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

### Context

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

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Idea

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

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

## Definition

###### Definition

The composite (∞,1)-functor

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

is the shape functor . Its value

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

on an $(\infty,1)$-topos $\mathbf{H}$ is the shape of $\mathbf{H}$.

Here

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

## Properties

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

$(LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd$

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

###### Proposition

The shape of $\mathbf{H}$ is the composite functor

$\Pi(\mathbf{H}) := \Gamma \circ LConst \;\;:\;\; \infty Grpd \stackrel{LConst}{\to} \mathbf{H} \stackrel{\;\;\Gamma\;\;}{\to} \infty Grpd$

regarded as an object

$\Pi(\mathbf{H}) \in Pro(\infty Grpd) = Lex(\infty Grpd, \infty Grpd)^{op} \,.$
###### 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

$Func(X,\infty Grpd) \simeq \infty Grpd/X$

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

$(LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \infty Grpd/X \stackrel{\overset{\pi^*}{\leftarrow}}{\underset{\pi_*}{\to}} \infty Grpd \,.$

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

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

$(* \to \pi^* X) := (X \stackrel{(Id,Id)}{\to} X \times X) \,.$

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

$* \to f^* \pi^* X = LConst X$

in $\mathbf{H}$.

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

$\mathbf{H} \simeq \mathbf{H}/* \to \mathbf{H}/LConst X \stackrel{\Gamma}{\to} \infty Grpd/X \,.$

One checks that these constructions establish an equivalence

$(\infty,1)Topos(\mathbf{H}, \infty Grpd/X) \simeq \mathbf{H}(*, LConst X) \,.$

Using this, we see that

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

## Examples

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

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

\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 $\Pi(*)$ as “the fundamental ∞-groupoid of $\mathbf{H}$” — which is reasonable since when $\mathbf{H}=Sh(X)$ consists of sheaves on a locally contractible topological space $X$, $\Pi_{\mathbf{H}}(*)$ is equivalent to the usual fundamental ∞-groupoid of $X$ — then we can regard the shape of an $(\infty,1)$-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 $\mathbf{H}$ 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 $(\infty,1)$-topos theoretic shape of $Sh_{(\infty,1)}(X)$ 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 $(f_! \dashv f^* \dashv f_*) : \mathbf{H} \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} \mathbf{B}$ be an essential geometric morphism of $(\infty,1)$-toposes that exhibits $\mathbf{B}$ as an essential retract of $\mathbf{H}$ in that

$(Id \dashv Id) \;\; \simeq \;\; \mathbf{B} \stackrel{\overset{f_!}{\leftarrow}}{\underset{f^*}{\to}} \mathbf{H} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \mathbf{B} \,.$

Then the shape of $\mathbf{B}$ is equivalent to that of $\mathbf{H}$.

###### Proof

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

\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 $\infty Grpd$ has the shape of the point.

###### Proof

By definition $\mathbf{H}$ is $\infty$-connected if the constant ∞-stack inverse image $f^* = L Const$ 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 $f_! f^* \to Id$ 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 $\Pi(*)$.

## References

The definition of shape of $(\infty,1)$-toposes as $\Gamma \circ LConst$ 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