PDIFF
In geometric topology, PDIFF, for piecewise differentiable, is the category of piecewise-smooth manifolds and piecewise-smooth maps between them. It properly contains DIFF (the category of smooth manifolds and smooth functions between them) and PL (the category of piecewise linear manifolds and piecewise linear maps between them), and the reason it is defined is to allow one to relate these two categories. Further, piecewise functions such as splines and polygonal chains are common in mathematics, and PDIFF provides a category for discussing them. (Wikipedia).
