In mathematics, specifically algebraic topology, the mapping cylinder[1] of a continuous function between topological spaces and is the quotient
where the denotes the disjoint union, and ~ is the equivalence relation generated by
That is, the mapping cylinder is obtained by gluing one end of to via the map . Notice that the "top" of the cylinder is homeomorphic to , while the "bottom" is the space . It is common to write for , and to use the notation or 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 , obtained by collapsing one end of the cylinder to a point. Mapping cylinders are central to the definition of cofibrations.