Please let me denote the following
- (TOP) topological manifolds https://en.wikipedia.org/wiki/Topological_manifold
- (PDIFF), for piecewise differentiable https://en.wikipedia.org/wiki/PDIFF
- (PL) piecewise-linear manifolds https://en.wikipedia.org/wiki/Piecewise_linear_manifold#Smooth_manifolds
- (DIFF) the smooth manifolds https://en.wikipedia.org/wiki/Differentiable_manifold#Definition
- (TRI) triangulable manifolds https://en.wikipedia.org/wiki/Triangulation_%28topology%29
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
