# liftMapAlongQuasiIsomorphism(ComplexMap,ComplexMap) -- lift a map of chain complexes along a quasi-isomorphism

## Synopsis

• Function: liftMapAlongQuasiIsomorphism
• Usage:
f' = liftMapAlongQuasiIsomorphism(f, g)
f' = f // g
• Inputs:
• f, , where each term in the source of $f$ is a free module
• g, , a quasi-isomorphism having the same target as $f$
• Outputs:
• f', , a map from the source of $f$ to the source of $g$
• Consequences:
• the homotopy relating $f$ and $g \circ f'$ is available as homotopyMap f'.

## Description

Let $f \colon P \to C$ be a morphism of chain complexes, where each term in $P$ is a free module. Given a quasi-isomorphism $g \colon B \to C$, this method produces a morphism $f' \colon P \to B$ such that there exists a map $h \colon P \to C$ of chain complexes having degree $1$ satisfying

$f - g \circ f' = h \circ \operatorname{dd}^P + \operatorname{dd}^C \circ h$.

Given a morphism between complexes, we can construct the corresponding map between their free resolutions using this method.

To be more precise, given a morphism $\phi \colon B \to C$ of complexes, let $\alpha \colon P \to B$ and $\beta \colon F \to C$ denote the free resolutions of the source and target complexes. Lifting the composite map $\phi \circ \alpha$ along the quasi-isomorphism $\beta$ gives a commutative diagram $\phantom{WWWW} \begin{array}{ccc} P & \!\!\rightarrow\!\! & F \\ \downarrow \, {\scriptstyle \alpha} & & \downarrow \, {\scriptstyle \beta} \\ B & \xrightarrow{\phi} & C \end{array}$

The following three assumptions are not checked: $f$ is a morphism, the source of $f$ is semifree, and $g$ is a quasi-isomorphism.