Holmstrom Total derived functor

Let $F: C \to D$ be a left Quillen functor. Define its total (left) derived functor $LF: Ho \ C \to Ho \ D$ as the composite of

and

Here $Q$ is the cofibrant replacement functor.

Given a natural transformation $\tau$ of left Quillen functors, define the total derived natural transformation $L \tau$ to be $Ho \ \tau \circ Ho \ Q$.

Similarly, for a right Quillen functor we define the total (right) derived functor to be

Consider the 2-category of model cats, left (resp. right) Quillen functors, and natural transformations. The triple: Homotopy category, total derived functor, and total derived natural transformation, define a pseudo-2-functor from this 2-category to the 2-category of categories. SImilar statement with Quillen adjunctions instead of Quillen functor.

Remark: One can define total derived functors in more general settings, I think. For example, Quillen describes the total left derived functor as a Kan extension.

Goerss-Jardine p 122: Quillen’s total derived functor thm, explains when adjoint functors between model cats induces adjoint functors on the homotopy cats. The total left derived functor here is defined for functors from a simplicial model cat to any cat, sending WEs between cofibrant objects to isomorphisms. A version of the theorem is: For an adjoint pair (F, G) between simplicial model cats, the conclusion is true of $F$ preserves WEs between cofibrant objects and $G$ WEs between fibrant objects. There are also some corollaries and variations on this theme.

In Goerss and Schemmerhornopology/0609537), page 17, there is a brief description of the spectral sequence total left derived functor of tensor product.

nLab page on Total derived functor