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following the discussion and suggestion in another thread I have created an entry
motivation for cohesive toposes .
This contains mainly just the result of copy-and-pasting a hasty forum-post. Feel free to edit it and tweak it.
Is cohesion more important for smooth as opposed to topological?
The thing is that, roughly, cohesive spaces are all locally contractible. The local contractions are those of the “basic cohesive droplets”, in the spirit of the above discussion.
So locally contractible topological infinity-groupoids are cohesive, as are Euclidean-topological infinity-groupoid.
But a geometry modeled on a small full subcategory of Top that contains locally non-contractible spaces will in general not be cohesive. In particular for instance general topological stacks do not live in a cohesive $(2,1)$-topos. But differentiable stacks do, because every manifold and hence in particular every smooth manifold is locally contractible.
So, I would say cohesion is as “important for” smooth geometry as it is to “locally contractible continuous geometry”. But it is inapplicable to more general continuous geometry.
I have now copied the above comment into the entry, and edited a bit more.
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