Let $A$ and $B$ be two cocomplete categories (i.e. closed under small colimits) and $A'$ be a dense subcategory of $A$ i.e. any object in $A$ is a colimit of objects in $A'$. Given a functor $F': A' \to B$, does there always exist an extension of functor $F :A \to B$ preserving all colimits?
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$\begingroup$ I think you mean "cocomplete", rather than "complete"? And $A'$ is a dense subcategory of $A$, rather than $B$? $\endgroup$– Jeremy RickardCommented Jun 1, 2020 at 16:41
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$\begingroup$ Do you want some condition on $F'$? Otherwise you could just take $A'=A$, with $F'$ some functor that doesn't preserve colimits. $\endgroup$– Jeremy RickardCommented Jun 1, 2020 at 16:45
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$\begingroup$ @JeremyRickard Uhhh, sorry for the typos.. right, I meant cocomplete and $A'$ a dense subcategory of $A$. Maybe I need to say $F'$ preserves colimits. $\endgroup$– gregodomCommented Jun 1, 2020 at 17:24
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$\begingroup$ If $A'$ is full in $A$ and skeletally small, then yes: even more is true, there is also an adjunction $[A',Set]\leftrightarrows B$; replace $A'$ with its small skeleton, and use Yoneda lemma. Otherwise, subtle set theory comes in, and I guess you might want to say a bit more on the context of the question :-) $\endgroup$– foscoCommented Jun 1, 2020 at 17:29
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1$\begingroup$ arxiv.org/abs/1501.02503 see 3.1.1 here; not because there are no other reference, just because that's the most convenient source to quote for me :) in case $A'$ is not small, see here: ncatlab.org/nlab/show/small+presheaf $\endgroup$– foscoCommented Jun 2, 2020 at 18:17
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1 Answer
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No. This is true only when $A$ is a free cocompletion of $A'$, i.e., the category of small contravariant functors from $A$ to $Set$.
answered Jun 1, 2020 at 19:39
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$\begingroup$ 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 $A$? $\endgroup$– gregodomCommented Jun 2, 2020 at 17:45
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$\begingroup$ There is no size problem if you take small presheaves, i.e., small colimits of representables. $\endgroup$ Commented Jun 3, 2020 at 8:57
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$\begingroup$ 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! $\endgroup$– gregodomCommented Jun 4, 2020 at 4:46
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$\begingroup$ 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). $\endgroup$ Commented Jun 5, 2020 at 8:43
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$\begingroup$ 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. $\endgroup$– gregodomCommented Jun 5, 2020 at 8:57