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.


1 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 $X’ = \bigsqcup U_{ij}$.


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