nLab cylinder on a presheaf

Idea

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Definition

Cylinder Object

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

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

• Two jointly monomorphic morphisms $\partial^0_X, \partial^1_X\in Hom_{Psh(A)}(X,I X)$ admitting a common retraction $\sigma_X:I X\to X$. That is, the induced map $\partial^0_X\coprod \partial^1_X:X\coprod X\to I X$ is a monomorphism, and $\sigma_X\circ \partial^j_X=id_X$ for $j\in \{0,1\}$.

That is, a cylinder is an object $I X$ 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 $\phi:X\to Y$ and $\psi:I X\to I Y$ making the following diagram commute:

$$\begin{matrix} &&\partial^j_X &&\sigma_X&&\\ &X&\to&I X&\to&X&\\ \phi&\downarrow&&\psi\downarrow&&\downarrow&\phi\\ &Y&\to&I Y&\to&Y&\\ &&\partial^j_Y&&\sigma_Y&&\end{matrix}$$

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

Cylinder Functor

A cylinder functor is a cylinder object for the identity functor $1_Psh(A)$ in the endofunctor category $[Psh(A),Psh(A)]$.

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