Here are two kind-of dual results that have now been proved after I asked them here:
The first arose from Induced map on path manifolds: is it a submersion?, which got an answer in very general terms involving generalised smooth spaces in this preprint:
- Andrew Stacey, Yet More Smooth Mapping Spaces and Their Smoothly Local Properties, arXiv:1301.5493.
and then again in this paper, as Lemma 2.4:
- Habib Amiri, Alexander Schmeding A differentiable monoid of smooth maps on Lie groupoids, Journal of Lie Theory 29 (2019), No. 4, 1167–1192, arXiv:1706.04816
in a way that only partially overlaps with Andrew Stacey's version; in one sense it's less general, but it seems to use a stronger notion of submersion/regular map.
The second came from Extension of functions from geodesically convex compact sets in a Riemannian manifold, which has now been answered in
- David Michael Roberts, Alexander Schmeding, Extending Whitney's extension theorem: nonlinear function spaces, to appear, Annales de l'Institut Fourier, arXiv:1801.04126
A different kind of question I asked now has three papers giving three different approaches, namely On a weak choice principle, which led to (in chronological order):
Benno van den Berg, WISC is independent of ZF, (pdf), also Theorem 5.1/Corollary 5.2 in Predicative toposes, arXiv:1207.0959. This uses Gitik's class forcing symmetric model of ZF, over ZFC with a reasonably strong large cardinal assumption.
Asaf Karagila, Embedding Orders Into Cardinals With $DC_\kappa$, Fund. Math. 226 (2014), 143-156, doi:10.4064/fm226-2-4, arXiv:1212.4396. This uses class forcing symmetric models, over ZFC with no large cardinals.
David Michael Roberts, The weak choice principle WISC may fail in the category of sets, Studia Logica Volume 103 (2015) Issue 5, pp 1005-1017, doi:10.1007/s11225-015-9603-6 arXiv:1311.3074. This uses topos-theoretic methods, over a well-pointed base topos with no Choice.