Singular chain complex

Definition

Symbol-free definition

The singular complex (or total singular complex, to distinguish it from the normalized singular complex) associated with a topological space is defined as the following chain complex of Abelian groups:

• The $n^{th}$ member of this complex is the $n^{th}$ chain group, or the group of singular n-chains. This is essentially the free Abelian group on the set of all singular n-simplices.

• The boundary map goes from the $n^{th}$ chain group to the $(n-1{)}^{th}$ chain group, and it essentially sends each singular simplex to a signed sum of its codimension one faces.

Variations

• Reduced total singular complex
• Normalized singular complex
• Relative singular complex

Functoriality

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 $\sigma <math>to<math>f\circ \sigma$, 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.