Talk:Section (category theory)

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[edit] Another notion of retract

Here's a different notion of retract of morphisms:

Given a category E, and two objects A,B, we say that A is a retract of B if there are maps i:A\rightarrow B, r:B\rightarrow A such that ri = idA.

A map u:A\rightarrow B is a retract of a map v:C\rightarrow D if u is a retract of v in the category of arrows of E. i.e., if there is a conmutative diagram

\begin{array}{ccccc} A & \stackrel{i}{\rightarrow} & C &\stackrel{r}{\rightarrow} & A \\ u\downarrow &&v\downarrow &&u\downarrow \\ B & \stackrel{j}{\rightarrow} & D &\stackrel{s}{\rightarrow} & B \end{array}

ri = IdA,sj = IdB