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In this article, Joyce defines a new kind of smooth map of manifolds with corners.

The standard requirement would be that a smooth map is smooth in every chart. He calls such maps weakly smooth. In order for a map to be smooth, he requires it to preserve something he calls "boundary defining functions", which look a bit like local Morse functions to me.

This additional requirement seems to have a few drawbacks, namely several basic functions aren't smooth anymore, such as $f\colon \mathbb{R} \to [0,\infty), x \mapsto x^2$.

He lists a lot of examples for smooth functions, e.g. compositions of smooth functions, products, the boundary inclusion, the diagonal map. His category of manifolds with corners and smooth functions has initial and terminal objects, and pullbacks for transversal maps. His diffeomorphisms are exactly homeomorphisms that preserve the maximal atlas.

Which of these features are lost when removing his additional requirement? What doesn't work for the naïve definition, in his words, of "weakly smooth maps"?

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  • $\begingroup$ The same definition is used by Bunke in arXiv:math/02012v4. Such a restriction is useful to define the index of a Dirac operator on a manifold with corners. Local triviality around the boundary is used to simplify the analysis, and I am not sure if a treatment in full generality without such restrictions exists. $\endgroup$ – Dmitri Pavlov Apr 26 '17 at 9:24
  • $\begingroup$ @DmitriPavlov, that link doesn't seem to work. Why do we need to worry about Dirac operators if we're not doing geometry yet, but just differential topology? $\endgroup$ – Manuel Bärenz Apr 26 '17 at 10:15
  • $\begingroup$ There are no links in my comment, so I'm not sure what you expect to work there. There is no mention of differential topology in Joyce's paper, so I am uncertain why you brought it up. Joyce himself explains that his "theory is tailored to future applications in Symplectic Geometry, and is part of a project to describe the geometric structure on moduli spaces of J-holomorphic curves in a new way". In other words, he is doing analysis, not differential topology. $\endgroup$ – Dmitri Pavlov Apr 26 '17 at 10:22
  • $\begingroup$ @DmitriPavlov, I mean the arxiv article you mention. There is no article behind the link arxiv.org/abs/math/02012v4. It appears you're referring to this one: arxiv.org/abs/math/0201112 I'm using the term "differential topology" in the sense of "theory of smooth manifolds". Of course you need to understand smooth manifolds before doing analysis. But I'm not asking about the geometric aspects, I just want to know whether the naïve definition of a category of smooth manifolds has products, diagonal, transversal pullbacks and so on. $\endgroup$ – Manuel Bärenz Apr 26 '17 at 10:38
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    $\begingroup$ The only difference that I see is that Bunke makes (a germ of) a boundary defining function part of a definition of a manifolds with corners, which makes sense for his applications, whereas Joyce only postulates existence. $\endgroup$ – Dmitri Pavlov Apr 26 '17 at 13:13

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