We provide a mathematical foundation for a biologically inspired hierarchical identification and classification system. We describe lines detected by the viual cortex in terms of natural basis elements, upon which simplicial complexes are erected to describe simple objects. Simple objects are basis elements in the next level of the hierarchy, and objects are descirbed simply in terms of them. This is continued through several levels of the hierarchy, and each level identifies somewhat more complex objects onto a manifold the previous level. Projections of the identified objects onto a manifold provides a natural and useful representation of the system's environment.