construction makes simplicial homology a functor from simplicial complexes to abelian groups. This is essential to applications of the theory, including the Brouwer fixed point theorem and the topological invariance of simplicial homology. Related homologies Singular homology is a related theory that is better adapted to theory rather than computation. Singular homology is defined for all topological spaces and depends only on the topology, not any triangulation; and it agrees with simplicial homology for spaces which can be triangulated. Nonetheless, because it is possible to compute the simplicial homology of a simplicial complex automatically and efficiently, simplicial homology has become important for application to real-life situations, such as image analysis, medical imaging, and data analysis in general. Another related theory is Cellular homology. Applications A standard scenario in many computer applications is a collection of points (measurements, dark pixels in a bit map, etc.) in which one wishes to find a topological feature. Homology can serve as a qualitative tool to search for such a feature, since it is readily computable from combinatorial data such as a simplicial complex. However, the data points have to first be triangulated, meaning one replaces the data with a simplicial complex approximation. Computation of persistent homology involves analysis of homology at different resolutions, registering homology classes (holes) that persist as the resolution is changed. Such features can be used to detect structures of molecules, tumors in X-rays, and cluster structures in complex data. More generally, simplicial homology plays a central role in topological data analysis, a technique in the field of data mining. Implementations A MATLAB toolbox for computing persistent homology, Plex (Vin de Silva, Gunnar Carlsson), is available at this site. Stand-alone implementations in C++ are available as part of the Perseus, Dionysus and PHAT software projects. For Python, there are libraries such as scikit-tda, Persim, giotto-tda and GUDHI, the latter aimed at generating topological features for machine learning. These can be found at the PyPI repository. See also Simplicial homotopy References External links Topological methods in scientific computing Computational homology (also cubical homology) Computational topology