Timeline for extending functor from a dense subcategory
Current License: CC BY-SA 4.0
6 events
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| Jun 5, 2020 at 8:57 | comment | added | gregodom | Thanks for the answer! I don't really know the precise definition but I guess small sheaves are by definition the sheafification of small presheaves. In some cases these small sheaves seems to be all sheaves. For example, if we consider small etale site over a scheme (which is an essentially large category), all etale sheaves here would be small because the category of sheaves on the small etale site coincides with the category of sheaves on small affine etale site (which turns out to be essentially small). But this does not seem to work for the big sites e.g. big fppf site. | |
| Jun 5, 2020 at 8:43 | comment | added | Jiří Rosický | I would expect that sheafifications of small presheaves are small sheaves. All what I know about this subject is contained in my two joint papers with Boris Chorny (arXiv:1110.0605 and arXiv:1110.4252). | |
| Jun 4, 2020 at 4:46 | comment | added | gregodom | Sorry for being vague. I mean I eventually want to deal with arbitrary SHEAF. My thought was that all sheaves come from presheaves (via sheafification) and hence it'd be nice if we can talk about arbitrary presheaf, small or not. Now my problem is (sorry for asking another question here), is it true that any fppf sheaf is the sheafification of some small presheaf ? If true, that will save my day. Thanks very much! | |
| Jun 3, 2020 at 8:57 | comment | added | Jiří Rosický | There is no size problem if you take small presheaves, i.e., small colimits of representables. | |
| Jun 2, 2020 at 17:45 | comment | added | gregodom | I think I have problem with the condition of the functors being small here. If A′ is small, such condition should be empty. However, in practice (in algebraic geometry), say if we consider A′ be the fppf site over a fixed base scheme S which I think is not really small, there is no category of presheaves because of the size issue. But does the category of sheaves make sense here? if so call it $A$, can we say anything about extending a functor from $A'$ to $A$? | |
| Jun 1, 2020 at 19:39 | history | answered | Jiří Rosický | CC BY-SA 4.0 |