# nLab shape of an (infinity,1)-topos

### Context

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

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Idea

If an (∞,1)-topos $H$ is that of (∞,1)-sheaves on (the site of open subsets of) a paracompact topological space$H={\mathrm{Sh}}_{\left(\infty ,1\right)}\left(X\right)$ – then its shape is the strong shape of $X$ in the sense of shape theory: a pro-object $\mathrm{Shape}\left(X\right)$ in the category of CW-complexes.

It turns out that $\mathrm{Shape}\left(X\right)$ may be extracted in a canonical fashion from just the (∞,1)-topos ${\mathrm{Sh}}_{\left(\infty ,1\right)}\left(X\right)$, and in a way that makes sense for any (∞,1)-topos. This then gives a definition of shape of general $\left(\infty ,1\right)$-toposes.

## Definition

###### Definition

The composite (∞,1)-functor

$\Pi :\left(\infty ,1\right)\mathrm{Topos}\stackrel{Y}{\to }\mathrm{Func}\left(\left(\infty ,1\right)\mathrm{Topos},\infty \mathrm{Grpd}{\right)}^{\mathrm{op}}\stackrel{\mathrm{Lex}\left(\mathrm{PSh}\left(-\right),\infty \mathrm{Grpd}\right)}{\to }\mathrm{AccLex}\left(\infty \mathrm{Grpd},\infty \mathrm{Grpd}{\right)}^{\mathrm{op}}\simeq \mathrm{Pro}\infty \mathrm{Grpd}$\Pi : (\infty,1)Topos \stackrel{Y}{\to} Func((\infty,1)Topos, \infty Grpd)^{op} \stackrel{Lex(PSh(-), \infty Grpd)}{\to} AccLex(\infty Grpd, \infty Grpd)^{op} \simeq Pro \infty Grpd

is the shape functor . Its value

$\Pi \left(H\right)=\left(\infty ,1\right)\mathrm{Topos}\left(H,\mathrm{PSh}\left(-\right)\right)$\Pi(\mathbf{H}) = (\infty,1)Topos(\mathbf{H}, PSh(-))

on an $\left(\infty ,1\right)$-topos $H$ is the shape of $H$.

Here

That this does indeed land in accessible left exact functors is shown below.

## Properties

Notice that for every (∞,1)-topos $H$ there is a unique geometric morphism

$\left(\mathrm{LConst}⊣\Gamma \right):H\stackrel{\stackrel{\mathrm{LConst}}{←}}{\underset{\Gamma }{\to }}\infty \mathrm{Grpd}$(LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd

where ∞Grpd is the $\left(\infty ,1\right)$-topos of ∞-groupoids, $\Gamma$ is the global sections (∞,1)-functor and $\mathrm{LConst}$ is the constant ∞-stack functor.

###### Proposition

The shape of $H$ is the composite functor

$\Pi \left(H\right):=\Gamma \circ \mathrm{LConst}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\infty \mathrm{Grpd}\stackrel{\mathrm{LConst}}{\to }H\stackrel{\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\Gamma \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}}{\to }\infty \mathrm{Grpd}$\Pi(\mathbf{H}) := \Gamma \circ LConst \;\;:\;\; \infty Grpd \stackrel{LConst}{\to} \mathbf{H} \stackrel{\;\;\Gamma\;\;}{\to} \infty Grpd

regarded as an object

$\Pi \left(H\right)\in \mathrm{Pro}\left(\infty \mathrm{Grpd}\right)=\mathrm{Lex}\left(\infty \mathrm{Grpd},\infty \mathrm{Grpd}{\right)}^{\mathrm{op}}\phantom{\rule{thinmathspace}{0ex}}.$\Pi(\mathbf{H}) \in Pro(\infty Grpd) = Lex(\infty Grpd, \infty Grpd)^{op} \,.
###### Proof

For $X\in$ ∞Grpd we have by the (∞,1)-Grothendieck construction-theorem and using that up to equivalence every morphism of $\infty$-groupoids is a Cartesian fibration (see there) that

$\mathrm{Func}\left(X,\infty \mathrm{Grpd}\right)\simeq \infty \mathrm{Grpd}/X$Func(X,\infty Grpd) \simeq \infty Grpd/X

is the over-(∞,1)-category. Moreover, by the theorem about limits in ∞Grpd we have that the terminal geometric morphism $\mathrm{Hom}\left(*,-\right):\left[X,\infty \mathrm{Grpd}\right]\to \infty \mathrm{Grpd}$ is the canonical projection $\infty \mathrm{Grpd}/X\to \infty \mathrm{Grpd}$. This means that it is an etale geometric morphism. So for any geometric morphism $f:H\to \left[X,\infty \mathrm{Grpd}\right]$ we have a system of adjoint (∞,1)-functors

$\left(\mathrm{LConst}⊣\Gamma \right):H\stackrel{\stackrel{{f}^{*}}{←}}{\underset{{f}_{*}}{\to }}\infty \mathrm{Grpd}/X\stackrel{\stackrel{{\pi }^{*}}{←}}{\underset{{\pi }_{*}}{\to }}\infty \mathrm{Grpd}\phantom{\rule{thinmathspace}{0ex}}.$(LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \infty Grpd/X \stackrel{\overset{\pi^*}{\leftarrow}}{\underset{\pi_*}{\to}} \infty Grpd \,.

whose composite is the global section geometric morphism as indicated, because that is terminal.

