Vladimir Voevodsky writes:
> The slice category over P is equivalent to presheaves on the category of elements of P. What is your definition of the category of elements of $P$? Objects are pairs $(X:C,p:P(X))$, and morphisms from $(X,p)$ to $(X’,p’)$ are…?
...morphisms $f : C(X',X)$ such that $Pf(p) = p'$.
The slice category of the category of functors $A -> Set$ over a functor $F$ is equivalent to functors $el(F) -> Set$, right? Now if you define presheaves as functors $C^{op} -> Set$, you get that presheaves on $C$ over a presheaf $P$ are equivalent to *functors* $el(P) -> Set$, not presheaves on $el(P)$.
That's why I was trying to be clever and use that unnatural definition of presheaves, but, as you pointed out, that doesn't work either, because the resulting category of presheaves is the opposite of what we want.
Paolo
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