# nLab over-(infinity,1)-topos

### Context

#### $\left(\infty ,1\right)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

bundles

# Contents

## Idea

For $H$ an (∞,1)-topos and $X\in H$ any object, the over-(∞,1)-category $H/X$ is itself an $\left(\infty ,1\right)$-topos: the over-$\left(\infty ,1\right)$-topos .

If we think of $H$ as a big topos, then for $X\in H$ we may think of $H/X\in$ (∞,1)-topos as the little topos-incarnation of $X$. The objects of $H/X$ we may think of as (∞,1)-sheaves on $X$.

This correspondence between objects of $X$ and their little-topos incarnation is entirly natural: $H$ is equivalently recovered as the (∞,1)-category whose objects are over-$\left(\infty ,1\right)$-toposes $H/X$ and whose morphisms are (∞,1)-geometric morphisms over $H$.

## Definition

###### Proposition

For $H$ an (∞,1)-topos and $X\in H$ an object also the over-(∞,1)-category $H/X$ is an $\left(\infty ,1\right)$-topos. This is the over-$\left(\infty ,1\right)$-topos of $H$ over $X$.

This is HTT, prop 6.3.5.1 1).

## Properties

### Base change to the point

###### Proposition

There is a canonical (∞,1)-geometric morphism

$H/X\stackrel{\stackrel{{X}_{!}}{\to }}{\stackrel{\stackrel{{X}^{*}}{←}}{\underset{{X}_{*}}{\to }}}H$\mathbf{H}/X \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}}} \mathbf{H}

where the extra left adjoint ${X}_{!}$ is the obvious projection ${X}_{!}:\left(Y\to X\right)↦X$, and ${X}_{*}$ is given by forming the (∞,1)-product with $X$.

This is called an etale geometric morphism. See there for more details.

###### Proof

The fact that $\left({X}_{!}⊣{X}^{*}\right)$ follows from the universal property of the products. The fact that ${X}^{*}$ preserves (∞,1)-colimits and hence has a further right adjoint ${X}_{*}$ by the adjoint (∞,1)-functor theorem follows from that fact that $H$ has universal colimits.

###### Corollary

If $H$ is a locally ∞-connected (∞,1)-topos then for all $X\in H$ also the over-$\left(\infty ,1\right)$-topos $H/X$ is locally $\infty$-connected.

###### Proof

The composite of (∞,1)-geometric morphisms

$H/X\stackrel{\stackrel{{X}_{!}}{\to }}{\stackrel{\stackrel{{X}^{*}}{←}}{\underset{{X}_{*}}{\to }}}H\stackrel{\stackrel{{\Pi }_{H}}{\to }}{\stackrel{\stackrel{{\mathrm{LConst}}_{H}}{←}}{\underset{{\Gamma }_{H}}{\to }}}\infty \mathrm{Grpd}$\mathbf{H}/X \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}}} \mathbf{H} \stackrel{\overset{\Pi_{\mathbf{H}}}{\to}}{\stackrel{\overset{LConst_{\mathbf{H}}}{\leftarrow}}{\underset{\Gamma_{\mathbf{H}}}{\to}}} \infty Grpd

is itself a geometric morphism. Since ∞Grpd is the terminal object in (∞,1)Topos this must be the global section geometric morphism for $H/X$. Since it has the extra left adjoint $\Pi \circ {X}_{!}$ it is locally $\infty$-connected.

###### Proposition

Let $\left(\left(\infty ,1\right)\mathrm{Topos}/H{\right)}_{\mathrm{et}}\subset \left(\infty ,1\right)\mathrm{Topos}/H$ be the full sub-(∞,1)-category on the etale geometric morphisms $H/X\to H$. Then there is an equivalence

$\left(\left(\infty ,1\right)\mathrm{Topos}/H{\right)}_{\mathrm{et}}\simeq H$((\infty,1)Topos/\mathbf{H})_{et} \simeq \mathbf{H}

See etale geometric morphism for more details.

### As $\left(\infty ,1\right)$-sheaves on the big $\left(\infty ,1\right)$-site of an object

We spell out how $H/X$ is the (∞,1)-category of (∞,1)-sheaves over the big site of $X$.

###### Proposition

(forming overcategories commutes with passing to presheaves)

Let $C$ be a small (∞,1)-category and $X:K\to C$ a diagram. Write ${C}_{/X}$ and $\mathrm{PSh}\left(C\right){/}_{X}$ for the corresponding over (∞,1)-categories, where – notationally implicitly – we use the (∞,1)-Yoneda embedding $C\to \mathrm{PSh}\left(C\right)$.

