Singular chain complex functor

Definition

The singular chain complex functor is a functor from the category of topological spaces with continuous maps to the category of chain complexes with chain maps defined as follows.

Aspect	Input	Output
Objects	topological space $X$	the singular chain complex $C_{*}(X)$
Morphisms	continuous map $f:X\to Y$	We first define an induced map $S_{n}(X)\to S_{n}(Y)$ between the sets of singular n-simplices. The map is defined as follows: $g\in S_{n}(X)$ (a continuous map from the standard n-simplex to $X$) is sent to $f\circ g$, which is an element of $S_{n}(Y)$. We extend this to a group homomorphism from $C_{n}(X)$ to $C_{n}(Y)$ using the fact that $C_{n}(X)$ and $C_{n}(Y)$ are the free abelian groups on $S_{n}(X)$ and $S_{n}(Y)$ respectively. This homomorphism is what we denote by $C_{n}(f)$ $C(f)$ is the bunch of $C_{n}(f)$ maps.

To show that this is well-defined, we need to show the following:

Compatibility condition in broad terms	Compatibility condition in explicit terms	Why it is true
well defined, i.e., $C(f)$ is a chain map	$\partial _{n}\circ C_{n}(f)=C_{n-1}(f)\circ \partial _{n}$	See continuous map of topological spaces induces chain map of corresponding singular chain complexes
identity goes to identity	$C_{n}(id)=id$ for all $n$, where $id$ represents the identity map.	Direct from definition
composites go to composites	$C_{n}(f\circ g)=C_{n}(f)\circ C_{n}(g)$ where $f,g$ are composable continuous maps.	composite of continuous maps induces composite of chain maps