Talk:Section (category theory)

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 $$ri=id_A$$.

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

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

$$ri=Id_A, sj=Id_B$$

Word derivations would be nice
'retract' seems obvious; if "g follows f" is the identity relation, then g is undoing or retracting what f did.

I have no intuition for why 'section' is called that, and the known lack of intuition is interfering with my head-space for the rest of the subject. ArthurDent006.5 (talk) 03:57, 3 February 2013 (UTC)


 * Both section (as in Caesarian section) and retraction (as in retract the skin after making an incision) are surgical/medical terms. But I have no reference for that, so won't add it in. It's gruesome, what mathematicians do to their fiber bundles. --Mark viking (talk) 01:01, 21 September 2013 (UTC)


 * In an non-citable conversation, I have been told something like this: suppose that a relation maps from an R*k space to an R*(k-1) space; eg from a square to a line.  The inverse of that relation, the section, maps from R*(k-1) to R*k.  The range (the output values) of that relation are a surface through (some topological rearrangement of) the R*k space; they are cutting it into sections.  Someone with more appreciation of the math would do a better job of explaining this. ArthurDent006.5 (talk) 00:08, 25 June 2015 (UTC)

Existence of homomorphism
Am I wrong or does there exist a non-trivial map from $$Z_4$$ to $$Z_2$$? Namely, $$0\mapsto 0, 1\mapsto 1, 2\mapsto 0, 3\mapsto 1$$. — Preceding unsigned comment added by 66.244.81.55 (talk) 17:56, 31 March 2014 (UTC)

boxed diagram on the right is not correct
The diagonal morphism should be 1_X instead of 1_Y. — Preceding unsigned comment added by 71.21.89.0 (talk) 10:56, 2 December 2014 (UTC)

Yes. It was a typo. Thanks for reminding! --IkamusumeFan (talk) 00:48, 25 June 2015 (UTC)

(Split) monos and epis
Any monic split epimorphism is an isomorphism. The proof is below.

$$f \circ (g \circ f)=(f \circ g) \circ f=1_Y \circ f=f=f \circ 1_X \rightarrow g \circ f=1_X$$

Dually, any epic split monomorphism is an isomorphism. GeoffreyT2000 (talk) 03:56, 18 February 2015 (UTC)

Mistake in the image?
Should the arrow between Y and Y be 1_Y instead of 1_X in the section/retraction image? — Preceding unsigned comment added by 77.95.242.32 (talk) 16:36, 14 March 2015 (UTC)

Yes. You are right. 1_Y should denote the identity morphism defined on Y, and 1_X should be with X. I have corrected the image. --IkamusumeFan (talk) 00:49, 25 June 2015 (UTC)