nLab fundamental theorem of topos theory

Idea

The “fundamental theorem” of topos theory, in the terminology of McLarty 1992, asserts that for $\mathcal{T}$ any topos and $X \,\in\, \mathcal{T}$ any object, also the slice category $\mathcal{T}_{/X}$ is a topos: the slice topos.

If $\mathcal{T} \,\simeq\, Sh(\mathcal{S})$ is a category of sheaves, hence a Grothendieck topos, then so its its slice: $\mathcal{T}_{/X} \,\simeq\, Sh\big( \mathcal{S}_{/X} \big)$ (SGA4.1, p. 295).

The analogous statement holds for slice $\infty$-categories of $\infty$-toposes: slice $\infty$-toposes (Lurie 2009, Prop. 6.3.5.1).

The archetypical special case is that slice categories $PSh(\mathcal{S})_{/y(s)}$ of categories of presheaves over a representable are equivalently categories of presheaves on the slice site $\mathcal{S}_{/s}$. This is exhibited by the functor which sends a bundle $E \to y(X)$ internal to presheaves to its system $U_X \mapsto \Gamma_U(E)$ of sets of local sections:

$$PSh(\mathcal{S})_{/y(X)} \underoverset {\sim} { \Gamma_{(-)}(-) } {\longrightarrow} PSh \big( \mathcal{S}_{/X} \big)$$

The sSet-enriched derived functor of this construction yields the analogous statement for $\infty$-categories of $\infty$-presheaves, see at slice of presheaves is presheaves on slice for details.

Related concepts

• slice of presheaves is presheaves on slice

• base change

• Giraud's theorem

References

Discussion for Grothendieck toposes:

• Michael Artin, Alexander Grothendieck, Jean-Louis Verdier, Théorie des Topos et Cohomologie Etale des Schémas (SGA4) Tome 1: Théorie des Topos Springer LNM 269 (1972) (doi:10.1007/BFb0081551, pdf)

  (In particular, exposé III.5 and exposé IV.5 on the “induced topos” - topos induit = slice topos)

Discussion in the generality of elementary toposes:

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

See also:

• Wikipedia, Fundamental theorem of topos theory

Discussion for slices of Grothendieck $\infty$-toposes:

• Jacob Lurie, Section 6.3.5 of: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press 2009 (pup:8957, pdf)

The terminology “fundamental theorem of $\infty$-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)