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topos theory

# Contents

## 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 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.

## References

Discussion for Grothendieck toposes:

Discussion in the generality of elementary toposes:

Discussion for slices of Grothendieck $\infty$-toposes:
The terminology “fundamental theorem of $\infty$-topos theory” for this is used in