Eric Forgy Notes on Grothendieck Topologies, Fibered Categories and Descent Theory

This pages represents notes to myself (anyone is more than welcome to comment) as I attempt to read through Chapter 2 of

• Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory (pdf)

I like to draw pictures to help me understand things, so as I go through the paper, I will try to draw some illustration for the main concepts.

$Hom_C(U,X)$

The set of morphisms in a category $C$ between objects $U$ and $X$ will be depicted by a ball of “strands”, i.e. morphisms stretching between the two objects.

$h_X = Hom_C(-,X)$

Instead of calling $h_X = Hom_C(-,X)$ a functor $h_X:C^{op}\to Set$, I’d like to see how far I can get by calling it a contravariant functor $h_X:C\to Set$ instead. Contravariant functors seem more intuitive to me than opposite categories.

This contravariant functor $h_X$ sends the object $U$ to the set of morphisms $Hom_C(U,X)$ as illustrated below.

$h_X\alpha: h_X U\to h_X U'$

A morphism $\alpha: U'\to U$ in $C$ gets sent to the function in the opposite direction $h_X\alpha: h_X U\to h_X U'$ in $Set$.

To see this, note that the morphism $\alpha: U'\to U$ effectively pulls (“combs”) the strands in $Hom_C(U,X)$ back to strands in $Hom_C(U',X)$ (kind of like a “ponytail”) via

$h_f: h_X\to h_Y$

A morphism $f:X\to Y$ in $C$ induces a natural transformation

To see this, note the morphism $f:X\to Y$ effectively pushes (“combs”) the strands in $Hom_C(U,X)$ forward to strands in $Hom_C(U,Y)$ via

For this to be a natural transformation, we need to have the commuting diagram

but this simply means that it doesn’t matter if we first “comb” the strands back to $U'$ and then comb the strands forward to $Y$, or comb the strands forward to $Y$ first and then comb the strands back to $U'$

which follows from associativity of morphisms in $C$.