nLab category of descent data



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.

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


Last revised on November 14, 2023 at 15:55:44. See the history of this page for a list of all contributions to it.