Initial & Terminal Objects

The first universal property: an object fixed not by what it contains but by there being exactly one arrow out of it, or into it.

The idea

Initial and terminal objects are the two simplest examples of a universal property: a description of an object not by what it contains, but by how every other object maps into it or out of it.

Definition (Initial and terminal objects).

An object $I$ of a category is initial when for every object $A$ there is exactly one arrow $I \to A$. An object $T$ is terminal when for every object $A$ there is exactly one arrow $A \to T$.

"Exactly one" is two demands at once: at least one such arrow must exist, and no two of them may differ.

In $\mathbf{Set}$ the definitions single out familiar objects. A function out of the empty set must name an output for each of its inputs, and it has no inputs to name, so for every set $A$ there is precisely one function $\emptyset \to A$ — the empty set is initial. Every input of a function $A \to \{\star\}$ has only one available output, so any one-element set is terminal. Neither conclusion came from looking inside the object; both came from counting arrows.

Reversing every arrow of a category turns an initial object into a terminal one, so the two notions are formal duals and one argument about either settles both. Neither is guaranteed to exist, and a category may contain several of either, so the first thing to establish in any example is the count.

Ways to work on it

• Walkthrough. Define initial and terminal objects and find them in familiar categories.
• Proof. Why an initial object is unique up to a unique isomorphism — the pattern every later universal property reuses.
• Practice. Classify objects of a category as initial, terminal, both, or neither.
• Hardest. Decide which objects, if any, are initial or terminal in trickier categories.

Not sure where to start? Take the ten-question placement test.