The paper Facets of Descent I by Janelidze and Tholen defines $\mathbb E$-descent maps as those for which $\Phi^p:\mathbb EB\longrightarrow \mathsf{Des}_\mathbb{E}(p)$ is an equivalence of categories.

I like the conceptual definition of descent data as a homotopy limit, as described in this answer, but am not sure how to translate $\mathbb E$-descent to this context. Are the sites on the category of spaces changing, or is the fibration changing?

So if $\text{Č}$ is the Čech nerve of an arrow $f:X\rightarrow Y$, and $F$ is a fibration, say $f$ is an effective descent morphism if $FY=\operatorname{holim}F(\text{Č})$.

My question is whether or not its true that $f$ is an $\mathbb E$-descent morphism iff we restrict to the subcategory of arrows in $\mathbb E$, or how to describe $\mathbb E$-descent in these terms.

What's the relationship between $\mathbb E$-descent for $\mathbb E$ the class of étale maps and descent for the étale topological site?

  • $\begingroup$ As it is pointed out in the answer you refer to, when $F$ is fibered in categories you can truncate the Cech nerve above the degree 2 to compute the homotopy limit. This gives the classical descent category. considered in JT. $\endgroup$ May 12, 2016 at 12:44
  • $\begingroup$ @DimitriChikhladze I'm not sure this answers my question. I'm asking whether the $\mathbb E$ in $\mathbb E$-descent refers to taking the Čech nerve over the subcategory of maps in $\mathbb E$, or something else? $\endgroup$
    – popo
    May 12, 2016 at 15:06
  • $\begingroup$ Eventually, $\mathbb{E}$ refers to the fibered category with respect to which descent is considered. In this case it is the fibration whose fibers are the subcategories of the slice categories $\mathcal{C}/X$ of maps in $\mathbb{E}$. $\endgroup$ May 12, 2016 at 15:28
  • $\begingroup$ To obtain a fibration from the class $\mathbb{E}$ in this way one should assume that $\mathbb{E}$ is pullback stable. Although for a fixed $p$ one can only assume pullback stability along $p$, then one does not have a fibration $\mathbb{E}$, but one can still define $\mathsf{Des}_\mathbb{E}(p)$. This what is done in the paper as far as I can see. $\endgroup$ May 12, 2016 at 15:32


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.