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In mathematics, specifically algebraic topology, the mapping cylinder of a continuous function f {\displaystyle f} between topological spaces X {\displaystyle X} and Y {\displaystyle Y} is the quotient
where the ⨿ {\displaystyle \amalg } denotes the disjoint union, and ∼ is the equivalence relation generated by
That is, the mapping cylinder M f {\displaystyle M_{f}} is obtained by gluing one end of X × {\displaystyle X\times } to Y {\displaystyle Y} via the map f {\displaystyle f}. Notice that the "top" of the cylinder { 1 } × X {\displaystyle \{1\}\times X} is homeomorphic to X {\displaystyle X} , while the "bottom" is the space f ⊂ Y {\displaystyle f\subset Y}. It is common to write M f {\displaystyle Mf} for M f {\displaystyle M_{f}} , and to use the notation ⊔ f {\displaystyle \sqcup _{f}} or ∪ f {\displaystyle \cup _{f}} for the mapping cylinder construction. That is, one writes
with the subscripted cup symbol denoting the equivalence. The mapping cylinder is commonly used to construct the mapping cone C f {\displaystyle Cf} , obtained by collapsing one end of the cylinder to a point. Mapping cylinders are central to the definition of cofibrations.