For a category, a cylinder functor on is a functor denoted
equipped with three natural transformations
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.
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.
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
Cylinder functors also form one of the key elements in Baues’ approach to algebraic homotopy:
Cylindrical model structures are discussed in
Last revised on January 20, 2015 at 21:02:47. See the history of this page for a list of all contributions to it.