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In summary, PDiff is more general than Diff because it allows pieces (corners), and one cannot in general smooth corners, while PL is no less general that PDiff because one can linearize pieces (more precisely, one may need to break them up into smaller pieces and then linearize, which is allowed in
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a PDiff manifold. Thus, going from Diff to PDiff and PL to PDiff are natural – they are just inclusion. The map PL to PDiff, while not an equality – not every piecewise smooth function is piecewise linear – is an equivalence: one can go backwards by linearize pieces. Thus it can for some purposes be
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However, while a smooth manifold is not a PL manifold, it carries a canonical PL structure – it is uniquely triangularizable; conversely, not every PL manifold is smoothable. For a particular smooth manifold or smooth map between smooth manifolds, this can be shown by breaking up the manifold into
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This relation between Diff and PL requires choices, however, and is more naturally shown and understood by including both categories in a larger category, and then showing that the inclusion of PL is an equivalence: every smooth manifold and every PL manifold
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PDIFF is mostly a technical point: smooth maps are not piecewise linear (unless linear), and piecewise linear maps are not smooth (unless globally linear) – the intersection is
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small enough pieces, and then linearizing the manifold or map on each piece: for example, a circle in the plane can be approximated by a triangle, but not by a
195:). The result is elementary and rather technical to prove in detail, so it is generally only sketched in modern texts, as in the brief proof outline given in (
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between them), and the reason it is defined is to allow one to relate these two categories. Further, piecewise functions such as
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113:(because not based) – so they cannot directly be related: they are separate generalizations of the notion of an affine map.
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These categories all sit inside TOP, the category of topological manifold and continuous maps between them.
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are piecewise-smooth, hence in PDIFF, but not globally smooth or piecewise-linear, hence not in DIFF or PL.
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are common in mathematics, and PDIFF provides a category for discussing them.
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187:) manifold has a unique PL structure was originally proven in (
164:{\displaystyle {\text{Diff}}\to {\text{PDiff}}\to {\text{PL}}.}
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PDIFF serves to relate DIFF and PL, and it is equivalent to PL.
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inverted, or considered an isomorphism, which gives a map
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between them. It properly contains DIFF (the category of
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203:), while a short but detailed proof is given in (
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273:, Annals of Mathematics Studies, vol. 54,
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327:, PDIFF described as "piecewise smooth"
311:Three-Dimensional Geometry and Topology
239:"Re: PL and DIFF manifolds: a question"
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73:between them) and PL (the category of
16:Category of piecewise-smooth manifolds
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199:). A very brief outline is given in (
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224:Whitehead Triangulations (Lecture 3)
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271:Elementary Differential Topology
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183:That every smooth (indeed,
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337:: CS1 maint: postscript (
315:Princeton University Press
297:: CS1 maint: postscript (
275:Princeton University Press
75:piecewise linear manifolds
356:The Annals of Mathematics
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221:(February 13, 2009),
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61:and piecewise-smooth
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349:(October 1940). "On
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109:, or more precisely
347:Whitehead, J. H. C.
46:erentiable, is the
395:Geometric topology
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32:geometric topology
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359:. Second Series.
307:Thurston, William
267:Munkres, James R.
247:sci.math.research
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324:978-0-69108304-9
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87:polygonal chains
71:smooth functions
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237:(21 Aug 1997).
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253:on Apr 8, 2013
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255:. Retrieved
251:the original
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219:Lurie, Jacob
193:Munkres 1966
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111:affine maps
107:linear maps
79:linear maps
211:References
205:Lurie 2009
93:Motivation
243:Newsgroup
151:→
143:→
59:manifolds
52:piecewise
42:iecewise
389:Category
333:citation
309:(1997),
293:citation
269:(1966),
175:PDiff).
48:category
377:1968861
257:May 10,
245::
179:History
83:splines
24:Splines
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321:
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56:smooth
36:PDIFF,
373:JSTOR
228:(PDF)
147:PDiff
119:2-gon
339:link
319:ISBN
299:link
279:ISBN
259:2012
139:Diff
85:and
69:and
63:maps
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38:for
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155:PL
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40:p
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