In this context, the whole sparse matrix is never created explicitly, though the decomposition of the matrix is stored. This approach was mainly used historically, when computers had little memory; in such implementations, only the front is in memory, while the factors in the decomposition are written into files.
For example, for matrices arising from the finite element method, one can structure the "assembly" of element matrices by assembling the matrix and eliminating equations only on a subset of elements at a time. This subset is called the front and it is essentially the transition region between the part of the system already finished and the part not touched yet.