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I've heard it claimed that the adjective "coherent" in logic/topos theory (i.e. coherent logic, coherent toposes, coherent categories) was adopted to fit in with the terminology of coherent sheaves in algebraic geometry.

What is this connection? Is the category of coherent sheaves on a scheme a coherent topos?

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    $\begingroup$ That claim is wrong and there is no connection (the category of coherent sheaves is not a topos). The correct analogue in algebraic geometry is that "coherent topos" is akin to the condition on a scheme that it be quasi-compact and quasi-separated, so this is just a rare case of poor terminology choice by Grothendieck (much as his use of the phrase "maximal point" to mean "generic point" is an unfortunate choice due to the fact that in the affine case maximal ideals correspond to closed points, sort of the opposite extreme from generic points). $\endgroup$ – user74230 Mar 25 '15 at 4:58
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    $\begingroup$ The definition of coherent topos is built on the definition of coherent object, which is essentially the same as the definition of coherent sheaf. $\endgroup$ – Zhen Lin Mar 25 '15 at 8:28
  • $\begingroup$ @ZhenLin Would you consider adding your comment as an answer? I think it reflects the true situation far more accurately than the comment before yours. $\endgroup$ – Todd Trimble Mar 25 '15 at 13:54
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    $\begingroup$ @ToddTrimble: Every qcqs scheme gives rise to a coherent topos, but generally on such schemes the structure sheaf is not coherent and so one cannot readily produce any interesting examples of coherent sheaves of modules. Also, the structure sheaf on a complex manifold is coherent (in Serre's sense) by Oda's theorem but is not coherent in the sense of coherent objects of the category of sheaves of modules on the complex manifold. Although the definition of coherent object of an abelian category is modeled on the notion of coherent sheaf, there are subtle but important differences. $\endgroup$ – user74230 Mar 25 '15 at 14:29
  • $\begingroup$ @user74230 I think you misunderstood my comment. The claim of the first paragraph of the OP has essentially to do with etymology, and it was the category theorists / categorical logicians who adapted the terminology, to the best of my knowledge. You are of course absolutely correct about the last question of the OP, but I think he really wants to know what is the terminological connection, and this is what Zhen Lin is addressing. $\endgroup$ – Todd Trimble Mar 25 '15 at 14:41

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