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)