  We study extensively the homotopy theory of coalgebras. By coalgebras, we
mean the full theory of coalgebras: with counits and not necessarily locally
conilpotent. For example $\mathcal E_\infty$-coalgebras, $\mathcal
A_\infty$-coalgebras, $\mathcal L_\infty$-coalgebras etc. To do so, we define
the category of complete curved algebras -- where the notion of
quasi-isomorphims does not make sense -- and endow it with a model category
structure, equivalent to that of the category of coalgebras.
