nLab
cylinder on a presheaf

Idea

---

Definition

Cylinder Object

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

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

• Two jointly monomorphic morphisms ${\partial }_X^0,{\partial }_X^1\in {\mathrm{Hom}}_{\mathrm{Psh}(A)}(X,IX)$ 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 $j\in \{0,1\}$.

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:

$$\begin{array}{ccccccc}& & {\partial }_X^j& & {\sigma }_X& & \\ & X& \to & IX& \to & X& \\ \varphi & \downarrow & & \psi \downarrow & & \downarrow & \varphi \\ & Y& \to & IY& \to & Y& \\ & & {\partial }_Y^j& & {\sigma }_Y& & \end{array}$$\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 $1_{\mathrm{Psh}}(A)$ in the endofunctor category $[\mathrm{Psh}(A),\mathrm{Psh}(A)]$.

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 ${\partial }_{(-)}^j:1_{\mathrm{Psh}(A)}\to I$ 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..)