Classification of sheaves

The concept of a sheaf requires classification, towards that end I have produced a tentative classification of sheafs and related concepts. One term which not be completely familiar is that of a "categorical morphism" or a morphism of categories. I simply introduced that term to create a class that includes both covariant and contravariant functors. Covariant and contravariant functors are similar in almost everyway, so they should belong to a single class.

The basic category-theoretic classification of a sheaf is that it is a special case of contravariant functor. The next concepts we ought to introduce is that of a presheaf to the topoi of sets, which is dual to the concept of a set-valued functor. From the perspective of topoi theory, it is sufficient to consider set-valued functors and presheafs because the functor categories constructed from either of them form elementary topoi.

After the class of presheafs, the classification splits up a bit as presheafs can be defined either on a site or the lattice of open sets of a topology. Site presheafs are more general then topological ones, as you can make a lattice of open sets into a site by defining the family of covers to be all common output morphism systems whose inputs join into the output. Sites generally just give selected categories more of a topological character which allows us to define sheaves over them.

The two different covering conditions (1) locality and (2) gluing complete our classification. At this point, I should briefly comment on concrete categories, which are defined by faithful functors to the category of sets. Although the basic concept of a presheaf is a contravariant functor to the topoi of sets, defining sheafs and presheafs to arbitrary concrete categories is also necessary for any real application, so presheafs are not always classified as set-valued contravariant functors. The underlying presheaf can clearly be derived by composition.

Topoi theory essentially splits up into the study of elementary topoi and of Grothendieck topoi. Elementary topoi are generalizations of the topoi of sets and Grothendeick topoi are functor categories of sheaves on a site. The monomorphisms in the topoi of sheaves, are precisely the natural transformations whose component morphisms in the output category object common to both functors are all monomorphisms. A key realisation is that the input action of these mono natural transformations induces a lattice of subobjects in the category of sheaves.