## Initial and terminal objects

Let’s look at something abstract. When a definition relies only on category theoretical notion (objects and arrows), it often reduces down to a form “given a diagram abc, there exists a unique x that makes another diagram xyz commute.” Commutative in this case mean that all the arrows compose correctly.Those defenitions are called *universal property* or *universal mapping property* (UMP).

Some of the notions have a counterpart in set theory, but it’s more powerful because of its abtract nature. Consider making the empty set and the one-element sets in **Sets** abstract.

**Definition 2.9.** In any category **C**, an object

- 0 is
*initial* if for any object *C* there is a unique morphism

0 => C
- 1 is
*terminal* if for any object *C* there is a unique morphism

C => 1

### Uniquely determined up to isomorphism

As a general rule, the uniqueness requirements of universal mapping properties are required only up to isomorphisms. Another way of looking at it is that if objects *A* and *B* are isomorphic to each other, they are “equal in some sense.” To signify this, we write *A ≅ B*.

**Proposition 2.10** Initial (terminal) objects are unique up to isomorphism.

Proof. In fact, if C and C’ are both initial (terminal) in the same category, then there’s a unique isomorphism C => C’. Indeed, suppose that 0 and 0’ are both initial objects in some category **C**; the following diagram then makes it clear that 0 and 0’ are uniquely isomorphic:

Given that isomorphism is defined by *g ∘ f = 1*_{A} and *f ∘ g = 1*_{B}, this looks good.

### Examples

In **Sets**, the empty set is initial and any singleton set {x} is terminal.

So apparently there’s a concept called an empty function that maps from an empty set to any set.

In a poset, an object is plainly initial iff it is the least element, and terminal iff it is the greatest element.

This kind of makes sense, since in a poset we need to preserve the structure using ≤.

There are many other examples, but the interesting part is that seemingly unrelated concepts share the same structure.

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Initial and terminal objects