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This result says in particular that any Grothendieck topos can equivalently be described as a category of sets with an equality relation taking truth-values in a Grothendieck quantale.
It follows from the arguments above that, if D is a class of etale maps in a Grothendieck topos [epsilon], then D/A is closed under arbitrary colimits in [epsilon]/A, for any A; so we might expect to be able to apply the Adjoint Functor Theorem to conclude that it is coreflective, and thus that D is a calibration in our sense.
If D is a calibration in a Grothendieck topos [epsilon], then the categories D/A, being coreflective in [epsilon]/A, are (co)complete and locally small; but, once again, there seems to be no a priori reason why they should possess generating sets (i.e.
In 2.7(d) below, we shall show by a different method that [D.sub.p] is a calibration when p is any local geometric morphism, provided [epsilon] is a Grothendieck topos.
If D is a calibration in a Grothendieck topos [epsilon] = Sh(C,J), then it follows immediately from (b) of 1.3 and (l) of 1.4 that a morphism f: B [right arrow] A belongs to D iff its pullback along every morphism l (X) [right arrow] A, with X [member of] ob C, does so.
Suppose we are given for each X [member of] ob C a connected geometric morphism [q.sub.X]: E/l(X) [right arrow] [F.sub.X] (where [F.sub.X] is a Grothendieck topos), and for each morphism [alpha]: Y [right arrow] X in C a geometric morphism la making
In [10], Joyal and Moerdijk showed that any class of etale maps in a Grothendieck topos [epsilon] satisfying their 'collection axiom' may be expressed as [f.sub.*] (D) for some geometric morphism f: [2, F] [right arrow] [epsilon], where D is the canonical class of 2.3.