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I call $(s, r)$ such that $r\circ s =\operatorname{id}$ a split morphism. (Other names are “section-retraction pair”, “split pair.”) Notice that this is not a section, because any section is one morphism. I want to make sure that no official name exists thus I can invent it myself. Do you know the name?

The same question about isomorphisms. “Isomorphism pair”, “isomorphic pair”?

The distinction may be quite apparent. Every split morphism is a categorical diagram whereas a section is not.

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There is a notion of a split epi/monomorphism, but it isn't defined as a pair of morphisms. This seems more useful in a algebraic or constructive setting... –  Zhen Lin Oct 17 '11 at 17:48
@Zhen Lin: Split monomorphism is just a synonym for section. I read Wikipedia before. :) –  beroal Oct 19 '11 at 11:33
Have you settled on any terminology? I want to use it. For the second concept, I think "isomorphism pair" is a good name. –  goblin Feb 12 '14 at 23:55
@user18921: No. I leave it to you. –  beroal Feb 14 '14 at 6:15

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