Why is this called the cocycle condition?

Question

My question is about why the condition below is called the cocycle condition. Surely it is named after some interpretation of it in cohomology.

Let $C$ be a site, and let $F$ be a fibered category over $C$; for $U \in \text{Obj}(C)$ in we write $F(U)$ for the category whose objects are objects in $F$ sent to $U$ in $C$, and whose morphisms are morphisms in $F$ sent to $1_U$ in $C$. Fix an open cover $\sigma : U_i \rightarrow U$ of an object $U$ in $C$, pullbacks $U_{ij} = U_i \times_U U_j$, and pullbacks $U_{ijk} = U_i \times_U U_j \times_U U_k$. Write $\pi_{ab} : U_i \times_U U_j \times_U U_k \rightarrow U_{a} \times_U U_b$ for the projection maps, for each $\{ a, b \} \subset \{ i, k, j \}$, and $\pi_a : U_i \times_U U_j \rightarrow U_a$ for the projection maps, for each $\{ a \} \subset \{i, j \}$.

For a map $\pi : U \rightarrow V$ in $C$, we have a pullback functor $\pi^* : F(V) \rightarrow F(U)$.

Definition: An object with descent data $( \{ V_i \}, \{ \phi_{ij} \} )$ is a collection of objects $\{ V_i \}$, with $V_i \in \text{Obj}(F(U_i))$, together with isomorphisms $\phi_{ij} : \pi_{j}^* V_j \rightarrow \pi_i^* V_i$, such that $\pi_{ik}^* \phi_{ik} = \pi_{ij}^* \phi_{ij} \circ \pi_{jk}^* \phi_{jk}$.

The last condition is called the cocycle condition. I am wondering if it has something to do with cohomology.

Answer 1

It is the same idea as the cocycle equation for gluing together a scheme from affine open subschemes. For instance if you take the big Zariski site, then an object with descent data is just a scheme.

The analogy with cocycles is that $u_{ik} = u_{ij} + u_{jk}$ is the cocycle equation for first Cech cohomology of a sheaf of abelian groups, so if the group operation were instead composition of functions then this will look identical to the gluing equation. A special case is the constant sheaf, which for many spaces computes the singular cohomology.

If you want to stretch the analogy as far as possible, I think you could say that the $\varphi_{ij}$ defines a nonabelian Cech 1-cocycle for a certain sheaf of automorphisms of $X’ = \bigsqcup U_{ij}$.