into it from any group.
• It turns out that the trivial group is also the terminal object in $\mathbf{Gpd}$, as you should verify.
• The terminal object of a preorder is an element which is above-or-equal-to every other.

We’ll also use the terms “universal” and “couniversal” for initial and terminal objects, respectively. We’ll see that many constructions we already know — and many we’ll come to know — consist of setting up an apropriate category and finding an initial or a terminal object. We say that such a construction satisfies a universal condition, and the result is well-defined up to isomorphism.