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This was the query in topologizing subcategory which I summarized shortly:
Mike: Where does the word ’topologizing’ come from?
Zoran Skoda: I am not completely sure anymore, but I think it is from ring theory, where people looked at the localizations at topologizing categories. There exist some topologies on various sets of ideals like Jacobson topology, so it is something of that sort in the language of subcategories instead of the language of filters of ideals. I’ll consult old references like Popescu, maybe I recall better. In any case it is pretty standard and has long history in usage: both classical and modern. No, it is not in Popescu…old related term is in fact talking about topologizing filters of ideals in a ring, so that must be the source…for example, the classical algebra by Faith, vol I, page 520 defines when the set of right ideals is topologizing. I am not good with that notion, but I can make an entry with quotation to be improved later.
New entry topologizing filter and also localizing subcategory.
Well topologizing filter is for me the same as Gabriel filter, but for some there are slight differences which I may treat later. If not I will eventually merge the two entries, but now better not.
Related new entry Gabriel localization (which is defined for every Gabriel filter). It will be needed when discussing an interesting example of Q-category which relates the Gabriel localization in ring/module theory with sheafification via the plus construction of Artin-Grothendieck.
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