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I added the sentence
The factorizing morphism $c \to im(f)$ is sometimes called the corestriction of $f$:
to image and made corestriction redirect to this page.
I was convinced that I have once written an entry about corestrictions.
The case above is just the most important but special case. Like a morphism can be restricted to any subobject, and a function to any subset of a domain, the function can be also corestricted to any set between the range and the original codomain; similarly a morphism can be corestricted to any subobject of the codomain which contains the image; and the corestriction to the image is a special case. I am not sure if the best place for all this in full generality is eventually at image or maybe a separate entry will be needed. There is also a notion of corestriction in representation theory/Hopf algebras.
Good point. I was wondering about creating a separate entry for it. Maybe we should. But I don’t have time for this right now. If you have a minute, please split it off!
Let’s wait with this for some other day :)
Namely, I got for hours absorbed in creating lots of new algebraic number theory entries. It will get interesting at some point, once I get to the role of fundamental group! Namely an analogue of the fact that the fundamental group which counts homotopy classes of based mappings from circle to a space is isomorphic to the group of deck transformation of the universal covering at the level of the maximal abelian quotient in the case of arithmetic geometry amounts to central facts of class field theory. In the function field case those are on the other hand analogous to the Takahashi-Ward identities for QFT on the algebraic curve, as shown by Lev Tahtajan.
Setting off to bed now…
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