Sheaves on Ringed Sites

The categories of small sets and small abelian groups are respectively denoted by Set and Ab. For example, M is an object of Ab if and only if M is a group whose underlying set is in V, and M is bijective to some set in U. 

In these notes, a site (i.e., a category with a Grothendieck topology, in the sense of [42]) is required to be a small category whose topology is defined by a pretopology (see [42]). 

Note that (ZY, ) is a functor from PA(X) to Ab, which is compatible with arbitrary limits. 

Note that the counit of adjunction ag — Id is given as the unique natural map such that qe qaq q is the inverse of uq. 

Thanks to the re-definition of sites and continuous functors, we have the following. 

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Sheaves on Ringed Sites Proof. Consider the diagram... 

Derived Categories and Derived Functors of Sheaves on Ringed Sites... 

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