An intuitive approach to basic descent theory for me started with open covers $ \left\{ U_i \right\}$, replaced them with a singleton covering $\coprod _iU_i\rightarrow X$, and then generalized to general morphisms $f:Y\rightarrow X$ using the Čech nerve.
However, as was pointed out to me here, the effective descent morphisms of a category (w.r.t the codomain fibration) do not depend on any external information like a choice of Grothendieck topology. The nlab says some things about effective descent morphisms being very special instances of stack conditions, while descent itself supposedly strictly generalizes sheaf conditions. Now, sheaf conditions depend on topology, so I was wondering whether the effective descent morphisms somehow point to the "right" topology to put on a category in some intrinsic sense?
Sorry if this question makes no sense, after thinking about it for a little I would really like to hear some answers.
