nLab
category of descent data

Idea

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Descent along a cover

This kind of descent is described in the article stack using the bar construction to write a sieve as a colimit of representables.

Let $C$ be a site, $p:D\to C$ a fibration, $x\in C_0$, $U_x$ a cover of $x$. $h_x\hookrightarrow C/x$ the full subcategory of arrows, that factor through some $u\rightarrow x$ of the cover; the corresponding functor $C^{op}\rightarrow \Cat$ is then a sieve. The category of descent data (sometimes shortcut to descent category) of $p$ along $U_x$ is defined to be the functor category $[h_x,D]$.

See also

descent

descent object

stack

References

• Angelo Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory (arXiv) (p.74)