nLab
cylinder functor

Definition

For CC a category, a cylinder functor on CC is a functor denoted

()×I:CC (-)\times I : C \to C

equipped with three natural transformations

e 0,e 1:Id C()×I e_0, e_1 : Id_C \to (-)\times I
σ:()×IId C \sigma : (-)\times I \to Id_C

such that σe 0=σe 1=Id C\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 II. 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

Revised on April 7, 2014 01:08:19 by Tim Porter (2.26.40.190)