## Definition

Suppose are topological spaces and and are continuous maps. The **double mapping cylinder** of and is defined as the quotient of via the relations and .

## Particular cases

- Mapping cylinder: Here and is the identity map
- Mapping cone: Here is a one-point space and is the map to that one point
- Join: The join of spaces and is the double mapping cylinder where , , and the maps are simply projections onto the coordinates