Notice that in $\infty \mathrm{Grpd}/X$ there is a canonical morphism

$\left(*\to {\pi }^{*}X\right):=\left(X\stackrel{\left(\mathrm{Id},\mathrm{Id}\right)}{\to }X×X\right)\phantom{\rule{thinmathspace}{0ex}}.$(* \to \pi^* X) := (X \stackrel{(Id,Id)}{\to} X \times X) \,.

The image of this under ${f}^{*}$ is (using that this preserves the terminal object) a morphism

$*\to {f}^{*}{\pi }^{*}X=\mathrm{LConst}X$* \to f^* \pi^* X = LConst X

in $H$.

Conversely, given a morphism of the form $*\to \mathrm{LConst}X$ in $H$ we obtain the base change geometric morphism

$H\simeq H/*\to H/\mathrm{LConst}X\stackrel{\Gamma }{\to }\infty \mathrm{Grpd}/X\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{H} \simeq \mathbf{H}/* \to \mathbf{H}/LConst X \stackrel{\Gamma}{\to} \infty Grpd/X \,.

One checks that these constructions establish an equivalence

$\left(\infty ,1\right)\mathrm{Topos}\left(H,\infty \mathrm{Grpd}/X\right)\simeq H\left(*,\mathrm{LConst}X\right)\phantom{\rule{thinmathspace}{0ex}}.$(\infty,1)Topos(\mathbf{H}, \infty Grpd/X) \simeq \mathbf{H}(*, LConst X) \,.

Using this, we see that

$\begin{array}{rl}\Pi \left(H\right):X↦& \left(\infty ,1\right)\mathrm{Topos}\left(H,X\right)\\ & \simeq H\left(*,\mathrm{LConst}X\right)\\ & \simeq H\left(\mathrm{LConst}*,\mathrm{LConst}X\right)\\ & \simeq \infty \mathrm{Grpd}\left(*,\Gamma \mathrm{LConst}X\right)\\ & \simeq \Gamma \mathrm{LConst}X\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\begin{aligned} \Pi (\mathbf{H}) : X \mapsto & (\infty,1)Topos(\mathbf{H}, X) \\ & \simeq \mathbf{H}(*,LConst X) \\ & \simeq \mathbf{H}(LConst *, LConst X) \\ & \simeq \infty Grpd(*, \Gamma LConst X) \\ & \simeq \Gamma LConst X \end{aligned} \,.
###### Remark

In particular this does show that $\Pi \left(H\right):\infty \mathrm{Grpd}\to \infty \mathrm{Grpd}$ does preserve finite $\left(\infty ,1\right)$-limits, since $\Gamma$ preserves all limits and $\mathrm{LConst}$ is a left exact functor. It also shows that it is accessible, since $\Gamma$ and $\mathrm{LConst}$ are both accessible.

## Examples

### Shape of a locally $\infty$-connected topos

Suppose that $H$ is locally ∞-connected, meaning that $\Gamma$ has a left adjoint $\Pi$ which constructs the homotopy ∞-groupoids of objects of $H$. Then $Shape\left(H\right)$ is represented by $\Pi \left(*\right)\in \infty \mathrm{Grpd}$, for we have

$\begin{array}{rl}\mathrm{Shape}\left(H\right)\left(A\right)& =\Gamma \left(\mathrm{LConst}\left(A\right)\right)\\ & ={\mathrm{Hom}}_{\infty \mathrm{Grpd}}\left(*,\Gamma \left(\mathrm{LConst}\left(A\right)\right)\right)\\ & ={\mathrm{Hom}}_{H}\left(\mathrm{LConst}\left(*\right),\mathrm{LConst}\left(A\right)\right)\\ & ={\mathrm{Hom}}_{H}\left(*,\mathrm{LConst}\left(A\right)\right)\\ & ={\mathrm{Hom}}_{\infty \mathrm{Grpd}}\left(\Pi \left(*\right),A\right).\end{array}$\begin{aligned} Shape(\mathbf{H})(A) &= \Gamma(LConst(A))\\ &= Hom_{\infty Grpd}(*, \Gamma(LConst(A)))\\ &= Hom_{\mathbf{H}}(LConst(*), LConst(A)) \\ &= Hom_{\mathbf{H}}(*, LConst(A)) \\ &= Hom_{\infty Grpd}(\Pi(*),A). \end{aligned}

Thus, if we regard $\Pi \left(*\right)$ as “the fundamental ∞-groupoid of $H$” — which is reasonable since when $H=\mathrm{Sh}\left(X\right)$ consists of sheaves on a locally contractible topological space $X$, ${\Pi }_{H}\left(*\right)$ is equivalent to the usual fundamental ∞-groupoid of $X$ — then we can regard the shape of an $\left(\infty ,1\right)$-topos as a generalized version of the “homotopy $\infty$-groupoid” which nevertheless makes sense even for non-locally-contractible toposes, by taking values in the larger category of “pro-$\infty$-groupoids.”

