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category theory

# Contents

## Definition

For $C$ a category, a cylinder functor on $C$ 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 $\sigma e_0 = \sigma e_1 = Id_C$.

## 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 $I$. 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).

## 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

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 26, 2021 at 20:37:12. See the history of this page for a list of all contributions to it.