What are the smooth manifolds in the topos of sheaves on a smooth manifold? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T19:44:32Z http://mathoverflow.net/feeds/question/79514 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79514/what-are-the-smooth-manifolds-in-the-topos-of-sheaves-on-a-smooth-manifold What are the smooth manifolds in the topos of sheaves on a smooth manifold? Dmitri Pavlov 2011-10-30T11:10:38Z 2012-12-09T19:16:32Z <p>The category of internal locales in the Grothendieck topos of sheaves on a locale X is equivalent to the slice category over X. In other words, internal locales over X are precisely morphisms of locales of the form Y→X.</p> <p><strong>Is the same statement true for smooth manifolds instead of locales and submersions instead of continuous maps?</strong></p> <p>More precisely, consider the site of smooth manifolds equipped with its standard Grothendieck topology of surjective submersions.</p> <p><strong>Is the category of internal smooth manifolds in the Grothendieck topos of sheaves on a smooth manifold M equivalent to the category whose objects are surjective submersions of the form N→M and morphisms are commuting triangles?</strong></p> <p>Definitions tend to branch when we internalize them, so an answer to this question should include the correct definition of a smooth manifold in an arbitrary Grothendieck topos.</p> <p>According to nLab, <a href="http://ncatlab.org/nlab/show/vector+bundle#sheaftheoretic_version_11" rel="nofollow">internal dualizable modules over the locale of real numbers</a> in the category of sheaves on a smooth manifold M are precisely finite-dimensional vector bundles over M. This can be seen as a further motivation for the above question.</p> http://mathoverflow.net/questions/79514/what-are-the-smooth-manifolds-in-the-topos-of-sheaves-on-a-smooth-manifold/115909#115909 Answer by Peter Michor for What are the smooth manifolds in the topos of sheaves on a smooth manifold? Peter Michor 2012-12-09T19:16:32Z 2012-12-09T19:16:32Z <p>See the book (a kind of culmination point of the theory: "synthetic differential geometry")</p> <p>MR1083355 Moerdijk, Ieke; Reyes, Gonzalo E. Models for smooth infinitesimal analysis. Springer-Verlag, New York, 1991.</p>