Timeline for A site is subcanonical if and only if its sheafification is fully faithful?
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| Dec 11, 2016 at 22:50 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Nov 20, 2016 at 10:59 | comment | added | Arrow | @MarcHoyois sorry, could you explain why that is true? | |
| Nov 11, 2016 at 22:41 | answer | added | Simon Henry | timeline score: 1 | |
| Nov 11, 2016 at 20:38 | comment | added | Marc Hoyois | Fix $c\in C$. Consider the class $S$ of presheaves $P$ such that the canonical map $Hom(P,yc)\to Hom(P,ayc)$ is an iso. You've shown that $S$ contains representable presheaves. It remains to observe that $S$ is closed under colimits. | |
| Nov 11, 2016 at 18:49 | history | migrated | from math.stackexchange.com (revisions) | ||
| Oct 24, 2016 at 19:04 | comment | added | Arrow | @VladimirSotirov I'd given up and started looking for a counterexample, but then I found the added exercise in Borceux vol III. The exercise involves the word "effective", but I am hopeful for a simple direct proof that circumvents the characterization you mention. | |
| Oct 24, 2016 at 18:44 | comment | added | Vladimir Sotirov | This might help. Representable functors are by definition presheaves. The sheaffification of a presheaf, or more precisely the presheaf morphism from the representable functor to its sheafficiation, is its reflection along the inclusion of sheaves into presheaves. Subcanonical means that the reflections of representable functors are isomorphisms. I don't immediately see how being fully faithful could force the reflections to be isos. Also, you may want to think through the effective-epimorphic characterization at ncatlab.org/nlab/show/subcanonical+coverage if you want counterexamples. | |
| Oct 24, 2016 at 17:17 | history | asked | Arrow | CC BY-SA 3.0 |