Double mapping cylinder

Definition
Suppose $$X,Y,Z$$ are topological spaces and $$f:X \to Z$$ and $$g: X \to Z$$ are continuous maps. The double mapping cylinder of $$f$$ and $$g$$ is defined as the quotient of $$X \times [0,1] \sqcup Y \sqcup Z$$ via the relations $$(x,0) \simeq f(x)$$ and $$(x,1) \simeq g(x)$$.

More specific constructions
Specific cases of the above arise either by setting $$X = Y$$ and $$f$$ the identity map (or correspondingly for $$Z$$ and $$g$$) or setting $$Y$$ or $$Z$$ to be a one-point space. If we impose only one constraint, the resultant construction is the construction corresponding to the other unspecified map. If we impose two constraints, then the resulting construction depends only on the input space $$X$$.

Note that the roles of $$Y$$ and $$Z$$ can be interchanged here.

The join
The join of two spaces $$A$$ and $$B$$ can be constructed as a double mapping cylinder as follow: Set $$X = A \times B$$, $$Y = A$$ and $$Z = B$$, and let $$f,g$$ be the coordinate projection maps.

Generalizations

 * Mapping torus

Related notions

 * Mapping telescope

Facts
There is a relation between the homology of the double mapping cylinder of $$f$$ and $$g$$, and the homologies of the spaces $$X$$, $$Y$$ and $$Z$$. The relation is described by the exact sequence for double mapping cylinder.