!include homotopy - contents?
The concept of a cylinder object in a category is an abstraction of the construction in Top which associates to any topological space the cylinder over , where is the standard interval. It is notably used to define the concept of left homotopy, say in a model category.
The standard topological cylinder naturally comes equipped with a continuous map
that identifies as the two ends and of the cylinder, and with a map
that collapses the cylinder back onto .
These properties are the characterizing properties of the cylinder that can be abstracted and realized in other categories.
The notion dual to cylinder object is path space object, which is thus sometimes alternatively called a cocylinder. Cylinder objects and path space objects are used to define left homotopies and right homotopies, respectively.
There are several views on the role of cylinders / cocylinders in homotopy theory. If there is a natural notion of weak equivalence or quasi-isomorphism then the cylinder is used to encode a notion of homotopy equivalence compatible with the weak equivalences. In some other situations, a ‘cylinder’ , often functorially given and well structured in some way, may be the primitive notion that allows a notion of ‘homotopy equivalence’ to be put forward. Below we give a definition optimised for the former situation. Some indication of the second context is given in the entry cylinder functor.
of the codiagonal out of the coproduct of with itself, such that is a weak equivalence and such that the morphism is “nice” in some way.
In some situations the assignment of cylinder objects may exist functorially, in which case one speaks of a cylinder functor.
If has the structure of a model category then “nice” means that is a cofibration. The factorization axiom of a model category ensures that for each object there is a cylinder object with this property; in fact, one with the additional property that is an acyclic fibration. Cylinder objects such that is a cofibration are sometimes called good, and those for which moreover is an acyclic fibration are then called very good.
In Top, the standard cylinder is a cylinder object for both the classical model structure on topological spaces (the one with Serre fibrations) as well as for the Strøm model structure (the one with Hurewicz fibrations).
This standard cylinder is generally a “good cylinder” in the above sense only for (in which case it is in fact a “very good cylinder”).
In a sufficient condition for the standard cylinder to be good is that is a CW-complex.
The precise argument that for a cell complex then also the standard cyclinder is a cell complex is spelled out for instance as prop. 2.9 in