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

tsakanikasnickos on Friday, October 30 2015, 07:51 PM
0

(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.

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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 $$g\circ f=id_A$$.

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

then \begin{align*}
\end{align*}
So, $$f$$ is monic.

2) Because $$f:A\longrightarrow B$$ is a retraction, exists $$g:B\longrightarrow A$$ such that $$f\circ g=id_B$$.

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

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)\\
\end{align*}
So, $$f$$ is epic.
1 vote
Grigorios Kostakos