It follows also that $H$ is not only locally ∞-connected but also ∞-connected, then it has the shape of a point.

### Shape of a topological space

For a discussion of how the $\left(\infty ,1\right)$-topos theoretic shape of ${\mathrm{Sh}}_{\left(\infty ,1\right)}\left(X\right)$ relates to the ordinary shape-theoretic strong shape of the topological space $X$ see shape theory.

### Shape of an essential retract

The following is trivial to observe, but may be useful to note.

###### Observation

Let $\left({f}_{!}⊣{f}^{*}⊣{f}_{*}\right):H\stackrel{\stackrel{{f}_{!}}{\to }}{\stackrel{\stackrel{{f}^{*}}{←}}{\underset{{f}_{*}}{\to }}}B$ be an essential geometric morphism of $\left(\infty ,1\right)$-toposes that exhibits $B$ as an essential retract of $H$ in that

$\left(\mathrm{Id}⊣\mathrm{Id}\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\simeq \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}B\stackrel{\stackrel{{f}_{!}}{←}}{\underset{{f}^{*}}{\to }}H\stackrel{\stackrel{{f}^{*}}{←}}{\underset{{f}_{*}}{\to }}B\phantom{\rule{thinmathspace}{0ex}}.$(Id \dashv Id) \;\; \simeq \;\; \mathbf{B} \stackrel{\overset{f_!}{\leftarrow}}{\underset{f^*}{\to}} \mathbf{H} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \mathbf{B} \,.

Then the shape of $B$ is equivalent to that of $H$.

###### Proof

Since $\infty \mathrm{Grpd}$ is the terminal object in the category of Grothendieck $\left(\infty ,1\right)$-toposes and geometric morphisms, we have

$\begin{array}{rl}\left(\infty \mathrm{Grpd}\stackrel{{\mathrm{LConst}}_{B}}{\to }B\stackrel{{\Gamma }_{B}}{\to }\infty \mathrm{Grpd}\right)& \simeq \left(\infty \mathrm{Grpd}\stackrel{{\mathrm{LConst}}_{B}}{\to }B\stackrel{{f}^{*}}{\to }H\stackrel{{f}_{*}}{\to }B\stackrel{{\Gamma }_{B}}{\to }\infty \mathrm{Grpd}\right)\\ & \simeq \left(\infty \mathrm{Grpd}\stackrel{{\mathrm{LConst}}_{H}}{\to }H\stackrel{{\Gamma }_{H}}{\to }\infty \mathrm{Grpd}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\begin{aligned} (\infty Grpd \stackrel{LConst_{\mathbf{B}}}{\to} \mathbf{B} \stackrel{\Gamma_\mathbf{B}}{\to} \infty Grpd) &\simeq (\infty Grpd \stackrel{LConst_{\mathbf{B}}}{\to} \mathbf{B} \stackrel{f^*}{\to} \mathbf{H} \stackrel{f_*}{\to} \mathbf{B} \stackrel{\Gamma_\mathbf{B}}{\to} \infty Grpd) \\ &\simeq (\infty Grpd \stackrel{LConst_\mathbf{H}}{\to} \mathbf{H} \stackrel{\Gamma_\mathbf{H}}{\to} \infty Grpd) \end{aligned} \,.
###### Example

Every

over $\infty \mathrm{Grpd}$ has the shape of the point.

###### Proof

By definition $H$ is $\infty$-connected if the constant ∞-stack inverse image ${f}^{*}=L\mathrm{Const}$ is

1. not only a left but also a right adjoint;

By standard properties of adjoint (∞,1)-functors we have that a right adjoint ${f}^{*}$ is a full and faithful (∞,1)-functor precisely if the counit ${f}_{!}{f}^{*}\to \mathrm{Id}$ is an equivalence.

Equivalently, we can observe that a locally ∞-connected (∞,1)-topos is ∞-connected precisely when $\Pi$ preserves the terminal object, and apply the above observation that the shape of a locally ∞-connected (∞,1)-topos is represented by $\Pi \left(*\right)$.

## References

The definition of shape of $\left(\infty ,1\right)$-toposes as $\Gamma \circ \mathrm{LConst}$ is due to

This and the relation to shape theory, more precisely the strong shape, of topological spaces is further discussed in section 7.1.6 of

Revised on February 11, 2011 21:44:25 by Zoran Škoda (161.53.130.104)