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:

- Colin McLarty,
*Elementary Categories, Elementary Toposes*, Oxford University Press 1992 (ISBN:9780198514732)

The beginning of the proof here was typeset in a way that made it all but unreadable. I have tried to add some more formatting. (Still room left to improve further…)

]]>I added a description of the category of points of a slice topos.

Jens Hemelaer

]]>Fixed a typo where $f_*$ was used as the left adjoint to $f^*$

Anonymous

]]>Thanks!!

]]>Added the reference to SGA.

]]>briefly added the general statement, with the reference that Thomas gave, here.

There is much room to expand and beautify that section. But I will leave it at that now.

]]>Thanks!!

]]>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.

]]>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.

]]>We should finally give a canonical reference here for the stated fact that

$Sh(\mathcal{C}/X) \simeq Sh(\mathcal{C})/y(X) \,.$Anyone has page and verse for this at hand, either in Johnstone or in Borceux, or elsewhere?

(Currently we keep pointing to Lurie for this statement…)

]]>Thanks Max! It’s good to see this spelled-out.

]]>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.

]]>I have added to your remark a pointer to *topos of coalgebras over a comonad*.

I added a remark on pullback-preserving comonads, with some references.

]]>Added the construction of the power object.

]]>have added to *slice topos* a basic remark, right in the Definition section, about the subobject classifier in the slice.

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.

created over-topos

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