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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 23rd 2015
• (edited Apr 23rd 2015)

I am beginning to work on a new chapter geometry of physics – manifolds and orbifolds.

The goal for today is to write detailed exposition of how the theory of manifolds and their frame bundles is set up using the infinitesimal shape modality on the Cahiers topos.

So far I have (only) the Introduction and the first two subsections Formal smooth Cartesian spaces and Formal smooth sets and some scattered material following that. But now first some lunch break.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeApr 23rd 2015

Now there is brief subsections in the “Model Layer”

that gradually go (or are meant to go) to seeing how the traditional definition of smooth manifolds is re-expressed in terms of infinitesimal shape on the Cahiers topos.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 23rd 2015

on to smooth groupoids. Added a section

with the proof that a Lie groupoid $\mathcal{G}_\bullet$ is an étale groupoid precisely if the infinitesimal shape unit of the atlas $\mathcal{G}_0 \to \mathcal{G}$ is a homotopy pullback.

It’s a simple proof, but I found that fun when I first saw it, back then. The same kind of proof (but with décalage for fibrant replacement instead of the default resolution by factorization lemma) gives the analogous result for étale $\infty$-groupoids, but I don’t have that in the $n$Lab entry at the moment. Will focus on 1-groupoids there for the time being.