this page is under construction
structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
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.
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
to be the (∞,1)-category of (∞,1)-sheaves on $CTop$.
This is an cohesive (∞,1)-topos.
The corresponding 1-cohesive topos over locally connected topological spaces was considered in
A decent account of the above $\infty$-topos is in prepation by David Carchedi…
Thanks to David Carchedi for highlighting this. ↩