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A question on relation of different triangulations of a triangulable space

Suppose we get two triangulations of a manifold with boundary $M$ such that the triangulation is compatible with boundary, i.e. the restriction on the boundary is itself a triangulation, is it these ...
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Category of Manifolds and Maps: TOP $\supseteq$ TRI $\supseteq$ PL $\supseteq$ DIFF? [closed]

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) ...