Sheafification of G
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My university doesn't let me teach anymore, so the rest of the world is my victim now.
Let g be the functor mapping source to *.o objects. If we sheafify g, we get a functor g++ that now satisfies gluing: a system of compatible objects can be uniquely linked!
In case you can't pronounce "sheafification", I also go by (g+)+ or GSheaf, or Faisceautisation de G. Whichever is easier.
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