The following excerpts come from the book Algebra: Chapter 0's section II.5 - Free groups. My question is about an easy exercise.
First, to define the free group $F(A)$ on $A$, the author introduces a category $\mathscr{F}^A$ as follows:
The exercise is 5.1 at the end of the section, which is shown below.
In the main text, the author proved that free group $F(A)$ on A will be (the group component of) an initial object in $\mathscr{F}^A$. The universal property is stated as follows:
When $A=\{a\}$ is a singleton. The author find this function $j: \{a\} \rightarrow \mathbb{Z}$ which sends $a$ to $1$, as an qualified initial object.
My attempt is also to consider the simplest situation when $A = \{a\}$. As the following diagram shows:
If $F(A)$ above is a final object, then we can find a set-function $j: A \rightarrow F(A)$ such that for all groups $G$ and set-functions $f: {a} \rightarrow G$ there exists an unique group homomorphism $\varphi: G \rightarrow F(A)$ such that the above diagram commutes.
Because the diagram must commute, I have to force $j(a) = \varphi \circ f(a)$. Because this property holds for all $f$'s, let me make $f(a)=1_G$. Because $\varphi$ is a group homomorphism, I have to force $\varphi(1_G) = \varphi \circ f(a) = 1_{F(A)} = j(a)$.
Then, I find I can't do any 'force' anymore, which means I can find many $\varphi$ satisfying the above requirements. For the specific function $f(a)=1_G$, $\varphi$ is not unique. So there is no such final object.
My question is:
- I only proved when $A= \{a\}$ is a singleton, there are no final objects. Is there a situation when $A$ is larger, the case is true?
- Maybe there is some generalization behind this exercise? I chose a very special function $f(a)=1_G$. If there is, could someone please indicate to me what kind of generalization it is?