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Talk:Section (category theory) - Wikipedia

Talk:Section (category theory)

Latest comment: 9 years ago by IkamusumeFan in topic Mistake in the image?

Another notion of retract

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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  ,   such that  .

A map   is a retract of a map   if u is a retract of v in the category of arrows of E. i.e., if there is a conmutative diagram

 

 

Word derivations would be nice

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'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)Reply

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)Reply
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)Reply

Existence of homomorphism

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Am I wrong or does there exist a non-trivial map from   to  ? Namely,  . — Preceding unsigned comment added by 66.244.81.55 (talk) 17:56, 31 March 2014 (UTC)Reply

boxed diagram on the right is not correct

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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)Reply

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

(Split) monos and epis

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Any monic split epimorphism is an isomorphism. The proof is below.

 

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

Mistake in the image?

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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)Reply

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)Reply









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