# Terminal objects

Similar to initial objects, a category can have terminal objects. A terminal object $T$ in a category $\mathcal C$ is one for which if we have any object $X$ of $\mathcal C$, then there is a unique map from $X$ to $T$. See the resemblance? Terminal objects are like the “anti-initial objects.”

• In $\textsc{Grp}$ (groups and group homomorphisms), the trivial group is a terminal object.
• In $k$$\textsc{Vec}$ (vector spaces over a field $k$ and $k$-linear transformations), the $0$-dimensional vector space is a terminal object.

Now you may be wondering: “How can they be terminal. They were initial?” Fair question. That’s just what happens sometimes. In that case, we call it a zero object; which is a good name, because we already used $0$ to denote a $0$-dimensional vector space, or the trivial group.

But terminal and initial objects don’t always align. For instance:

• In $\textsc{Set}$ (sets and functions), any one element set is a terminal object (why?). This is distinct from $\varnothing$, the initial object in $\textsc{Set}$.

Remember that initial objects are unique up to isomorphism. A slight modification to our proof will show that terminal objects are unique up to isomorphism as well. I leave it as an exercise.

Whenever I encounter a new category, my first thought’s are toward initial and terminal objects. Do they exist in this category? If so what do they look like? This is probably true for you too, even if you didn’t know it. Initial and terminal objects tend to be “the simplest” objects in some sense. Look at the examples we have. The trivial group, empty sets and sets with one element. It’s pretty sweet that we can say this sort of stuff in the full generality of categories.