Homotopy theory of unital algebras

In this talk, I will describe the homotopy theory of differential graded unital associative algebras. We already know that they are organized into a model category whose weak equivalences are quasi-isomorphisms. However, the computations of cofibrant resolutions of algebras make this framework unwieldy. I will show that the category of dg unital associative algebras may be embedded into the category of curved coalgebras whose homotopy theory is equivalent but more manageable. Then, I will generalize this method to the case of dg operads and to the case of algebras over an operad.