Singular chain complex

Definition with coefficients over integers (default, if no coefficients specified)
The singular chain complex (or total singular chain complex, to distinguish it from the normalized singular complex) associated with a topological space $$X$$ is defined as the following chain complex of abelian groups:

Variations

 * Augmented singular chain complex
 * Normalized singular chain complex
 * Relative singular chain complex

On the category of topological spaces
The total singular complex is a functor from the category of topological spaces with continuous maps to the category of chain complexes with chain maps. The functor associates to a continuous map $$f:X \to Y$$ to a map $$C_n(f):C_n(X) \to C_n(Y)$$ as follows. $$C_n(f)$$ sends a singular $$n$$-simplex $$\sigma$$ to $$f \circ \sigma$$, and more generally sends $$\sum a_\sigma \sigma$$ to $$\sum a_\sigma f\circ \sigma$$.

On the 2-category of topological spaces
Consider the 2-category of topological spaces with continuous maps and homotopies. Then the total singular complex is a 2-functor from this category to the 2-category of chain complexes with chain maps and chain homotopies.

This fact implies in particular that the homology of the total singular complex is homotopy-invariant.