In mathematics (especially algebraic topology and abstract algebra), homology is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object.

The procedure works as follows: Given the object X, one first defines a chain complex that encodes information about X. A chain complex is a sequence of abelian groups or modules A_{0}, A_{1}, A_{2}... connected by homomorphisms d_{n} : A_{n} > A_{n1}, such that the composition of any two consecutive maps is zero: d_{n} o d_{n+1} = 0 for all n. This means that the image of the n+1th map is contained in the kernel of the nth, and we can define the nth homology group of X to be the factor group (or factor module)
A chain complex is said to be exact if the image of the n+1th map is always equal to the kernel of the nth map. The homology groups of X therefore measure "how far" the chain complex associated to X is from being exact.
The motivating example comes from algebraic topology: the simplicial homology of a simplicial complex X. Here A_{n} is the free abelian group or module whose generators are the ndimensional oriented simplexes of X. The mappings are called the boundary mappings and send the simplex with vertices (a[1], a[2], ..., a[n]) to the sum of (1)^{i} (a[1], ..., a[i1], a[i+1], ..., a[n]) from i = 0 to i = n. If we take the modules to be over a field, then the dimension of the nth homology of X turns out to be the number of "holes" in X at dimension n.
Using this example as a model, one can define a simplicial homology for any topological space X. We define a chain complex for X by taking A_{n} to be the free abelian group (or free module) whose generators are all continuous maps from ndimensional simplices into X. The homomorphisms d_{n} arise from the boundary maps of simplices.
In abstract algebra, one uses homology to define derived functors[?], for example the Tor functors[?]. Here one starts with some covariant additive functor F and some module X. The chain complex for X is defined as follows: first find a free module F_{1} and a surjective homomorphism p_{1} : F_{1} > X. Then one finds a free module F_{2} and a surjective homomorphism p_{2} : F_{2} > ker(p_{1}). Continuing in this fashion, a sequence of free modules F_{n} and homorphisms p_{n} can be defined. By applying the functor F to this sequence, one obtains a chain complex; the homology H_{n} of this complex depends only on F and X and is, by definition, the nth derived functor of F, applied to X.
Chain complexes form a category: A morphism from the chain complex (d_{n} : A_{n} > A_{n1}) to the chain complex (e_{n} : B_{n} > B_{n1}) is a sequence of homomorphisms f_{n} : A_{n} > B_{n} such that f_{n1} o d_{n} = e_{n1} o f_{n} for all n. The nth homology H_{n} can be viewed as a covariant functor from the category of chain complexes to the category of abelian groups (or modules).
If the chain complex depends on the object X in a covariant manner (meaning that any morphism X > Y induces a morphism from X's chain complex to Y's), then the H_{n} are covariant functors from the category that X belongs to into the category of abelian groups (or modules).
The only difference between homology and cohomology is that in cohomology the chain complexes depend in a contravariant manner on X, and that therefore the homology groups (which are called cohomology groups in this context and denoted by H^{n}) form contravariant functors from the category that X belongs to into the category of abelian groups or modules.
If (d_{n} : A_{n} > A_{n1}) is a chain complex such that all but finitely many A_{n} are zero, and the others are finitely generated abelian groups (or finite dimensional vector spaces), then we can defined the Euler characteristic
Every short exact sequence
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