We first describe the category $\mathcal{CT}^{k}$, $1\leq k\leq\omega$, whose objects $V$ are locally compact subsets of a Euclidean space possessing a finite, $C^{k}$ Whitney regular stratification \[ V=M_{1}\cup M_{2}\cup\cdot\cdot\cdot\cup M_{s}, \] with morphisms $\varphi:V\rightarrow W$ that are $C^{k}$ maps respecting the stratifications of $V$ and $W$. It is shown that this category is especially well suited to the formulation and solution of a wide range of problems in computational topology involving geometric objects that are piecewise differentiable manifolds or non-manifolds (varieties).

For one thing, the category $\mathcal{CT}^{k}$ can be seen to include just about all of the geometric objects that one is likely to encounter in computational toplogy applications. Moreover, this category enjoys the property of having analogs of many of the most important and useful structures and operations associated with differentiable manifolds. For example, it is demonstrated that the objects in $\mathcal{CT}^{k}$ have tubular-like neighborhoods, which can be used to reduce various shape invariant analyses to local calculations (in these neighborhoods). More particularly, they can be employed - as will be explained in a brief description of current joint work with R. Kopperman and T. Peters - to simplify certain questions pertaining to embedding equivalence or ambient isotopy to questions of homeomorphism equivalence in tubular-like neighborhoods.