In generalization to how a topological space has a fundamental groupoid whose morphisms are homotopy-classes of paths in and whose composition operation is the concatenation of paths, a directed space has a fundamental category whose morphisms are directed paths in .
A stratified space has a ‘fundamental n-category with duals’, which generalizes the fundamental n-groupoid of a plain old space. When a path crosses a codimension- stratum, “something interesting happens” – i.e., a catastrophe. So, we say such a path gives a noninvertible morphism. The idea is that going along such a path and then going back is not “the same” as having stayed put. So, going back along such a path is not its inverse, just its dual.
The left adjoint of the nerve functor , which takes a simplicial set to a category, is sometimes called the fundamental category functor. One notation for it is . Explicitly, for a simplicial set , is the category freely generated by the directed graph whose vertices are 0-simplices of and whose edges are 1-simplices (the source and target are defined by the face maps), modulo the relations for and for . Here and denote the degeneracy and face maps, respectively.
fundamental category, fundamental (∞,1)-category