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fundamental theorem of topos theory

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Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

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. 6.5.3.1).

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.

References

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

Created on October 12, 2021 at 04:14:57. See the history of this page for a list of all contributions to it.