nLab cylinder object



Homotopy Theory

!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 XX the cylinder X×[0,1]X \times [0,1] over XX, where [0,1][0,1] is the standard topological interval. It is notably used to define the concept of left homotopy, say in a model category.

The standard topological cylinder X×[0,1]X \times [0,1] naturally comes equipped with a continuous map

XXX×[0,1] X \coprod X \to X \times [0,1]

that identifies XX as the two ends X×{0}X \times \{0\} and X×{1}X \times \{1\} of the cylinder, and with a map

X×[0,1]X X \times [0,1] \to X

that collapses the cylinder back onto XX.

The composite of these two maps is the codiagonal (Id,Id):XXX(Id,Id) : X \coprod X \to X. Moreover, the cylinder X×[0,1]X \times [0,1] is homotopy equivalent to XX.

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.


In a category with weak equivalences CC that has coproducts a cylinder object Cyl(X)Cyl(X) for an object XX is a factorization

XXCyl(X)X X \coprod X \to Cyl(X) \stackrel{\simeq}{\to} X

of the codiagonal XXXX \coprod X \to X out of the coproduct of XX with itself, such that Cyl(X)XCyl(X) \to X is a weak equivalence and such that the morphism XXCyl(X)X \coprod X \to Cyl(X) 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 CC has the structure of a model category then “nice” means that XXCyl(X)X \coprod X \hookrightarrow Cyl(X) 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 Cyl(X)XCyl(X) \to X is an acyclic fibration. Cylinder objects such that XXCyl(X)X \coprod X \hookrightarrow Cyl(X) is a cofibration are sometimes called good, and those for which moreover Cyl(X)XCyl(X) \to X is an acyclic fibration are then called very good.



The general definition in model categories is due to:

Lecture notes:

The precise argument that for XX a cell complex then also the standard cyclinder X×IX\times I is a cell complex is spelled out in

  • Ottina, Prop. 2.9 in: An A-based cofibrantly generated model category (arXi:1405.2086)

Last revised on July 13, 2021 at 11:30:19. See the history of this page for a list of all contributions to it.