5

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?

saolof
  • 1,823
  • 4
    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 https://ncatlab.org/nlab/show/torsion+theory has some material and references to get you started. – Todd Trimble Apr 25 '22 at 19:53
  • 4
    More search terms: Gabriel topology on an abelian category, universal closure operator. – Todd Trimble Apr 25 '22 at 20:25
  • 1
    Related: https://mathoverflow.net/questions/128446/general-theory-of-left-exact-localization/349948#349948 – Ivan Di Liberti Apr 25 '22 at 20:40
  • 1
    @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. – Zhen Lin Apr 25 '22 at 23:09

0 Answers0