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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 similarr to ETop∞Grpd, which models cohesibion 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 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 $U \in CTop$ 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
to be the (∞,1)-category of (∞,1)-sheaves on $CTop$.
This is a cohesive (∞,1)-topos.
Proposition. For $X \in LCTop \hookrightarrow LC\infty Grpd$ a locally contractible space (…maybe with some local size restriction, depending on the details of $CTop$…), regarded as a 0-truncated locally contractible topological $\infty$-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 $X$.
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 $[CTop^{op}, sSet]_{proj,loc}$. 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 $CTop$. By ArtinMazur there is a weak homotopy equivalence $\lim_\to Y\to Sing X$.
(…)