cylinder on a presheaf
Let be a small category, and let denote the functor category . An object is called a presheaf (of sets).
A cylinder on a presheaf is a presheaf with the following data:
- Two jointly monomorphic morphisms admitting a common retraction . That is, the induced map is a monomorphism, and for .
That is, a cylinder is an object with the morphisms above making the diagram below commute:
A morphism of cylinders on presheaves and is given by a pair of morphisms and making the following diagram commute:
In particular, is a retract of .
A cylinder functor is a cylinder object for the identity functor in the endofunctor category .
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 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