nLab cylinder functor



Category theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




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.


  • 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 cylinder 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 (Williamson 2012).


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 27, 2021 at 01:37:12. See the history of this page for a list of all contributions to it.