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