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It is known that the category of schemes is not cocomplete (e.g. see this question: Colimits of schemes). However, do diagrams of schemes for which every morphism is etale have colimits? More generally, I'd really like to know if, given a scheme $X$, if the category of etale schemes over $X$ is cocomplete. I should mention that I know very little of algebraic geometry. I'm interested for "categorical reasons".

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    $\begingroup$ The answer to you first question ("do diagrams of schemes for which every morphism is etale have colimits?") is no: there are examples in which you have two étale maps that have no coequalizer in the category of schemes. You can find one of those in the book "Algebraic spaces" by Knutson (on page 14). The point is that there is a scheme $X$ with an action of $Z/2Z$ that doesn't have a quotient in the category of schemes, so the two maps $X\times Z/2Z \to X$ given by the action and the projection will not have a coequalizer (but they are étale). $\endgroup$
    – Mattia Talpo
    Commented Jun 22, 2010 at 19:43
  • $\begingroup$ Ok, thanks. If you make this into an answer instead of a comment, I'll accept it. $\endgroup$
    – David Carchedi
    Commented Jun 22, 2010 at 20:20
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    $\begingroup$ @Mattia: You seem to be making the same mistake that many made in response to the question linked in the statement of the question: just because the quotient doesn't exist as an algebraic space, it does not follow that there cannot be a categorical co-equalizer in the category of schemes (which may not be a quotient). Being a quotient is a stronger property. (Admittedly this is a picky point, since if such a co-equalizer exists it would be useless since it lacks other good properties, but nonetheless it is not evident, strictly speaking, if your suggested example is really an example. $\endgroup$
    – Boyarsky
    Commented Jun 22, 2010 at 21:56
  • $\begingroup$ @Boyarsky: I think you're right, the fact is that when I read of this counterexample I always thought of "quotient" as of "categorical quotient", and was thus convinced that it was proved that there is no categorical quotient (my bad). $\endgroup$
    – Mattia Talpo
    Commented Jun 22, 2010 at 23:00
  • $\begingroup$ But does there exist a real counterxample? It'd be nice if these colimits existed, but, I'm guessing they don't... $\endgroup$
    – David Carchedi
    Commented Jun 22, 2010 at 23:55

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