# Nerve of a covering

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Constructing the nerve of an open good cover containing 3 sets in the plane.

In topology, the nerve of an open covering is a construction of an abstract simplicial complex from an open covering of a topological space X that captures many of the interesting topological properties in an algorithmic or combinatorial way. It was introduced by Pavel Alexandrov[1] and now has many variants and generalisations, among them the Čech nerve of a cover, which in turn is generalised by hypercoverings.

## Alexandrov's definition

Given an index set I, and open sets Ui contained in X, the nerve N is the set of finite subsets of I defined as follows:

• a finite set JI belongs to N if and only if the intersection of the Ui whose subindices are in J is non-empty,
${\displaystyle \bigcap _{j\in J}U_{j}\neq \varnothing .}$

Obviously, if J belongs to N, then any of its subsets is also in N. Therefore N is an abstract simplicial complex.

In general, the complex N need not reflect the topology of X accurately. For example we can cover any n-sphere with two contractible sets U and V, in such a way that N is an abstract 1-simplex. However, if we also insist that the open sets corresponding to every intersection indexed by a set in N are also contractible, the situation changes. In particular, the nerve lemma states that if ${\displaystyle {\mathcal {U}}=\{U_{i}\}_{i\in I}}$ is a good cover, in that for each ${\displaystyle \sigma \subset I}$, the set ${\displaystyle \bigcap _{i\in \sigma }U_{i}}$ is contractible if it is nonempty, then the nerve ${\displaystyle N({\mathcal {U}})}$ is homotopy equivalent to ${\displaystyle \bigcup _{i\in I}U_{i}}$. [2]

This means for instance that a circle covered by three open arcs, intersecting in pairs in one arc, is modelled by a homeomorphic complex, the geometrical realization of N.

## The Čech nerve

Given an open cover ${\displaystyle {\mathcal {U}}=\{U_{i}\}}$ of a topological space ${\displaystyle X}$, or more generally a cover in a site, we can regard the pairwise fibre products ${\displaystyle U_{ij}=U_{i}\times _{X}U_{j}}$, which in the case of a topological space is precisely the intersection ${\displaystyle U_{i}\cap U_{j}}$. The collection of all such intersections can be referred to as ${\displaystyle {\mathcal {U}}\times _{X}{\mathcal {U}}}$ and the triple intersections as ${\displaystyle {\mathcal {U}}\times _{X}{\mathcal {U}}\times _{X}{\mathcal {U}}}$. By considering the natural maps ${\displaystyle U_{ij}\to U_{i}}$ and ${\displaystyle U_{i}\to U_{ii}}$, we can construct a simplicial object ${\displaystyle C({\mathcal {U}})_{\bullet }}$ defined by ${\displaystyle C({\mathcal {U}})_{n}={\mathcal {U}}\times _{X}\cdots \times _{X}{\mathcal {U}}}$, n-fold fibre product. This is the Čech nerve, and by taking connected components we get a simplicial set, which we can realise topologically: ${\displaystyle |C(\pi _{0}({\mathcal {U}}))|}$. If the covering and the space are sufficiently nice, for instance if ${\displaystyle X}$ is compact and all intersections of sets in the cover are contractible or empty, then this space is weakly equivalent to ${\displaystyle X}$. This is known as the nerve theorem.