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.

${\mathrm{Hom}}_C(U,X)$

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.

$h_X={\mathrm{Hom}}_C(-,X)$

Instead of calling $h_X={\mathrm{Hom}}_C(-,X)$ a functor $h_X:C^{\mathrm{op}}\to \mathrm{Set}$, I’d like to see how far I can get by calling it a contravariant functor $h_X:C\to \mathrm{Set}$ 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 ${\mathrm{Hom}}_C(U,X)$ as illustrated below.

$h_X\alpha :h_{X}U\to h_{X}U\prime$

A morphism $\alpha :U\prime \to U$ in $C$ gets sent to the function in the opposite direction $h_X\alpha :h_{X}U\to h_{X}U\prime$ in $\mathrm{Set}$.

To see this, note that the morphism $\alpha :U\prime \to U$ effectively pulls (“combs”) the strands in ${\mathrm{Hom}}_C(U,X)$ back to strands in ${\mathrm{Hom}}_C(U\prime ,X)$ (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

$$h_f:h_X\to h_Y.$$h_f:h_X\to h_Y.

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

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

$$\begin{array}{ccc}{h}_{X}U& \stackrel{h_{f}U}{\to }& h_{Y}U\\ h_X\alpha \downarrow & & \downarrow h_Y\alpha \\ h_{X}U\prime & \stackrel{h_{f}U\prime }{\to }& h_{Y}U\prime \end{array}$$\array{ h_X U & \stackrel{h_f U}{\rightarrow} & h_Y U \\ \mathllap{h_X\alpha\quad}{\downarrow} & {} & \mathrlap{\downarrow}{\quad h_Y\alpha} \\ h_X U' & \stackrel{h_f U'}{\rightarrow} & h_Y U' }

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

which follows from associativity of morphisms in $C$.