# nLab fundamental infinity-groupoid of a locally infinity-connected (infinity,1)-topos

### Context

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

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Idea

Every (∞,1)-topos $E$ has a shape $\mathrm{Shape}\left(E\right)\in \mathrm{Pro}\infty \mathrm{Grpd}$. When $E$ is locally ∞-connected then this is a genuine ∞-groupoid $\Pi \left(E\right)\in$ ∞Grpd. We may think of this as the fundamental ∞-groupoid of the $\left(\infty ,1\right)$-topos regarded as a generalized space.

But also every locally ∞-connected (∞,1)-topos has an internal notion of fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos for objects of $E$, denoted ${\Pi }_{E}:E\to \infty \mathrm{Grpd}$. Applied to its terminal object this does agree with the fundamental ∞-groupoid of the topos:

$\Pi \left(E\right)\simeq {\Pi }_{E}\left(*\right)\phantom{\rule{thinmathspace}{0ex}}.$\Pi(E) \simeq \Pi_E(*) \,.

Conversely, for an object $X\in E$, the fundamental ∞-groupoid ${\Pi }_{E}\left(X\right)$ internal to $E$ can be identified with the fundamental ∞-groupoid of the locally ∞-connected (∞,1)-topos $E/X$.

## Definition

###### Definition

For $\left({\Pi }_{E}⊣{\Gamma }_{E}⊣{\mathrm{LConst}}_{E}\right):E\to \infty \mathrm{Grpd}$ a locally ∞-connected (∞,1)-topos we say its fundamental $\infty$-groupoid is

$\Pi \left(E\right):={\Pi }_{E}\left(*\right)\phantom{\rule{thinmathspace}{0ex}},$\Pi(E) := \Pi_E(*) \,,

where $*$ is the terminal object of $E$.

In other words, it is the internal fundamental ∞-groupoid of the terminal object of $E$.

## Properties

Let $H$ be a locally $\infty$-connected $\left(\infty ,1\right)$-topos and $X\in H$ an object. Then also the over-(∞,1)-topos $H/X$ is locally $\infty$-connected (as discussed there).

We have then two different definitions of the fundamental $\infty$-groupoid of $X$: once as ${\Pi }_{H}\left(X\right)$ – the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos – and once as $\Pi \left(H/X\right)$.

###### Proposition

These agree:

${\Pi }_{H}\left(X\right)\simeq \Pi \left(H/X\right)\phantom{\rule{thinmathspace}{0ex}}.$\Pi_{\mathbf{H}}(X) \simeq \Pi(\mathbf{H}/X) \,.
###### Proof

Since $X\stackrel{\mathrm{Id}}{\to }X$ is the terminal object in $H/X$ we have by definition

$\Pi \left(H/X\right)={\Pi }_{H/X}\left({\mathrm{Id}}_{X}\right)\phantom{\rule{thinmathspace}{0ex}}.$\Pi(\mathbf{H}/X) = \Pi_{\mathbf{H}/X}(Id_X) \,.

Now observe that ${\Pi }_{H/X}={\Pi }_{H}\circ {X}_{!}$ since the terminal global section geometric morphism of the over-topos is

$H/X\stackrel{\stackrel{{X}_{!}}{\to }}{\stackrel{\stackrel{{X}^{*}}{←}}{\underset{{X}_{*}}{\to }}}H\stackrel{\stackrel{{\Pi }_{H}}{\to }}{\stackrel{\stackrel{{\mathrm{LConst}}_{H}}{←}}{\underset{{\Gamma }_{H}}{\to }}}\infty \mathrm{Grpd}$\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

and that ${X}_{!}$ in the etale geometric morphism is the projection map that sends $Y\stackrel{}{\to }X$ to $Y$.

###### Definition

Let $\mathrm{LC}\left(\infty ,1\right)\mathrm{Topos}$ denote the full sub-(∞,1)-category of (∞,1)Topos determined by the locally ∞-connected objects.

###### Proposition

The $\left(\infty ,1\right)$-category $\infty \mathrm{Gpd}$ (as the category of local homeomorphisms over $\infty \mathrm{Gpd}$) is reflective in $\mathrm{LC}\left(\infty ,1\right)\mathrm{Topos}$,

$\infty \mathrm{Grpd}\stackrel{\stackrel{\Pi }{←}}{↪}\mathrm{LC}\left(\infty ,1\right)\mathrm{Topos}$\infty Grpd \stackrel{\overset{\Pi}{\leftarrow}}{\hookrightarrow} LC(\infty,1)Topos

with the reflector given by forming the fundamental $\infty$-groupoid.

