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 $\infty$-groupoids forms a cohesive (∞,1)-topos $LCTop\infty Grpd$.

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

Definition

Let $CTop$ be some small version (…details missing…) of the site of locally contractible contractible topological spaces with continuous maps betwen them and equipped 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 $U \in CTop$ 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 as pointed out on MO here.¹

Define then

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

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

This is an cohesive (∞,1)-topos.

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

Related concepts

• topological infinity-groupoid

  • Euclidean-topological infinity-groupoid

References

The corresponding 1-cohesive topos over locally connected topological spaces was considered in

• Peter Johnstone, example 1.5 of Remarks on punctual local connectedness, Theory and Applications of Categories, Vol. 25, 2011, No. 3, pp 51-63 (web)

A decent account of the above $\infty$-topos is in prepation by David Carchedi…

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1. Thanks to David Carchedi for highlighting this. ↩