The category of descent data

Question

This is from Angelo Vistoli’s notes http://homepage.sns.it/vistoli/descent.pdf page $71$.

Let $\mathcal{C}$ be a site and $\mathcal{U}=\{\sigma_i:U_i\rightarrow U\}$ be a covering of $U$. Let $\mathcal{F}$ be a fibered category over $\mathcal{C}$.

An object with descent data $(\{\xi_i\},\{\phi_{ij}\})$ on $\mathcal{U}$ is

1. a collection of objects $\xi_i\in \mathcal{F}(U_i)$
2. a collection of isomorphisms $\phi_{ij}:pr_2^*\xi_j\rightarrow pr_1^*\xi_i$ in $\mathcal{F}(U_i\times_U U_j)$

such that the following cocycle condition is satisfied. $$pr_{13}^*\phi_{ik}=pr_{12}^*\phi_{ij}\circ pr_{23}^*\phi_{jk}:pr_3^*\xi_k\rightarrow pr_1^*\xi_i$$

I do not understand the notation in this cocylce relation. Please explain what it means.

I am adding photo of diagrams relavent to the set up.

Answer 1

So first we have the projections :

• $pr_{12}:U_i\times U_j\times U_k\to U_i\times U_j$
• $pr_{13}:U_i\times U_j\times U_k\to U_i\times U_k$
• $pr_{23}:U_i\times U_j\times U_k\to U_j\times U_k$

These define functors $pr_{12}^*:\mathcal{F}(U_i\times U_j)\to\mathcal{F}(U_i\times U_j\times U_k)$ and similarly for $pr_{13}$ and $pr_{23}$.

Now $\phi_{ij}$ are morphisms between two objects $A$ and $B$ in $\mathcal{F}(U_i\times U_j)$. So we can apply the functor $pr_{12}$ to get a new morphism $pr_{12}^*\phi_{ij}:pr_{12}^*A\to pr_{12}^*B$. This will be a morphism in $\mathcal{F}(U_i\times U_j\times U_k)$.

You get similarly three morphisms $pr_{12}^*\phi_{ij}, pr_{13}^*\phi_{ik}$ and $pr_{23}^*\phi_{jk}$. The three of them being in the same category $\mathcal{F}(U_i\times U_j\times U_k)$.

What are their source and target ? $\phi_{ij}$ is a morphism $pr_2^*\xi_j\to pr_1^*\xi_i$, so pulling back you get $pr_{12}^*\phi_{ij}=pr_{12}^*pr_2^*\xi_j\to pr_{12}^*pr_1^*\xi_i$. But $pr_2pr_{12}:U_i\times U_j\times U_k\to U_k$ is the projection onto the second factor, and $pr_{12}^*pr_2^*=(pr_2pr_{12})^*$. So the source of $pr_{12}^*\phi_{ij}$ is $\xi_j$ pulled-back to $U_i\times U_j\times U_k$ through the second projection. So call it $pr_2^*\xi_j$, though here $pr_2$ means the second projection from the triple product $U_i\times U_j\times U_k$ onto the second factor.

Similarly you will have three morphisms :

• $pr_{12}^*\phi_{ij}:pr_2^*\xi_j\rightarrow pr_1^*\xi_i$
• $pr_{13}^*\phi_{ik}:pr_3^*\xi_k\rightarrow pr_1^*\xi_i$
• $pr_{23}^*\phi_{jk}:pr_3^*\xi_k\rightarrow pr_2^*\xi_j$

Note that $pr_{12}^*\phi_{ij}$ and $pr_{23}^*\phi_{jk}$ can be composed to get a new morphism $$pr_{12}^*\phi_{ij}\circ pr_{23}^*\phi_{jk}:pr_3^*\xi_k\rightarrow pr_2^*\xi_j\rightarrow pr_1^*\xi_i$$

The cocycle condition is that this composition is equal to $pr_{13}^*\phi_{ik}$.

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A quick note to get your hand on this : I recommend to write all of this using the topological notion of restriction. Instead of $pr_{12}:U_i\times U_j\times U_k\to U_i\times U_j$, think about the inclusion of $U_i\cap U_j\cap U_k\subset U_i\cap U_j$. So instead of writing $pr_{12}^*\xi$, write $\xi|_{U_{ijk}}$.

All of this can be written as follow :

• take objects $\xi_i\in\mathcal{F}_{U_i}$
• isomorphisms $\phi_{ij}:\xi_i|_{U_i\cap U_j}\to \xi_j|_{U_i\cap U_j}$
• such that on the triple intersection $U_{ijk}=U_i\cap U_j\cap U_k$ : $\phi_{ik}|_{U_{ijk}}=\phi_{ij}|_{U_{ijk}}\circ \phi_{jk}|_{U_{ijk}}$ as morphisms $\xi_k|_{U_{ijk}}\to\xi_i|_{U_{ijk}}$.