- * indices of simplices resp. their faces are of type IT. To each simplex a
- * value is assigend, which is of type VT. When a simplicial_complex is
- * instantiated, a single (-1) dimensional simplex is automatically created.
- * Each 0-dimensional simplex automatically has this simplex as its face.
- * Consequently, the innner class simplex_order gives the extended boundary
- * matrix. */
-template<int MAXDIM, class IT, class VT>
+ * indices of simplices and their faces are of type IT. When a
+ * simplicial_complex is instantiated, a single (-1) dimensional simplex is
+ * automatically created. (When we compute persistent homology, we actually
+ * compute reduced homology.) Each 0-dimensional simplex automatically has this
+ * simplex as its face.
+ *
+ * Based on a simplicial complex, we define a simplicial order. An order
+ * can be seen as a permutation of the simplices of the complex. If the order
+ * has the property that every prefix of the permutation is again a complex,
+ * i.e., all faces of all simplices of each prefix are contained in the prefix,
+ * then we call it a filtration. The innner class simplex_order gives the
+ * extended boundary matrix, which can then be used in the persistence class to
+ * compute persistent homology.
+ *
+ * One way to obtain a filtration is to define a monotone simplicial function
+ * on the complex. That is, each simplex gets a value of type VT assigned. Then
+ * one can obtain a sub-level set filtration from the complex w.r.t. to this
+ * simplicial function.
+ */
+template<int MAXDIM, class IT>