###### Proof

Any ∞-groupoid $G$ gives rise to an (∞,1)-presheaf (∞,1)-topos $\mathrm{PSh}\left(G\right)=\left[{G}^{\mathrm{op}},\infty \mathrm{Gpd}\right]$, which by the (∞,1)-Grothendieck construction is equivalent to the over-(∞,1)-topos $\infty \mathrm{Gpd}/G$. The $\left(\infty ,1\right)$-toposes of this form are, by definition, those for which the unique (∞,1)-geometric morphism to $\infty \mathrm{Gpd}$ is a local homeomorphism of toposes. This construction embeds $\infty \mathrm{Gpd}$ as a full sub-(∞,1)-category of (∞,1)Topos:

$\mathrm{PSh}\left(-\right):\infty \mathrm{Grpd}↪\mathrm{LC}\left(\infty ,1\right)\mathrm{Topos}\phantom{\rule{thinmathspace}{0ex}}.$PSh(-) : \infty Grpd \hookrightarrow LC(\infty,1)Topos \,.

since in particular the $\left(\infty ,1\right)$-toposes $\mathrm{PSh}\left(G\right)$ are locally ∞-connected.

To show that $\Pi$ is a left adjoint (∞,1)-functor to $\mathrm{PSh}\left(-\right)$ we demonstrate a natural hom-equivalence

$\mathrm{LC}\left(\infty ,1\right)\mathrm{Topos}\left(E,\left(\infty ,1\right)\mathrm{PSh}\left(A\right)\right)\simeq \infty \mathrm{Grpd}\left({\Pi }_{E}\left(*\right),A\right)$LC(\infty,1)Topos(E,(\infty,1)PSh(A)) \simeq \infty Grpd(\Pi_E(*), A)

for $E\in \mathrm{LC}\left(\infty ,1\right)\mathrm{Topos}$ and $A\in \infty \mathrm{Grpd}$.

At shape of an (∞,1)-topos it is shown that we have a natural equivalence

$\left(\infty ,1\right)\mathrm{Topos}\left(E,\mathrm{PSh}\left(A\right)\right)\simeq {\Gamma }_{E}{\mathrm{LConst}}_{E}G=:\mathrm{Shape}\left(E\right)\left(A\right)\phantom{\rule{thinmathspace}{0ex}}.$(\infty,1)Topos(E, PSh(A)) \simeq \Gamma_E LConst_E G =: Shape(E)(A) \,.

Now observe that furthermore we have a sequence of natural equivalences

$\begin{array}{rl}\mathrm{Shape}\left(E\right)\left(A\right)& =\Gamma \left(\mathrm{LConst}\left(A\right)\right)\\ & \simeq \infty \mathrm{Grpd}\left(*,\Gamma \left(\mathrm{LConst}\left(A\right)\right)\right)\\ & \simeq E\left(\mathrm{LConst}\left(*\right),\mathrm{LConst}\left(A\right)\right)\\ & \simeq E\left(*,\mathrm{LConst}\left(A\right)\right)\\ & \simeq \infty \mathrm{Grpd}\left({\Pi }_{E}\left(*\right),A\right).\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\begin{aligned} Shape(E)(A) &= \Gamma(LConst(A))\\ &\simeq \infty Grpd(*, \Gamma(LConst(A)))\\ &\simeq E(LConst(*), LConst(A)) \\ &\simeq E(*, LConst(A)) \\ &\simeq \infty Grpd(\Pi_E(*),A). \end{aligned} \,.
###### Remark

So equivalently, one may say that a locally ∞-connected (∞,1)-topos $E$ has a shape which is representable, and its fundamental ∞-groupoid $\Pi \left(H\right)$ is the representing object.

## Examples

###### Proposition

For $X$ a locally contractible topological space, we have an equivalence

$\Pi \left(\left(\infty ,1\right)\mathrm{Sh}\left(X\right)\right)\simeq \mathrm{Sing}X$\Pi ((\infty,1)Sh(X)) \simeq Sing X

between the ordinary fundamental ∞-groupoid of $X$ defined by the singular simplicial complex and the topos-theoretic fundamental $\infty$-groupoid of the (∞,1)-sheaf (∞,1)-topos $\left(\infty ,1\right)\mathrm{Sh}\left(X\right)$ over $X$.

###### Proof

Details are at geometric homotopy groups in an (∞,1)-topos.

More generally the shape of an (∞,1)-topos of $\left(\infty ,1\right)\mathrm{Sh}\left(X\right)$ reproduces the shape theory of $X$.

Revised on December 6, 2010 23:10:52 by Urs Schreiber (131.211.233.37)