Category of descent data is a particular presentation of a 2-categorical limit in Cat, i.e. the descent object, intended to solve the descent problem. It can be defined in several setups: in a formulation of descent via fibered categories and Grothendieck topologies, as well as in the (co)monadic descent. In the first case, the descent category appears as a presentation of the descent object, while in the (co)monadic descent the descent category is given by the Eilenberg-Moore category of the (co)monad. In a particular setup, the two approaches give the same result, see Bénabou-Roubaud theorem.
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 $[h_x,D]$.
Angelo Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory (arXiv) (p.74)
Last revised on November 14, 2023 at 15:55:44. See the history of this page for a list of all contributions to it.