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I did a little bit of rewriting and cleaning up at reflective subcategory, in an effort to make things clearer for the neophyte. Part of the cleaning-up was to remove a query initiated by Zoran under the section Characterizations (I rewrote a bit to make the question vanish altogether).
There’s another query of Zoran at the bottom which I think was answered by Mike, but let me ask before removing it.
This were the discussions:
Zoran: Gabriel–Zisman neglect the set theoretical issues on the EXISTENCE of localizations. Is the last conditions really equivalent or we need to make some set-theoretical assumptions ?
Urs Schreiber: the point here is that the localization is not at any arbitrary set of morphisms, but at precisely the class that the left adjoint sends to equivalences. This is a very special class with very nice properties and is what makes the localization come out nicely. More details on this happen to be at reflective sub-(infinity,1)-category.
About word localization:
Zoran: this is not universally accepted. In topos theory community yes. But in the setup of abelian categories, like categories of modules, people often use word localization even if left exactness is not met. If it is it is often said flat localization in those circles (though sometimes one says flat localization only if the stronger condition is satisfied: composed endofunctor is flat). The localization of the underlying ring (in the case of module categories) is the component of adjunction at that ring, and for Gabriel localizations (where T is flat) the arget module is canonically a ring and the component of the adjunction is a ring morphism. But only if the localization is perfect this morphism of rings tell you all information about the localization functor.
Mike Shulman: I changed it, how’s that?
Yes, those were the discussions, but I’m asking whether it’s alright by you to remove those discussions, i.e., have your queries been answered to your satisfaction?
For example, Urs answered your first query, and I went ahead and rewrote the statement to say that the reflection $r: A \to B$ exhibits $B$ as the localization with respect to the class of arrows $s: a \to a'$ for which $i r(s)$ is an isomorphism. I suppose it would be good to write down a formal proof, but the point is that there is no need to worry about set-theoretical assumptions since the statement just gives you the localization explicitly. (It may be clearer if I write something down.)
In the other case, apparently Mike changed the wording in response to your comment, and asked if you were now satisfied.
I removed the boxes at least half an hour before you were asking me if that is OK…sorry
Oh, sorry. You were just recording those discussions in the Forum; I get it now. Makes sense.
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