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

Idea

Every (∞,1)-topos $E$ has a shape $\mathrm{Shape}(E)\in \mathrm{Pro}\mathrm{\infty }\mathrm{Grpd}$. When $E$ is locally ∞-connected then this is a genuine ∞-groupoid $\Pi (E)\in$ ∞Grpd. We may think of this as the fundamental ∞-groupoid of the $(\mathrm{\infty },1)$-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 ${\Pi }_E:E\to \mathrm{\infty }\mathrm{Grpd}$. Applied to its terminal object this does agree with the fundamental ∞-groupoid of the topos:

$$\Pi (E)\simeq {\Pi }_E(*).$$\Pi(E) \simeq \Pi_E(*) \,.

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

Definition

Definition

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

$$\Pi (E):={\Pi }_E(*),$$\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 $\mathrm{\infty }$-connected $(\mathrm{\infty },1)$-topos and $X\in H$ an object. Then also the over-(∞,1)-topos $H/X$ is locally $\mathrm{\infty }$-connected (as discussed there).

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

Proposition

These agree:

$${\Pi }_H(X)\simeq \Pi (H/X).$$\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

$$\Pi (H/X)={\Pi }_{H/X}({\mathrm{Id}}_X).$$\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

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

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

Definition

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

Proposition

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

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

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

Proof

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

$$\mathrm{PSh}(-):\mathrm{\infty }\mathrm{Grpd}\hookrightarrow \mathrm{LC}(\mathrm{\infty },1)\mathrm{Topos}.$$PSh(-) : \infty Grpd \hookrightarrow LC(\infty,1)Topos \,.

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

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

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

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

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

$$(\mathrm{\infty },1)\mathrm{Topos}(E,\mathrm{PSh}(A))\simeq {\Gamma }_E{\mathrm{LConst}}_{E}G=:\mathrm{Shape}(E)(A).$$(\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

$$\begin{array}{rl}\mathrm{Shape}(E)(A)& =\Gamma (\mathrm{LConst}(A))\\ & \simeq \mathrm{\infty }\mathrm{Grpd}(*,\Gamma (\mathrm{LConst}(A)))\\ & \simeq E(\mathrm{LConst}(*),\mathrm{LConst}(A))\\ & \simeq E(*,\mathrm{LConst}(A))\\ & \simeq \mathrm{\infty }\mathrm{Grpd}({\Pi }_E(*),A).\end{array}.$$\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 $\Pi (H)$ is the representing object.

Examples

Proposition

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

$$\Pi ((\mathrm{\infty },1)\mathrm{Sh}(X))\simeq \mathrm{Sing}X$$\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 $\mathrm{\infty }$-groupoid of the (∞,1)-sheaf (∞,1)-topos $(\mathrm{\infty },1)\mathrm{Sh}(X)$ over $X$.

Proof

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

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

Related concepts

• fundamental groupoid, fundamental ∞-groupoid,

• homotopy group, homotopy groups in an (∞,1)-topos

• fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos / of a locally $\mathrm{\infty }$-connected $(\mathrm{\infty },1)$-topos