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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In brief, singular homology is constructed by taking maps of the standard n-simplex to a topological space, and composing them into formal sums, called singular chains. The boundary operation on a simplex induces a singular chain complex. The singular homology is then the homology of the chain complex. The resulting homology groups are the same for all homotopically equivalent spaces, which is the reason for their study. These constructions can be applied to all topological spaces, and so singular homology can be expressed in terms of category theory, where the homology group becomes a functor from the category of topological spaces to the category of graded abelian groups. These ideas are developed in greater detail below.