A reason for the success of the C++ standard template library (STL) is the concept of iterators, which allow for the processing of different kinds of data structures in a uniform way. In this paper, Heinzl generalizes this idea to the application area of physical simulation, by providing a unifying theoretical framework and corresponding C++ library for the representation of spatial domain decompositions.

The core idea is to apply concepts from algebraic topology to the structured representation of spatial cells, using their topological properties; thus, a three-dimensional (3D) cell is related to its two-dimensional (2D) facets, which are related to their one-dimensional (1D) boundary lines, which are ultimately related to their corner points. This representation is applied to complexes of cells whose topologies are described by the adjacency of each element. For example, topological operators decompose a complex into its boundary elements or evaluate real-valued functions over a complex. Based on these concepts, a C++ template library is developed that allows the definition of various kinds of representations of n-dimensional spaces that can be processed in a uniform way by topological operators.

The paper presents an interesting and useful approach. Unfortunately, it is not easily accessible without prior background knowledge. This is partially due to sub-optimal organization; in particular, the introduction does not describe well the concrete context of the work, and some basic notions are used before they are defined. However, the paper refers to a number of more detailed resources, in particular Heinzl’s dissertation. I advise the reader to consult these for more information.