nLab cylinder object

Idea

The concept of a cylinder object in a category is an abstraction of the construction in Top which associates to any topological space $X$ the cylinder $X \times [0,1]$ over $X$, where $[0,1]$ 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 $X \times [0,1]$ naturally comes equipped with a continuous map

$$X \coprod X \to X \times [0,1]$$

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

$$X \times [0,1] \to X$$

that collapses the cylinder back onto $X$.

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

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.

Definition

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

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

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

Examples

• In sSet equipped with the standard model structure on simplicial sets the standard cylinder object for any $S \in sSet$ is $S \times \Delta[1]$.

• In Top, the standard cylinder $X\times [0,1]$ is a cylinder object for both the classical model structure on topological spaces $Top_{Quillen}$ (the one with Serre fibrations) as well as for the Strøm model structure $Top_{Strom}$ (the one with Hurewicz fibrations).

  This standard cylinder is generally a “good cylinder” in the above sense only for $Top_{Stron}$ (in which case it is in fact a “very good cylinder”).

  In $Top_{Quillen}$ a sufficient condition for the standard cylinder $X\times I$ to be good is that $X$ is a CW-complex.

• In any elementary topos, the Lawvere cylinder is a sort of minimal cylinder functor

Related concepts

• cylinder functor

• left homotopy

• mapping cylinder

• path space object

• reduced cylinder

• cylinder spectrum

• h-cobordism

References

The general definition in model categories is due to:

• Daniel Quillen, Section I.1, Def. 4, p. 9 (15 of 165) in: Axiomatic homotopy theory in: Homotopical Algebra, Lecture Notes in Mathematics 43, Springer 1967 (doi:10.1007/BFb0097438)

Lecture notes:

• Introduction to Homotopy Theory, around this Def.

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

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