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 in a category is one so that has exactly one element for each object . Many of the categories we’ve considered have initial objects. For example:

• The empty set is the initial object of , since there’s exactly one function from the empty set to any other set.
• The trivial group is the initial object of , since (again) there’s only one group homomorphism from the trivial group to any other group.
• The initial object of the category 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 and are both initial. Then there is exactly one arrow , and exactly one . They compose to give arrows and , 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 so that every 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 , since there’s only one function from any other set into it: send all elements of to the same point.
• The trivial group is also the terminal object of , since there’s only one homomorphism into it from any group.
• It turns out that the trivial group is also the terminal object in , 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.