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

Idea

Every (∞,1)-topos $E$ has a shape $Shape(E) \in Pro\infty Grpd$. 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 $(\infty,1)$-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 $\Pi_E : E \to \infty Grpd$. (If in addition $\Delta \colon \infty Grpd \to E$ is fully faithful then we call $\Pi$ the shape modality.) Applied to its terminal object this does agree with the fundamental ∞-groupoid of the topos:

$$\Pi(E) \simeq \Pi_E(*) \,.$$

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 $(\Pi_E \dashv \Gamma_E \dashv LConst_E) : E \to \infty Grpd$ a locally ∞-connected (∞,1)-topos we say its fundamental $\infty$-groupoid is

$$\Pi(E) := \Pi_E(*) \,,$$

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

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

Properties

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

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

Proposition

These agree:

$$\Pi_{\mathbf{H}}(X) \simeq \Pi(\mathbf{H}/X) \,.$$

Proof

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

$$\Pi(\mathbf{H}/X) = \Pi_{\mathbf{H}/X}(Id_X) \,.$$

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

$$\mathbf{H}/X \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}}} \mathbf{H} \stackrel{\overset{\Pi_{\mathbf{H}}}{\to}}{\stackrel{\overset{LConst_{\mathbf{H}}}{\leftarrow}}{\underset{\Gamma_{\mathbf{H}}}{\to}}} \infty Grpd$$

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

Definition

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

Proposition

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

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

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

Proof

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

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

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

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

$$LC(\infty,1)Topos(E,(\infty,1)PSh(A)) \simeq \infty Grpd(\Pi_E(*), A)$$

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

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

$$(\infty,1)Topos(E, PSh(A)) \simeq \Gamma_E LConst_E G =: Shape(E)(A) \,.$$

Now observe that furthermore we have a sequence of natural equivalences

$$\begin{aligned} Shape(E)(A) &= \Gamma(LConst(A))\\ &\simeq \infty Grpd(*, \Gamma(LConst(A)))\\ &\simeq E(LConst(*), LConst(A)) \\ &\simeq E(*, LConst(A)) \\ &\simeq \infty Grpd(\Pi_E(*),A). \end{aligned} \,.$$

Remark

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

Examples

Proposition

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

$$\Pi ((\infty,1)Sh(X)) \simeq Sing X$$

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 $(\infty,1)Sh(X)$ over $X$.

Proof

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

More generally the shape of an (∞,1)-topos of $(\infty,1)Sh(X)$ 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 $\infty$-connected $(\infty,1)$-topos

• étale homotopy theory, pro-left adjoint