Sites and Grothendieck Topologies, Sites and Sheaves

We give the minimum of category theory necessary for understanding the definition and applications of Grothendieck topos. We state the basic properties of sites and sheaves, and give applications to the theory of moduli. We prove that for categories $\mathbf{C}$ with explicit stated properties, we can construct a category of schemes of objects in $\mathbf{C}$ which is a proper category rather than a $2$-category as in the case of stacks.