For a category, a cylinder functor on 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 .
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 . While this is the motivating example, the interval functor need not be of that form.
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).
Revised on November 23, 2013 09:37:33