nLab fundamental theorem of topos theory



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

If 𝒯Sh(𝒮)\mathcal{T} \,\simeq\, Sh(\mathcal{S}) is a category of sheaves, hence a Grothendieck topos, then so its its slice: 𝒯 /XSh(𝒮 /X)\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.

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

PSh(𝒮) /y(X)Γ ()()PSh(𝒮 /X) 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.


Discussion for Grothendieck toposes:

Discussion in the generality of elementary toposes:

See also:

Discussion for slices of Grothendieck \infty -toposes:

The terminology “fundamental theorem of \infty-topos theory” for this is used in

Last revised on August 28, 2023 at 08:44:41. See the history of this page for a list of all contributions to it.