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created over-topos
I have added to over-topos…
in the section etale geometric morphism statement and proof of the general case induced over any morphism in the topos;
in the section Slice geometric morphism statement and proof of how every geoemtric morphism lifts to one to the slice by any object of the codomain;
in the section Topos points statement and proof of the fact that every slice topos inherits topos points given by points of the original topos and points in the stalk of the object being sliced over.
have added to slice topos a basic remark, right in the Definition section, about the subobject classifier in the slice.
Added the construction of the power object.
I added a remark on pullback-preserving comonads, with some references.
I have added to your remark a pointer to topos of coalgebras over a comonad.
I added a little more detail to the equivalence between a slice of a presheaf topos and the presheaf topos of the category of elements.
Thanks Max! It’s good to see this spelled-out.
It’s mentioned at the end of chapter VII on page 416 of Sheaves in Geometry and Logic by Mac Lane and Moerdijk. Not stated explicitly as a result though.
One finds in Verdier’s exposé III.5 prop.5.4 (SGA4, p.295) the result that $Sh(\mathcal{C}/X)\simeq Sh(\mathcal{C})/a(X)$ where $\mathcal{C}$ is a site and $a$ is the associated sheaf functor. I suppose your formula is the special case that the topology that comes with $\mathcal{C}$ is canonical.
Thanks!!
Thanks!!
Where the Idea-section asserted that slices of toposes being topos has “also been called” the fundamental theorem of topos theory I am making that link to to a stand-alone entry and I have added pointer to Theorem 17.4 in:
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