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Please let me denote the following

we consider the category of manifolds and their maps based on what I learned from Wikipedia above links.

Is it true that the above categories have the following relations:

(1) TOP $\supseteq$ TRI ?

Namely, every TRI must be TOP manifolds?

(2) TRI $\supseteq$ PL ?

Namely, every PL must be TRI manifolds?

(3) TRI $\supseteq$ DIFF ?

Namely, every DIFF must be TRI manifolds?

(4) PL $\supseteq$ DIFF ?

Namely, every DIFF must be PL manifolds?

(5) So in a short summary, is it true that

$$\text{ TOP $\supseteq$ TRI $\supseteq$ PL $\supseteq$ DIFF} ?$$

(If what I said in (5) is false, what are their intersections, unions and complements of these categories?)

p.s. This is based on an improved unanswer question from MSE a week ago. I am sorry I hope more experts can fill in this question. Thanks! <3

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    $\begingroup$ Perhaps include a link to the MSE question. i.e. An question that is unanswered on MSE may still be closed here. $\endgroup$
    – Ryan Budney
    Commented Jul 4, 2019 at 1:20
  • $\begingroup$ thanks, vote up, a more basic MSE question is math.stackexchange.com/questions/3278365 $\endgroup$
    – annie marie cœur
    Commented Jul 4, 2019 at 18:29
  • $\begingroup$ There is no such thing as a PDIFF manifold. PDIFF is only a type of map from a PL manifold to a Diff manifold. Also, while people say that DIFF$\subset$PL, that is technically false and the true statement is subtle. Finally, no one considers TRI as a category. What are the morphisms? Is the double suspension of a homology sphere an object in this category not isomorphic to the sphere? Maybe they should consider a category, but probably they shouldn't. $\endgroup$
    – Ben Wieland
    Commented Sep 6, 2019 at 15:28

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