# Contents

## Definition

### Cylinder Object

Let $A$ be a small category, and let $\mathrm{Psh}\left(A\right)$ denote the functor category $\left[{A}^{\mathrm{op}},\mathrm{Set}\right]$. An object $X\in \mathrm{Ob}\left(\mathrm{Psh}\left(A\right)\right)$ is called a presheaf (of sets).

A cylinder on a presheaf $X$ is a presheaf $IX$ with the following data:

• Two jointly monomorphic morphisms ${\partial }_{X}^{0},{\partial }_{X}^{1}\in {\mathrm{Hom}}_{\mathrm{Psh}\left(A\right)}\left(X,IX\right)$ admitting a common retraction ${\sigma }_{X}:IX\to X$. That is, the induced map ${\partial }_{X}^{0}\coprod {\partial }_{X}^{1}:X\coprod X\to IX$ is a monomorphism, and ${\sigma }_{X}\circ {\partial }_{X}^{j}={\mathrm{id}}_{X}$ for $j\in \left\{0,1\right\}$.

That is, a cylinder is an object $IX$ with the morphisms above making the diagram below commute:

(PLACEHOLDER!)

A morphism of cylinders on presheaves $X$ and $Y$ is given by a pair of morphisms $\varphi :X\to Y$ and $\psi :IX\to IY$ making the following diagram commute:

$\begin{array}{ccccccc}& & {\partial }_{X}^{j}& & {\sigma }_{X}& & \\ & X& \to & IX& \to & X& \\ \varphi & ↓& & \psi ↓& & ↓& \varphi \\ & Y& \to & IY& \to & Y& \\ & & {\partial }_{Y}^{j}& & {\sigma }_{Y}& & \end{array}$\begin{matrix} &&\partial^j_X &&\sigma_X&&\\ &X&\to&I X&\to&X&\\ \phi&\downarrow&&\psi\downarrow&&\downarrow&\phi\\ &Y&\to&I Y&\to&Y&\\ &&\partial^j_Y&&\sigma_Y&&\end{matrix}

In particular, $\varphi$ is a retract of $\psi$.

### Cylinder Functor

A cylinder functor is a cylinder object for the identity functor ${1}_{\mathrm{Psh}}\left(A\right)$ in the endofunctor category $\left[\mathrm{Psh}\left(A\right),\mathrm{Psh}\left(A\right)\right]$.

#### Elementary Homotopy Data

A presheaf category is said to have a elementary homotopy data if it is equipped with a cylinder functor I such that

• the functor I commutes with all small colimits;
• the functor I respects monomorphisms;
• the natural transformation ${\partial }_{\left(-\right)}^{j}:{1}_{\mathrm{Psh}\left(A\right)}\to I$ sends arrows of Psh(A) to commutative squares in Psh(A) in the obvious way. We require that it sends all monomorphisms to cartesian squares.

(More to come..)

Revised on June 6, 2010 03:48:29 by Harry (76.242.100.74)