I have often come across the idea of placing different topologies on the category of schemes $Sch$. This can be useful in proving that certain functors are representable. However, whenever I read about these constructions I become lost as to when/how exactly the topology is actually placed on $Sch$.
I imagine that if we place a (Grothendieck) topology on a category the object of the category should act as open sets of the topological space. However, wikipedia write that: "A Grothendieck topology $J$ on a category $C$ is a collection, for each object $c$ of $C$, of distinguished sieves on $c$, denoted by $J(c)$ and called covering sieves of $c$. " So, what exactly are the open subsets?
Obviously I am confused as I don't understand the big picture underlying these constructions and thus I am getting caught up in the sea of definitions.
I am hoping somebody can give a big picture description of how one goes about placing a topologies on $Sch$.
A site, that is, a small category with a Grothendieck topology, abstracts part of the theory of topological spaces: specifically the fact that every topological space gives rise to a theory of sheaves. But the theory of sheaves only requires that in the category of open sets of a topological space, we know what it means for some collection of morphisms to be "covering". There is no reliance of sheaf theory on the concrete realization of objects of that category as subsets of a fixed set.
To make concrete how a topological space $X$ gives rise to a site: the poset of open sets of $X$ admits a Grothendieck topology in which a family of inclusions $U_i\to U$ is covering if and only if $\cup U_i=U$. But in topologies on categories of schemes, the maps of an open covering are rarely subset inclusions (to the extent that even makes sense for schemes.) Rather, they might be etale maps, or something more exotic. It doesn't matter what the maps are, only that we can define a notion of "covering" for a family of maps which obeys the appropriate axioms. Then we can define a sheaf as a contravariant functor which sends covering families to limit diagrams.