Initial and final objects
The description of universal properties is typically done by stating that an object of a category is terminal: it is either initial or final. These concepts are defined here.
Initial objects
An object \( I \) in category \( \cat{C} \) is said to be initial in \( \cat{C} \) iff for every object \( A \) in \( \cat{C} \) there is exactly one morphism from \( I \) to \( A \). That is:
Final objects
An object \( F \) in category \( \cat{C} \) is said to be final in \( \cat{C} \) iff for every object \( A \) in \( \cat{C} \) there is exactly one morphism from \( A \) to \( F \). That is:
An object is said to be a terminal object iff if is either an initial object or a final object.
Unique up to a unique isomorphism. Proposition.
For any two initial objects in a category there is a single isomorphism between them. This statement is often phrased as: "initial objects are unique up to a unique isomorphism". The same is true for final objects.
Proof of this proposition is on the reverse side; I'd recommend trying to think of the proof before looking at it.
Here is different way of presenting the proposition, from Aluffi:
Let \( \cat{C} \) be a category.
- If \( I_1 \) and \( I_2 \) are both initial objects in \( \cat{C} \), then \( I_1 \cong I_2 \).
- If \( F_1 \) and \( F_2 \) are both final objects in \( \cat{C} \), then \( F_1 \cong F_2 \).
In addition, these isomorphisms are unique.
Unique up to a unique isomorphism. Proof.
First, a lemma: terminal objects have a single endomorphism which is the identity morphism.
Proof. Let \( T \) be a terminal object in category \( C \). Then there is exactly one morphism in \( \cathom{C}(T, T) \), as \( T \) being initial or final asserts this. The definition of a category asserts that the identity morphism \( 1_T \) is a morphism of \( \cathom{C}(T, T) \), so the single endomorphism is the identity morphism.
Now the main proof. Consider two initial objects \( I_1 \) and \( I_2 \). Consider possible morphisms between \( I_1 \) and \( I_2 \): being initial objects, there is a unique morphism \( f : I_1 \to I_2 \) and a unique morphism \( g: I_2 \to I_1 \). We can compose these morphisms to obtain endomorphisms, for example, \( g \, f : I_1 \to I_1 \), and \( f \, g : I_2 \to I_2\). But it was just established in the above lemma that the identities \( 1_{I_1} \) and \( 1_{I_2} \) are the only endomorphisms in \( \cathom{C}(I_1, I_1) ) \) and \( \cathom{C}(I_2, I_2) \), so \( g \, f = 1_{I_1} \) and \( f \, g = 1_{I_2} \), so \( f \) and \( g \) are isomorphic inverses.
Example
\( \mathbb{Z} \) with relation \( \le \), no terminal objects.
The category formulated by considering the relation \( \le \) on \( \mathbb{Z} \) (morphisms being the pairs of the relation's graph), has no initial or final object: an initial object one be an integer less that all other integers, and a final object would be an integer which all other integers are less than it.
In comparison, the slice category formulated from the relation \( \le \) on \( \mathbb{R} \) where objects are the morphisms of the category just mentioned–this category has a final object, namely \( (3, 3) \).
This example is mentioned by Aluffi in "Example 5.2".
Terminal objects of \( \mathrm{Set} \)
The category \( \mathrm{Set} \) has just one initial object, the empty set \( \emptyset \); the empty graph defines the unique morphism from \( \emptyset \) to all other sets.
\( \mathrm{Set} \) has many final objects: all singleton sets are a final object; for any set \( A \) and any singleton set \( \{p\} \), there is a unique function, the constant function, from \( A \) to \( \{p\} \).
It is a useful exercise to confirm that all these singleton sets are isomorphic.