  Families of objects appear in several contexts, like algebraic topology,
theory of deformations, theoretical physics, etc. An unified coordinate-free
algebraic framework for families of geometrical quantities is presented here,
which allows one to work without introducing ad hoc spaces, by using the
language of differential calculus over commutative algebras. An advantage of
such an approach, based on the notion of sliceable structures on cylinders, is
that the fundamental theorems of standard calculus are straightforwardly
generalized to the context of families. As an example of that, we prove the
universal homotopy formula.
