nLab
cylinder functor

Definition

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

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

equipped with three natural transformations

$$e_0,e_1:{\mathrm{Id}}_C\to (-)\times I$$e_0, e_1 : Id_C \to (-)\times I

$$\sigma :(-)\times I\to {\mathrm{Id}}_C.$$\sigma : (-)\times I \to 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.

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).