In algebraic topology and homotopy theory, a cocylinder is a dual construction to a cylinder. In contexts where spatial intuition is involved, it is perhaps more often called a path space or a path space object. In general, however, a cocylinder, , may not involve any object nor use a mapping space in its construction, see cylinder functor for the discussion of the dual point.
Similarly, the mapping cocylinder, which is dual to the mapping cylinder, is equally called the mapping path space or mapping path fibration. It provides a canonical way to factor any map as a homotopy equivalence followed by a fibration.
For a topological space , its cocylinder is simply the path space . More generally, in a cartesian closed category with an interval object , the cocylinder of an object is the exponential object . Even more generally, in a model category the cocylinder of any object is the path space object — the factorization of the diagonal morphism as an acyclic cofibration followed by a fibration.
In any of these cases:
where is the cocylinder.
The mapping cocylinder is sometimes denoted or .
If we interchange and then we have an upside-down version of a cylinder, sometimes called inverse (or inverted) mapping cocylinder; but usually it is clear just from the context which version is used. They are homotopy equivalent, so usually it does not matter.
In homotopy type theory the mapping cocylinder is expressed as
In the case of topological spaces, the mapping cocylinder is the subspace whose elements are pairs such that .
In homotopy type theory, cocylinders represent identity types, and the mapping cocylinder represents the dependent type . This is used crucially in the definition of equivalence in homotopy type theory.
The mapping cocylinder is the central ingredient in the factorization lemma.
One usage is discussed at Hurewicz connection.
The mapping path fibration is used in the construction of the Strøm model structure on topological spaces.
(This uses the terminology mapping path space.)