Then we have an equivalence of (∞,1)-categories

$\mathrm{PSh}\left(C/X\right)\stackrel{\simeq }{\to }\mathrm{PSh}\left(C\right)/X\phantom{\rule{thinmathspace}{0ex}}.$PSh(C/X) \stackrel{\simeq}{\to} PSh(C)/X \,.

This appears as HTT, 5.1.6.12. For more on this see (∞,1)-category of (∞,1)-presheaves.

###### Remark

Here we may think of $C/X$ as the big site of the object $c\in \mathrm{PSh}\left(C\right)$, hence of $\mathrm{PSh}\left(C/X\right)$ as presheaves on $X$.

###### Proposition

Let $C$ be equipped with a subcanonical coverage, let $X\in C$ and regard $C/X$ as an (∞,1)-site with the big site-coverage. Then we have

$\mathrm{Sh}\left(C/X\right)\simeq \mathrm{Sh}\left(C\right)/X\phantom{\rule{thinmathspace}{0ex}}.$Sh(C/X) \simeq Sh(C)/X \,.
###### Proof

$\left(F⊣i\right):\mathrm{Sh}\left(C\right)\stackrel{\stackrel{F}{←}}{↪}\mathrm{PSh}\left(C\right)$(F \dashv i) : Sh(C) \stackrel{\overset{F}{\leftarrow}}{\hookrightarrow} PSh(C)

passes to an adjunction on the over-(∞,1)-categories

$\left(F/X⊣i/X\right):\mathrm{Sh}\left(C\right)/X\stackrel{\stackrel{F}{←}}{↪}\mathrm{PSh}\left(C\right)/X\phantom{\rule{thinmathspace}{0ex}},$(F/X \dashv i/X) : Sh(C)/X \stackrel{\overset{F}{\leftarrow}}{\hookrightarrow} PSh(C)/X \,,

(where we are using that $FiX\simeq X$ by the assumption that the coverage is subcanonical, so that the representable $X$ is a (∞,1)-sheaf), such that $i/X$ is still a full and faithful (∞,1)-functor (where we are using that the unit $X\to FiX$ is an equivalence, since $X$ is a sheaf).

Since moreover the (∞,1)-limits in $\mathrm{Sh}\left(C\right)/X$ are computed as limits in $\mathrm{Sh}\left(C\right)$ over diagrams with a bottom element adjoined (as discussed at limits in over-(∞,1)-categories) it follows that with $F$ preserving all finite limits, so does $F/X$.

In summary we have that $\left(F/X⊣i/X\right)$ is a reflective sub-(∞,1)-category of $\mathrm{PSh}\left(C/X\right)$ hence is the (∞,1)-category of (∞,1)-sheaves on the category $C/X$ for some (∞,1)-site-structure. But since $F/X$ inverts precisely those morphisms that are inverted by $F$ under the projection $\mathrm{PSh}\left(C\right)/X\to \mathrm{PSh}\left(C\right)$, it follows that this is the big site structure on $C/X$ (this is the defining property of the big site).

Specifically for the $\left(\infty ,1\right)$-topos $H=$ ∞Grpd we also have the following characterization.

###### Proposition

For $H=$ ∞Grpd we have that for $X\in \infty \mathrm{Grp}$ any ∞-groupoid the corresponding over-$\left(\infty ,1\right)$-topos is equivalent to the (∞,1)-category of (∞,1)-presheaves on $X$:

$\infty \mathrm{Grpd}/X\simeq \mathrm{PSh}\left(X\right)\simeq \left[X,\infty \mathrm{Grpd}\right]\phantom{\rule{thinmathspace}{0ex}}.$\infty Grpd/X \simeq PSh(X) \simeq [X, \infty Grpd] \,.
###### Proof

This is a special case of the (∞,1)-Grothendieck construction. See the section (∞,0)-fibrations over ∞-groupoids.

The following proposotion asserts that the over-$\left(\infty ,1\right)$-topos over an $n$-truncated object indeed behaves like a generalized n-groupoid

###### Proposition

For $n\in ℕ$ and $𝒳$ an n-localic (∞,1)-topos, then the over-$\left(\infty ,1\right)$-topos $𝒳/U$ is $n$-localic precisely if the object $U$ is $n$-truncated.

This is (StrSp, lemma 2.3.16).

### Object classifier

If ${\mathrm{Obj}}_{\kappa }\in H$ is an object classifier for $\kappa$-small objects, then the projection ${\mathrm{Obj}}_{Κ}×XoX$ regarded as an object in the slice is a $\kappa$-small object classifier in ${H}_{/X}$.

### In homotopy type theory

If a homotopy type theory is the internal language of $H$, then then theory in context $x:X⊢\cdots$ is the internal language of ${H}_{/X}$.

## References

The general notion is discussed in section 6.3.5 of

Some related remarks are in

Revised on January 18, 2013 03:56:08 by Urs Schreiber (203.116.137.162)