Relating dgLa and cdga: $C^*$ and its adjoint

In this talk we will see that the functor $C^*$ from dgLa to cdga admits a right adjoint up to homotopy $D$. In order to understand $D$, we will use the free-forgetful adjunction between dgLa and chain complexes: We will see that up to homotopy, $C^*$ composed with the free functor $F$ has a particularly simple form, which in the following week will let us write a simple expression for $D$ (up to homotopy, when computed on certain nice enough algebras).

The talk will include a discussion on some basics of $\infty$-categories used to prove the existence of $D$ - we'll mostly discuss homotopy pullbacks, computing some mapping spaces, and how to detect whether certain objects are homotopy limits.