The “fundamental theorem” of topos theory, in the terminology of McLarty 1992, asserts that for any topos and any object, also the slice category is a topos: the slice topos.
If is a category of sheaves, hence a Grothendieck topos, then so its its slice: (SGA4.1, p. 295).
The analogous statement holds for slice -categories of -toposes: slice -toposes (Lurie 2009, Prop. 6.3.5.1).
The archetypical special case is that slice categories of categories of presheaves over a representable are equivalently categories of presheaves on the slice site . This is exhibited by the functor which sends a bundle internal to presheaves to its system of sets of local sections:
The sSet-enriched derived functor of this construction yields the analogous statement for -categories of -presheaves, see at slice of presheaves is presheaves on slice for details.
Discussion for Grothendieck toposes:
(In particular, exposé III.5 and exposé IV.5 on the “induced topos” - topos induit = slice topos)
Discussion in the generality of elementary toposes:
See also:
Discussion for slices of Grothendieck -toposes:
The terminology “fundamental theorem of -topos theory” for this is used in
Marco Vergura, Rem. 2.2.10 in: Localization Theory in an Infinity Topos, 2019 [pdf, ir.lib.uwo.ca:etd/6257]
(in a context of localizations motivated by modal homotopy type theory)
Last revised on August 28, 2023 at 08:44:41. See the history of this page for a list of all contributions to it.