Holmstrom Sheaf theory

I think it is true in general that sheafification is left adjoint to inclusion, and commutes with all limits and colimits.

To a sheaf of sets, one can associate a sheaf of abelian groups: Compose with the free abelian group functor, and then sheafify (Voevodsky thesis, p 2). This is the left adjoint to the forgetful functor from abelian sheaves to sheaves of sets. This is an example of a functor from sheaves of sets to sheaves of something else, which is needed if one wants to embed a category into a category behaving like “something else”, in this case an abelian category. For basic properties of these abelian sheaves, see p 2 in the thesis (right exactness, preserves monos, “no torsion”, flatness, respects prod and coprod). More basic properties of these sheaves, in section 2.

http://mathoverflow.net/questions/4474/assumptions-on-the-category-c-for-sheafification-of-c-valued-presheaves

http://mathoverflow.net/questions/80013/given-a-small-category-with-some-colimits-can-the-rest-of-the-colimits-be-added

http://ncatlab.org/nlab/show/sheafification

http://mathoverflow.net/questions/51467/naturality-of-the-associated-sheaf

http://www.ncatlab.org/nlab/show/local+isomorphism

http://ncatlab.org/nlab/show/abelian+sheaf

http://ncatlab.org/nlab/show/quasicoherent+sheaf

http://ncatlab.org/nlab/show/IPC-property

http://mathoverflow.net/questions/38454/what-information-does-the-completion-of-a-stalk-at-its-maximal-ideal-give-us-in-t

http://www.ncatlab.org/nlab/show/restriction+and+extension+of+sheaves

http://ncatlab.org/nlab/show/category+of+sheaves

http://www.ncatlab.org/nlab/show/local+epimorphism