When is the unit map $\eta$ for a free object an epimorphism?

Question

The following is taken from "Arrows, Structures and Functors the categorical imperative" by Arbib and Manes.

$\color{Green}{Background:}$

$\textbf{Definition for “free over an object with respect to a functor":}$ Let $G:\textbf{A}\to \textbf{B}$ be any functor, and $B$ an object of $\textbf{B}.$ We say the pair $(A,\eta),$ where $A$ is an object of $\textbf{A}$ and $\eta:B\to GA$ is a morphism of $\textbf{B},$ is $\textbf{free over}$ $B$ $\textbf{with respect to}$ $G$ just in case $\eta;B\to GA$ has the couniversal property that given any morphism $f:B\to GA'$ with $A'$ any object of $\textbf{A,}$ there exists a unique $\textbf{A-}$morphism $\psi:A\to A';$ such that

Diagram 1

We refer to $\eta$ as the $\textbf{inclusion of generators;}$ and call the unique $\psi$ satisfying the above diagram the $\textbf{A-morphic extension of } f$ (with respect to $G$).

$\color{Red}{Questions:}$

In the definition aboe, assuming that $G$ is the forgetful functor $G:\textbf{C}\to \textbf{Set}$ does the $\eta$ map have to be surjective for certain categories, algebraic or non algebraic? I know it is an injective map, since $B\subset GA.$

Thank you in advance.

Answer 1

$\eta$ is almost never surjective, and also does not have to be injective.

Arbib and Manes call it the "inclusion of generators" for a reason: if $G$ is the forgetful functor from an algebraic category like groups, rings, or modules, then $A$ is the free group, ring, or module on the set $B$, which is typically much larger than $B$, since it must include elements obtained from $B$ by freely applying the group, ring, or module operations to elements of $B$. So for example if $B = \{ 1 \}$ is a one-element set and we consider the free group on $B$, this is the group $\mathbb{Z} = \{ \dots -1, 0, 1, 2 \dots \}$ which is much larger.

In most familiar examples $\eta$ is injective but it doesn't have to be. Here is a fairly degenerate example: we can consider the category of modules over the zero ring $0$. This category is actually the terminal category; it has a single object, namely $0$ itself, and a single morphism, namely the zero map $0 \to 0$. The free module over any set exists but it is just $0$ again, and so is $\eta$.