nLab nonabelian cocycle

A nonabelian cocycle is, generally, a cocycle in nonabelian cohomology.

This means usually that it is modeled in either of the following two ways:

• a descent datum, i.e. an object in the category of descent data (sometimes briefly called a descent category);
• an $\omega$-anafunctor.

Notice that for $n\geq 2$, descent data in a setup of strict $n$-categories still form a strict $(n+1)$-category, while $n$-anafunctors form weak $(n+1)$-category.