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.
Basic definition Let I {\displaystyle I} be a set of indices and C {\displaystyle C} be a family of sets ( U i ) i ∈ I {\displaystyle (U_{i})_{i\in I}} . The nerve of C {\displaystyle C} is a set of finite subsets of the index set I {\displaystyle I} . It contains all finite subsets J ⊆ I {\displaystyle J\subseteq I} such that the intersection of the U i {\displaystyle U_{i}} whose subindices are in J {\displaystyle J} is non-empty:
N ( C ) := { J ⊆ I : ⋂ j ∈ J U j ≠ ∅ , J finite set } . {\displaystyle N(C):={\bigg \{}J\subseteq I:\bigcap _{j\in J}U_{j}\neq \varnothing ,J{\text{ finite set}}{\bigg \}}.}
In Alexandrov's original definition, the sets ( U i ) i ∈ I {\displaystyle (U_{i})_{i\in I}} are open subsets of some topological space X {\displaystyle X} . The set N ( C ) {\displaystyle N(C)} may contain singletons (elements i ∈ I {\displaystyle i\in I} such that U i {\displaystyle U_{i}} is non-empty), pairs (pairs of elements i , j ∈ I {\displaystyle i,j\in I} such that U i ∩ U j ≠ ∅ {\displaystyle U_{i}\cap U_{j}\neq \emptyset } ), triplets, and so on. If J ∈ N ( C ) {\displaystyle J\in N(C)} , then any subset of J {\displaystyle J} is also in N ( C ) {\displaystyle N(C)} , making N ( C ) {\displaystyle N(C)} an abstract simplicial complex. Hence N(C) is often called the nerve complex of C {\displaystyle C} .
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