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Lawvere-Tierney topologies generalize the notion of local operators on a Topos from Sheaf toposes over a Grothendieck site to arbitrary Toposes.

However, while the special case of Sheaves of sets or Sheaves of spaces in a topos is one of the most common cases, Sheaves of modules over a ring R or more generally sheaves of algebraic objects in an Abelian category are also a very important case.

Now, Abelian categories are in some ways very similar to toposes with a slightly different behaviour of the initial and final objects. Is there a concept similar to the LT topology that generalizes the notion of "locally" from abelian categories of sheaves on a site to general abelian categories without explicitly referring to their construction?

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    $\begingroup$ I don't have time at the moment to give much more than a cursory reply, but a good search term might be "torsion theory". The nLab article ncatlab.org/nlab/show/torsion+theory has some material and references to get you started. $\endgroup$
    – Todd Trimble
    Commented Apr 25, 2022 at 19:53
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    $\begingroup$ More search terms: Gabriel topology on an abelian category, universal closure operator. $\endgroup$
    – Todd Trimble
    Commented Apr 25, 2022 at 20:25
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    $\begingroup$ Related: mathoverflow.net/questions/128446/… $\endgroup$
    – Ivan Di Liberti
    Commented Apr 25, 2022 at 20:40
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    $\begingroup$ @saolof In a sense, no, because there is no subobject classifier nor anything analogous that "makes representable" the notion of locality. But you don't need a representing object to study left exact localisations. $\endgroup$
    – Zhen Lin
    Commented Apr 25, 2022 at 23:09

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