# Connection between homotopy category and derived category

Let $A$ be differential graded algebra (we abbreviate to dga) and $K(A)$ (resp., $D(A)$) the homotopy category (resp., derived category) of $A$-Mod.

A dg A-module $P$ is cofibrant if $$\textrm{Hom}_{K(A)} (P,L) \to \textrm{Hom}_{K(A)} (P,M)$$ is surjective for any quasi-isomorphism $L \to M$ which is surjective. See Section 2.12 in Derived equivalents from mutations of quivers with potential.

I have two questions:

1. I would like to know how the projection functor $K(A) \to D(A)$ induces an equivelance $A-\textrm{cofib} \sim D(A)$, where $A-\textrm{cofib}$ is the full subcategory of $K(A)$ on the cofibrant objects.
2. The cannonical projection from $K(A) \to D(A)$ admits a left adjoint functor ${\bf p}$ sending a dg $A$-module to a cofibrant dg $A$-module $_{\bf p}M$ quasi-isomorphic to $M$. What's the left adjoint functor ${\bf p}$ or how to understand the left adjoint functor ${\bf p}$?
• The notation $K(A)$ for the homotopy category is unusual, and there is no reasonable functor from the homotopy category to $D(A)$. By $K(A)$ do you mean the category of cofibrant objects? – Dmitry Vaintrob Dec 17 '16 at 15:21
• I mean that $K(A)$ is a homotopy category, see en.wikipedia.org/wiki/Derived_category $A$-cofib is the full subcategory of $K(A)$ consisting of cofibrant objects. – bing Dec 18 '16 at 12:07