this page is under construction
structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
see also algebraic topology, functional analysis and homotopy theory
Basic concepts
topological space (see also locale)
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Basic homotopy theory
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. ↩