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