The Unapologetic Mathematician

Mathematics for the interested outsider

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.

June 10, 2007 Posted by | Category theory | 1 Comment