nLab
over-(infinity,1)-topos

Idea

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

$$H/X\stackrel{\stackrel{X_{!}}{\to }}{\stackrel{\stackrel{X^{*}}{\leftarrow }}{\underset{X_{*}}{\to }}}H$$\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 $X_{!}:(Y\to X)\mapsto X$, 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 $(X_{!}⊣X^{*})$ 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-$(\mathrm{\infty },1)$-topos $H/X$ is locally $\mathrm{\infty }$-connected.

Proof

The composite of (∞,1)-geometric morphisms

$$H/X\stackrel{\stackrel{X_{!}}{\to }}{\stackrel{\stackrel{X^{*}}{\leftarrow }}{\underset{X_{*}}{\to }}}H\stackrel{\stackrel{{\Pi }_H}{\to }}{\stackrel{\stackrel{{\mathrm{LConst}}_H}{\leftarrow }}{\underset{{\Gamma }_H}{\to }}}\mathrm{\infty }\mathrm{Grpd}$$\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 $\mathrm{\infty }$-connected.

Proposition

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

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

See etale geometric morphism for more details.

General base change

See base change geometric morphism.

As $(\mathrm{\infty },1)$-sheaves on the big $(\mathrm{\infty },1)$-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 $\mathrm{PSh}(C){/}_X$ for the corresponding over (∞,1)-categories, where – notationally implicitly – we use the (∞,1)-Yoneda embedding $C\to \mathrm{PSh}(C)$.

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

$$\mathrm{PSh}(C/X)\stackrel{\simeq }{\to }\mathrm{PSh}(C)/X.$$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 $c\in \mathrm{PSh}(C)$, hence of $\mathrm{PSh}(C/X)$ 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

$$\mathrm{Sh}(C/X)\simeq \mathrm{Sh}(C)/X.$$Sh(C/X) \simeq Sh(C)/X \,.

Proof

By the discussion of adjoint (∞,1)-functors on over-(∞,1)-categories adjoint (∞,1)-functors we have that the adjunction

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

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

$$(F/X⊣i/X):\mathrm{Sh}(C)/X\stackrel{\stackrel{F}{\leftarrow }}{\hookrightarrow }\mathrm{PSh}(C)/X,$$(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 $\mathrm{Sh}(C)/X$ are computed as limits in $\mathrm{Sh}(C)$ 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 $(F/X⊣i/X)$ is a reflective sub-(∞,1)-category of $\mathrm{PSh}(C/X)$ 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 $\mathrm{PSh}(C)/X\to \mathrm{PSh}(C)$, it follows that this is the big site structure on $C/X$ (this is the defining property of the big site).

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

Proposition

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

$$\mathrm{\infty }\mathrm{Grpd}/X\simeq \mathrm{PSh}(X)\simeq [X,\mathrm{\infty }\mathrm{Grpd}].$$\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-$(\mathrm{\infty },1)$-topos over an $n$-truncated object indeed behaves like a generalized n-groupoid

Proposition

For $n\in \mathbb{N}$ and $𝒳$ an n-localic (∞,1)-topos, then the over-$(\mathrm{\infty },1)$-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 ${\mathrm{Obj}}_{{\rm K}}\times XoX$ 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 $H_{/X}$.

Related concepts

• arrow category

• over-category

  • slice category

  • under category

  • over topos

• over (∞,1)-category,

  • model structure on an over category

  • over-(∞,1)-topos

• arrow (∞,1)-category, arrow (∞,1)-topos

References

The general notion is discussed in section 6.3.5 of

• Jacob Lurie, Higher Topos Theory

Some related remarks are in

• Jacob Lurie, Structured Spaces