#2
Post
by Grigorios Kostakos »
(I) 1) Because \(f:A\longrightarrow B\) is a section, exists \(g:B\longrightarrow A\) such that \(g\circ f=id_A\).
$$\xymatrix{ A \ar[r]^f \ar[dr]_{id_A} & B \ar[d]^g \\ & A }$$
If for \(g_1,g_2:C\longrightarrow A\) holds \(f\circ g_1=f\circ g_2\),
$$\begin{xy} (0,0)*+{C}="a", (20,0)*+{A}="b", (40,0)*+{B}="c"
\ar @{->}^{g_1} "a";"b" < 2pt>
\ar @{->}_{g_2} "a";"b" <-2pt> \ar @{->}^f "b";"c"
\end{xy}$$ then \begin{align*}
g\circ (f\circ g_1)=g\circ (f\circ g_2)\quad&\Rightarrow\quad (g\circ f)\circ g_1=(g\circ f)\circ g_2\\
&\Rightarrow\quad id_A\circ g_1=id_A\circ g_2\\
&\Rightarrow\quad g_1=g_2\,.
\end{align*}
So, \(f\) is monic.
2) Because \(f:A\longrightarrow B\) is a retraction, exists \(g:B\longrightarrow A\) such that \( f\circ g=id_B\).
$$\xymatrix @C=3.0pc @R=1pc{ A \ar[r]^{f} & B \\\\
B \ar[uu]^g \ar[uur]_{id_B} & }$$
If for \(g_1,g_2:B\longrightarrow C\) holds \( g_1\circ f=g_2\circ f\), $$\begin{xy} (0,0)*+{A}="c", (20,0)*+{B}="b", (40,0)*+{ C}="a"
\ar @{->}^f "c";"b" \ar @{->}^{g_1} "b";"a" < 2pt>
\ar @{->}_{g_2} "b";"a" <-2pt>
\end{xy}$$ then \begin{align*}
(g_1\circ f)\circ g =(g_2\circ f)\circ g \quad&\Rightarrow\quad g_1\circ( f\circ g) =g_2\circ( f\circ g)\\
&\Rightarrow\quad g_1\circ id_B=g_2\circ id_B\\
&\Rightarrow\quad g_1=g_2\,.
\end{align*}
So, \(f\) is epic.
Grigorios Kostakos