category of descent data



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 CC be a site, p:DCp:D\to C a fibration, xC 0x\in C_0, U xU_x a cover of xx. h xC/xh_x\hookrightarrow C/x the full subcategory of arrows, that factor through some uxu\rightarrow x of the cover; the corresponding functor C opCatC^{op}\rightarrow \Cat is then a sieve. The category of descent data (sometimes shortcut to descent category) of pp along U xU_x is defined to be the functor category [h x,D][h_x,D].

See also


descent object



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

Last revised on September 29, 2011 at 03:15:11. See the history of this page for a list of all contributions to it.