PL, or more precisely PDIFF, sits between DIFF and TOP : it is categorically "better behaved" than DIFF — for example, the Generalized Poincaré conjecture is true in PL, but is false generally in DIFF — but is "worse behaved" than TOP, as elaborated in surgery theory.
Smooth manifolds have canonical PL structures — they are uniquely triangulizable, by Whitehead's theorem on triangulation — but PL manifolds do not always have smooth structures — they are not always smoothable. This relation can be elaborated by introducing the category PDIFF, which contains both DIFF and PL, and is equivalent to PL. One way in which PL is better behaved than DIFF is that one can take cones in PL, but not in DIFF — the cone point is acceptable in PL. A consequence is that the Generalized Poincaré conjecture is true in PL for dimensions greater than four — the proof is to take a homotopy sphere, remove two balls, apply the h-cobordism theorem to conclude that this is a cylinder, and then attach cones to recover a sphere. This last step works in PL but not in DIFF, giving rise to exotic spheres.
Topological manifolds
Not every topological manifold admits a PL structure, and of those that do, the PL structure need not be unique — it can have infinitely many. This is elaborated at Hauptvermutung. The Kirby–Siebenmann class is an obstruction for giving a topological manifold a PL-structure. The obstruction to placing a PL structure on a topological manifold is the Kirby–Siebenmann class. To be precise, the Kirby-Siebenmann class is the obstruction to placing a PL-structure on M x R and in dimensions n > 4 this ensures that M has a PL-structure.
Real algebraic sets
An A-structure on a PL manifold is a structure which gives an inductive way of resolving the PL manifold to a smooth manifold. Compact PL manifolds admit A-structures. Compact PL manifolds are homeomorphic to real-algebraic sets. Put another way, A-category sits over the PL-category as a richer category with no obstruction to lifting, that is BA → BPL is a product fibration with BA = BPL × PL/A, and PL manifolds are real algebraic sets because A-manifolds are real algebraic sets.
