Categorically algebraic foundations for homotopical algebra

We investigate a structure for an abstract cylinder endofunctor I which produces a good basis for homotopical algebra. It essentially consists of the usual operations (faces, degeneracies, connections, symmetries, vertical composition) together with a transformation w: I-2 --> I-2, which we call lens collapse after its realisation in the standard topological case. This structure, somewhat heavy, has the interest of being ''categorically algebraic'', i.e., based on operations on functors. Consequently, it can be naturally lifted from a category A to its categories of diagrams A(S) and its slice categories A\X, A/X. Further, the dual structure, based on a cocylinder (or path) endofunctor P can be lifted to the category of A-valued sheaves on a site, whenever the path functor P preserves limits, and to the category Mon A of internal monoids, with respect to any monoidal structure of A consistent with P.