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Recently, I have been studying about quasi-coherator and I have some doubts.

1) I know that quasi-coherator of a sheaf on a scheme exists if the scheme is quasi-compact and semi-separated. Could you please justify whether or not it exists for any arbitrary scheme ?

2) What is the quasi-coherator in case we consider sheaves on a ringed site ? At least for the simplest case if we consider the site $X_{etale}$ where $X$ is a scheme, what is the exact construction of the quasi-coherator ? My guess is that in this case it is the same construction as in case of schemes, since here the site under consideration is the one obtained from a scheme. However, what if we are working with an arbitrary ringed site ?

3) Now, suppose we understand the meaning of quasi-coherator on a ringed site, I would like to ask what is the quasi-coherator on a fibred category (say, a gerb) on a site ?

Consider, $$\chi \xrightarrow{p} X_{etale}$$ be a fibred category. Now, $\chi$ has a site structure inherited from $X_{etale}$ and also has a structure sheaf $O_\chi = p^{-1}(O_X)$ inherited from that of $X$. Now, what is the quasi-coherator of a sheaf on this fibred category?

Thanks in advance !

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    $\begingroup$ Regarding question 1: it works in general, see stacks.math.columbia.edu/tag/077P. $\endgroup$
    – pbelmans
    Commented Mar 26, 2016 at 10:45
  • $\begingroup$ @pbelmans. I think that tag in the Stacks Project is discussing the coherator, not the quasi-coherator. The primary reference is Appendix B of Thomason-Trobaugh. $\endgroup$
    – Jason Starr
    Commented Mar 26, 2016 at 12:19
  • $\begingroup$ @pbelmans. That was my mistake. What the OP calls the quasi-coherator seems to be what everybody else calls the coherator. Anyway, Thomason-Trobaugh is the primary source. $\endgroup$
    – Jason Starr
    Commented Mar 26, 2016 at 12:25
  • $\begingroup$ @Sam. To steal a suggestion from Remy van Dobben de Bruyn: why not write up carefully a generalization to sites and then contribute that to the Stacks Project? $\endgroup$
    – Jason Starr
    Commented Mar 26, 2016 at 13:49
  • $\begingroup$ Thanks! I was actually studying quasi-coherator from Thomason-Trobaugh. However, it doesn't give any idea about coherator on sites. I just thought perhaps some of you know a source where the generalization is done. Now I am trying to generalize it myself. But you see unlike schemes there is no concept of affine cover on a site ! And the way Thomason-Trobaugh has constructed the coherator, uses affine cover. So may be I have to find some sort of replacement. $\endgroup$
    – Sam
    Commented Mar 26, 2016 at 17:56

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