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.
Alexandrov's definition.
Let "X" be a topological space. Let "formula_1" be an index set. Let "formula_2" be a family indexed by "formula_1" of open subsets of "X": formula_4. The nerve of "formula_2" is a set of finite subsets of the index-set "formula_1". It contains all finite subsets "formula_7" such that the intersection of the "Ui" whose subindices are in "J" is non-empty:
N("C") may contain singletons (elements "i" in "formula_1 "such that "Ui" is non-empty), pairs (pairs of elements of i,j in "formula_1" such that "Ui" intersects "Uj"), triplets, and so on. If "J" belongs to N"(C)", then any of its subsets is also in "N(C)". Therefore "N(C)" is an abstract simplicial complex and it is often called the nerve complex of "C".
Examples.
1. Let "X" be the circle S1 and "C" = {"U"1, "U"2}, where "U"1 is an arc covering the upper half of S1 and "U"2 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 S1). Then "N"(C) = { {1}, {2}, {1,2} }, which is an abstract 1-simplex.
2. Let "X" be the circle S1 and "C" = {"U"1, "U"2, "U"3}, where each "U"i is an arc covering one third of S1, with some overlap with the adjacent "Ui". Then "N"(C) = { {1}, {2}, {3}, {1,2}, {2,3}, {3,1} }. Note that {1,2,3} is not in N("C") since the common intersection of all three sets is empty.
The Čech nerve.
Given an open cover formula_11 of a topological space formula_12, or more generally a cover in a site, we can consider the pairwise fibre products formula_13, which in the case of a topological space are precisely the intersections formula_14. The collection of all such intersections can be referred to as formula_15 and the triple intersections as formula_16.
By considering the natural maps formula_17 and formula_18, we can construct a simplicial object formula_19 defined by formula_20, n-fold fibre product. This is the Čech nerve. 
By taking connected components we get a simplicial set, which we can realise topologically: formula_21.
Nerve theorems.
In general, the complex N("C") need not reflect the topology of "X" accurately. For example, we can cover any "n"-sphere with two contractible sets "U1" and "U2" that have a non-empty intersection, as in example 1 above. In this case, "N"(C) is an abstract 1-simplex, which is similar to a line but not to a sphere.
However, in some cases N("C") does reflect the topology of "X". For example, if a circle is covered by three open arcs, intersecting in pairs as in example 2 above, then N("C") is a 2-simplex (without its interior) and it is homotopy-equivalent to the original circle.
A nerve theorem (or nerve lemma) is a theorem that gives sufficient conditions on "C" guaranteeing that N("C") reflects, in some sense, the topology of "X". 
The basic nerve theorem of Leray says that, if any intersection of sets in "N(C)" is contractible , then N("C") is homotopy-equivalent to "X".
Another nerve theorem relates to the Čech nerve above: if formula_12 is compact and all intersections of sets in "C" are contractible or empty, then the space formula_21 is homotopy-equivalent to formula_12.
Homological nerve theorem.
The following nerve theorem uses the homology groups of intersections of sets in the cover. For each finite formula_25, denote formula_26 the "j"-th reduced homology group of formula_27.
If "HJ,j" is the trivial group for all "J" in the "k"-skeleton of N("C") and for all "j" in {0, ..., "k"-dim("J")}, then N("C") is "homology-equivalent" to "X" in the following sense: