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Added header. —Nils von Barth (nbarth) (talk) 22:44, 5 May 2009 (UTC)[reply]

This article needs more context. A short introduction what the topic is about can help. I came here in search for a software called pdiff. Does this explain some math used in pdiff, or is this unrelated?

77.24.237.81 (talk) 22:38, 7 June 2008 (UTC)[reply]

Sorry for the lack of context – I’ve substantially expanded this, which should provide better context.

AFAICT there is no relation whatsoever between the category PDIFF and any computer program by that name; this is a very technical point in math, and of likely very little interest outside it.

As for other uses of the term, I’ve create a disambiguation page at pdiff; hope this helps!

—Nils von Barth (nbarth) (talk) 22:44, 5 May 2009 (UTC)[reply]

This article is completely wrong. PDiff does not form a category and thus almost every else stated is false. Just read the first page of the first reference by Jacob Lurie! Or [this MO answer by Tom Goodwillie](https://mathoverflow.net/a/27673). The only reputable source that says otherwise is McMullen, but he, in turn, cites Thurston, who is the very clearest against him. I can't link Thurston, but I quote exercise 3.10.1(a):

"Show that such maps [pdiff] don’t form a group. Thus, piecewise smooth maps serve as bridges between piecewise linear and smooth structures; they don’t work well alone."

A stub consisting of just the bibliography (ex McMullen) would be an improvement on this article. 108.52.140.137 (talk) 16:12, 29 June 2019 (UTC)[reply]

I agree with this comment. In fact, this page is seemingly doing more harm than good. People or quoting the content of this page, but this content seems blatantly wrong to me. I believe there is a way to allow composition of piecewise smooth maps namely, define a map ${\displaystyle f:X\to Y}$ to be piecewise smooth if there exists a triangulation of ${\displaystyle X}$ such that ${\displaystyle f}$ is smooth on each simplex. Then, the composition of piecewise maps will be piecewise smooth by subdividing the triangulation on ${\displaystyle X}$. Then, one can even define a piecewise smooth manifold via a piecewise smooth atlas. We can even define an inclusion functor ${\displaystyle {\textbf {PL}}\to {\textbf {PDiff}}}$ which will be faithful and essentially surjective, but it won't be full. For example, one can come with examples of piecewise smooth functions ${\displaystyle \mathbb {R} \to \mathbb {R} }$ which can be approximated arbitrarily well by piecewise linear functions but never quite made piecewise linear no matter how much you subdivide.

I don't believe the content of this page is true and I believe it should be corrected so that it does not lead people astray. The Surgeon of Death (talk) 16:04, 18 April 2026 (UTC)[reply]

If we work up to homotopy, then I do believe there is an equivalence of categories. The Surgeon of Death (talk) 16:10, 18 April 2026 (UTC)[reply]