Nerve complex

In topology, the nerve complex of a set family is an abstract complex that records the pattern of intersections between the sets in the family. It was introduced by Pavel Alexandrov and now has many variants and generalisations, among them the Čech nerve of a cover, which in turn is generalised by hypercoverings. It captures many of the interesting topological properties in an algorithmic or combinatorial way. Let be a set of indices and be a family of sets . The nerve of is a set of finite subsets of the index set . It contains all finite subsets such that the intersection of the whose subindices are in is non-empty: In Alexandrov's original definition, the sets are open subsets of some topological space . The set may contain singletons (elements such that is non-empty), pairs (pairs of elements such that ), triplets, and so on. If , then any subset of is also in , making an abstract simplicial complex. Hence N(C) is often called the nerve complex of . Let X be the circle and , where is an arc covering the upper half of and is an arc covering its lower half, with some overlap at both sides (they must overlap at both sides in order to cover all of ). Then , which is an abstract 1-simplex. Let X be the circle and , where each is an arc covering one third of , with some overlap with the adjacent . Then . Note that {1,2,3} is not in since the common intersection of all three sets is empty; so is an unfilled triangle. Given an open cover of a topological space , or more generally a cover in a site, we can consider the pairwise , which in the case of a topological space are precisely the intersections . The collection of all such intersections can be referred to as and the triple intersections as . By considering the natural maps and , we can construct a simplicial object defined by , n-fold fibre product. This is the Čech nerve. By taking connected components we get a simplicial set, which we can realise topologically: . The nerve complex is a simple combinatorial object.