# 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

See also

Revised on June 4, 2015 06:17:08 by Urs Schreiber (50.207.161.2)