Mapping cylinder

In mathematics, specifically algebraic topology, the mapping cylinder^[1] of a continuous function ${\displaystyle f}$ between topological spaces ${\displaystyle X}$ and ${\displaystyle Y}$ is the quotient

${\displaystyle M_{f}=(([0,1]\times X)\amalg Y)\,/\,\sim }$

where the ${\displaystyle \amalg }$ denotes the disjoint union, and ~ is the equivalence relation generated by

${\displaystyle (0,x)\sim f(x)\quad {\text{for each }}x\in X.}$

That is, the mapping cylinder ${\displaystyle M_{f}}$ is obtained by gluing one end of ${\displaystyle X\times [0,1]}$ to ${\displaystyle Y}$ via the map ${\displaystyle f}$. Notice that the "top" of the cylinder ${\displaystyle \{1\}\times X}$ is homeomorphic to ${\displaystyle X}$, while the "bottom" is the space ${\displaystyle f(X)\subset Y}$. It is common to write ${\displaystyle Mf}$ for ${\displaystyle M_{f}}$, and to use the notation ${\displaystyle \sqcup _{f}}$ or ${\displaystyle \cup _{f}}$ for the mapping cylinder construction. That is, one writes

${\displaystyle Mf=([0,1]\times X)\cup _{f}Y}$

with the subscripted cup symbol denoting the equivalence. The mapping cylinder is commonly used to construct the mapping cone ${\displaystyle Cf}$, obtained by collapsing one end of the cylinder to a point. Mapping cylinders are central to the definition of cofibrations.