nLab
locally contractible topological infinity-groupoid

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Idea

A locally contractible topological ∞-groupoid is an ∞-groupoid equipped with cohesion in the form of locally contractible topology.

The collection of all these cohesive $\mathrm{\infty }$-groupoids forms a cohesive (∞,1)-topos $\mathrm{LCTop}\mathrm{\infty }\mathrm{Grpd}$.

This is similarr to ETop∞Grpd, which models cohesibion in the form of Euclidean topology.

Definition

Let $\mathrm{CTop}$ be some small version (…details missing…) of the site of locally contractible contractible topological spaces with continuous maps betwen them and equpped with the standard open cover coverage.

This is a cohesive site (…for the evident generalization of that definitions where Cech covers are generalized to hypercovers…). The key axiom to check is that for $Y\to U$ a hypercover of $U\in \mathrm{CTop}$ degreewise by a coproduct of contractibles, also the simplicial set ${\lim }_{\to }Y$ obtained by sending each contractible to a point is contractible. This follows with ArtinMazur. See the proposition below

Define then

$$\mathrm{LCTop}\mathrm{\infty }\mathrm{Grpd}:={\mathrm{Sh}}_{(\mathrm{\infty },1)}(\mathrm{CTop})$$LCTop\infty Grpd := Sh_{(\infty,1)}(CTop)

to be the (∞,1)-category of (∞,1)-sheaves on $\mathrm{CTop}$.

This is a cohesive (∞,1)-topos.

$$(\Pi ⊣\mathrm{Disc}⊣\Gamma ⊣\mathrm{coDisc}):\mathrm{LCTop}\mathrm{\infty }\mathrm{Grpd}\to \mathrm{\infty }\mathrm{Grpd}.$$(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : LCTop\infty Grpd \to \infty Grpd \,.

Proposition. For $X\in \mathrm{LCTop}\hookrightarrow \mathrm{LC}\mathrm{\infty }\mathrm{Grpd}$ a locally contractible space (…maybe with some local size restriction, depending on the details of $\mathrm{CTop}$…), regarded as a 0-truncated locally contractible topological $\mathrm{\infty }$-groupoid, we have that the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos-functor applied to $X$ coincides, up to equivalence, with the standard fundamental ∞-groupoid of $X$.

$$\Pi (X)\simeq \mathrm{Sing}X.$$\Pi(X) \simeq Sing X \,.

Proof. By the analogous arguments as at ETop∞Grpd we may present $\Pi$ by the left derived functor of the colimit functor on simplicial presheaves. This is the ordinary colimit applied to a cofibrant resolution of $X$ in $[{\mathrm{CTop}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{loc}}$. By Dugger’s cofibrant replacement theorem recalled at model structure on simplicial presheaves, such is given by a split hypercover $Y\to X$ degreewise a coproduct of objects in $\mathrm{CTop}$. By ArtinMazur there is a weak homotopy equivalence ${\lim }_{\to }Y\to \mathrm{Sing}X$.

(…)

References

• Artin-Mazur, Étale Homotopy , SLNM 100