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Category Theory For Beginners - 2 [entire topic moved to new forum]

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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.




    1. 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.




    1. \( \displaystyle f \) is epic and a section.




    1. \( \displaystyle f \) is an isomorphism.




       
  •  
    Replied by Grigorios Kostakos on Saturday, November 07 2015, 06:04 AM · Hide · #1
    (I) 1) Because \(f:A\longrightarrow B\) is a section, exists \(g:B\longrightarrow A\) such that \(g\circ f=id_A\).

    http://www.mathimatikoi.org/index.php/archives/finish/14-images/304-section02

    If for \(g_1,g_2:C\longrightarrow A\) holds \(f\circ g_1=f\circ g_2\),

    http://www.mathimatikoi.org/index.php/archives/finish/14-images/305-monic02

    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 \( f\circ g=id_B\).

    http://www.mathimatikoi.org/index.php/archives/finish/14-images/306-retraction02

    If for \(g_1,g_2:B\longrightarrow C\) holds \( g_1\circ f=g_2\circ f\),

    http://www.mathimatikoi.org/index.php/archives/finish/14-images/307-epic02

    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.
    1 vote by tsakanikasnickos
    Grigorios Kostakos
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