nLab
cylinder functor

Definition

For $C$ a category, a cylinder functor on $C$ is a functor denoted

$$(-)\times I : C \to C$$

equipped with three natural transformations

$$e_0, e_1 : Id_C \to (-)\times I$$

$$\sigma : (-)\times I \to Id_C$$

such that $\sigma e_0 = \sigma e_1 = Id_C$.

Remarks

• A cylinder functor functorially provides cylinder objects used for talking about homotopy.

• The notation is supposed to be suggestive of a product with an object $I$. While this is the motivating example, the interval functor need not be of that form.

Cylindrical model structures

Richard Williamson has developed a way to build a model structure from the simple point of departure of a structured interval in a monoidal category - more generally, a structured cylinder and a structured co-cylinder in a category. This is given in his thesis and also in the ArXiv article listed below.

References

A very brief introduction to cylinder functors is given starting on page 9 of Abstract Homotopy Theory.

A fuller development of their properties is given in

• K. H. Kamps and T. Porter, Abstract Homotopy and Simple Homotopy Theory (GoogleBooks)

Cylinder functors also form one of the key elements in Baues’ approach to algebraic homotopy:

• H. J. Baues: Algebraic Homotopy, Cambridge studies in advanced mathematics 15, Cambridge University Press, (1989).
• H. J. Baues: Combinatorial Homotopy and 4-Dimensional Complexes, de Gruyter Expositions in Mathematics 2, Walter de Gruyter, (1991).
• H. J. Baues: Homotopy Types, in I.M.James, ed., Handbook of Algebraic Topology, 1–72, Elsevier, (1995).

Cylindrical model structures are discussed in

• Cylindrical model structures: ArXiv 3104.0867