cylinder on a presheaf




Cylinder Object

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

A cylinder on a presheaf XX is a presheaf IXI X with the following data:

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

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


A morphism of cylinders on presheaves XX and YY is given by a pair of morphisms ϕ:XY\phi:X\to Y and ψ:IXIY\psi:I X\to I Y making the following diagram commute:

X j σ X X IX X ϕ ψ ϕ Y IY Y Y j σ Y \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, ϕ\phi is a retract of ψ\psi.

Cylinder Functor

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

Last revised on June 6, 2010 at 03:48:29. See the history of this page for a list of all contributions to it.