  This paper studies the homotopy theory of algebras and homotopy algebras over
an operad. It provides an exhaustive description of their higher homotopical
properties using the more general notion of morphisms called
infinity-morphisms. The method consists in using the operadic calculus to endow
the category of coalgebras over the Koszul dual cooperad or the bar
construction with a new type of model category structure, Quillen equivalent to
that of algebras. We provide an explicit homotopy equivalence for
infinity-morphisms, which gives a simple description of the homotopy category,
and we endow the category of homotopy algebras with an infinity-category
structure.
