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

Definition

For $(\Pi \dashv \Gamma \dashv LConst) : \mathbf{H} \to \infty Grpd$ a locally ∞-connected (∞,1)-topos and $X \in \mathbf{H}$ an object, we say that $\Pi(X)$ is the fundamental $\infty$-groupoid of $X$ in $\mathbf{H}$.

Properties

• This is the object that encodes the geometric homotopy groups in an (∞,1)-topos.

• For any $X \in \mathbf{H}$ the $\infty$-groupoid $\Pi(X)$ coincides with the fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos (see there) of the over-(∞,1)-topos $\mathbf{H}/X$.

Examples

• In the cohesive (∞,1)-topos Top the intrinsic fundamental $\infty$-groupoid functor coincides with the ordinary fundamental ∞-groupoid of a topological space. See discrete ∞-groupoid for details.

• In ETop∞Grpd the intrinsic fundamental $\infty$-groupoid is the generalization of that on Top to ∞-groupoids in paracompact spaces.

Related concepts

• geometric realization of cohesive ∞-groupoids

References

• differential cohomology in a cohesive topos

See also

• David Carchedi, On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces (arXiv:1504.02394)

for further discussion of the smooth shape modality of cohesion (the etale homotopy type operation in the context of smooth infinity-stacks) as applied to orbifolds and étale groupoids and generally étale ∞-groupoids.