# General Theory of Left-Exact Localization?

A left-exact localization of a category is a reflective subcategory such that the reflector preserves finite limits. There are several prominent examples of such localizations, such as sheafification, and localization of module categories. Is there a general theory of such localizations?

I don't have any particular type of result in mind, but given the prominence of the two examples I mentioned, it seems like the topic of left-exact localizations must have been studied for its own sake.

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Special cases of reflective localisation in locally presentable categories are studied in § 1.C of [Adámek and Rosický, Locally presentable and accessible categories] and, of course, the theory of left exact localisations of presheaf toposes is just the theory of Grothendieck topologies. –  Zhen Lin Apr 23 '13 at 7:47
Basics of the general theory are reviewed in the first thirteen pages of arXiv:0806.1324 Localization theory for triangulated categories. –  dhagbert Apr 23 '13 at 10:00
I don't see where Krause addresses left-exactness. –  arsmath Apr 23 '13 at 10:31
The first volume of Borceux's "Handbook of categorical algebra" has some discussion of left exact localizations. –  Ricardo Andrade Apr 23 '13 at 11:07
Thanks, Ricardo, that does help. (The terminology is slightly different than I used above, so if anyone is curious it's section 5.6.) –  arsmath Apr 23 '13 at 22:51