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    $\begingroup$ Isn't this question equivalent to asking whether the category of local rings and local ring homomorphisms is coherent, as in ncatlab.org/nlab/show/coherent+category? $\endgroup$ Commented Feb 28, 2012 at 21:57
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    $\begingroup$ Is it? I can't say I know enough about categorical algebra to see why this is plausible/implausible. $\endgroup$ Commented Feb 29, 2012 at 0:17
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    $\begingroup$ @Andrej: No, that doesn't seem to be the case. The theory of abelian groups is algebraic, hence coherent, but the category of abelian groups is not coherent (since coherent categories have a strict initial object, while $\textbf{Ab}$ has a zero object). $\endgroup$ Commented Mar 2, 2012 at 7:46
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    $\begingroup$ You may want to read about the concept of a "geometry" introduced in Lurie's DAG V. It was invented precisely to get around this problem. $\endgroup$ Commented May 18, 2013 at 1:16
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    $\begingroup$ David is right, see here ncatlab.org/nlab/show/… (and notice that the condition imposed there is reaLLy simple and has as such nothing much to do with the oo-category theory in which it is formulated). $\endgroup$ Commented Jun 11, 2013 at 22:59