Nerve of a covering

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. and now has many variants and generalisations, among them the Čech nerve of a cover, which in turn is generalised by hypercoverings.

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

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. 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.

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.

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