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PostPosted: Sat Jan 16, 2016 11:04 pm 
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(I) Let \( \displaystyle f : A \longrightarrow B \) be a morphism in a category \( \mathcal{C} \). Show that:

  1. If \( \displaystyle f \) is a section, then \( \displaystyle f \) is monic.
  2. If \( \displaystyle f \) is a retraction, then \( \displaystyle f \) is epic.

(II) If \( \displaystyle f \) is a morphism in a category \( \mathcal{C} \), then prove that the following are equivalent:

  1. \( \displaystyle f \) is monic and a retraction.
  2. \( \displaystyle f \) is epic and a section.
  3. \( \displaystyle f \) is an isomorphism.


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PostPosted: Sat Jan 16, 2016 11:05 pm 
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(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 \( f\circ g=id_B\).
$$\xymatrix{ A \ar[r]^{f} & B \\
B \ar[u]^g \ar[ur]_{id_B} & }$$
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


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PostPosted: Sat Jan 16, 2016 11:54 pm 
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(II) Let \(\displaystyle{f\in\rm{Mor}(A,B)}\) where \(\displaystyle{A\,,B\in C}\) .

\(\displaystyle{i)\implies ii)}\) : Suppose that \(\displaystyle{f}\) is monic and retraction.

According to (I) , \(\displaystyle{f}\) is epic.

Also, there exists \(\displaystyle{g:B\to A}\) such that \(\displaystyle{f\circ g=Id_{B}}\).

Now, \(\displaystyle{(f\circ g)\circ f=Id_{B}\circ f\implies f\circ (g\circ f)=f=f\circ Id_{A}}\)

and since \(\displaystyle{f}\) is monic, we get : \(\displaystyle{g\circ f=Id_{A}}\), so it is a section.

\(\displaystyle{ii)\implies iii)}\) : Since \(\displaystyle{f}\) is section, there exists

\(\displaystyle{g:B\to A}\) such that \(\displaystyle{g\circ f=Id_{A}}\). Now,

\(\displaystyle{f\circ (g\circ f)=f\circ Id_{A}\implies (f\circ g)\circ f=f=Id_{B}\circ f}\) and

using the fact that \(\displaystyle{f}\) is epic, we get \(\displaystyle{f\circ g=Id_{B}}\).

Then, \(\displaystyle{f}\) is an isomorphism.

\(\displaystyle{iii)\implies i)}\) : There exists \(\displaystyle{g:B\to A}\) such that

\(\displaystyle{g\circ f=Id_{A}\,\,,f\circ g=Id_{B}}\) .

The first relation gives us that \(\displaystyle{f}\) is section, so monic. The second relation

gives us that \(\displaystyle{f}\) is retraction

and the exercise comes to an end.


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