# Derived functor of the realisation in a simplicial model category

Given a model category $$C$$, a simplicial symmetric monoidal model category $$D$$ (in the sense of Goerss-Jardine) and a left Quillen functor $$F:C\to D$$, define $$|F|:C\to\mathsf{sSet}$$ to be the functor $$|F|=\mathsf{Map}_D(I,F(-))$$ where $$I$$ is the unit of the monoidal structure of $$D$$ which we suppose to be cofibrant.

Then, is there a link between $$|F|$$ and $$|\mathbb LF|$$, like $$\mathbb L|F|\simeq |\mathbb L F|$$ ? The problem is that $$|F|$$ may not verify the classical conditions to have a left derived functor, as we want fibrations on the right argument of the mapping space.

In my case $$D=C(k)$$ for $$k$$ a field of characteristic $$0$$, so every object is fibrant, which might help. My functor $$F$$ computes something which has the good homotopy type only for cofibrant objects, so I'd want $$\mathsf{Map}_D(I,F(Q(-)))$$ to yield a functor $$\mathsf{Ho}(C)\to\mathsf{Ho}({\mathsf{sSet}})$$ and I'd like it to be the total left derived functor of $$|F|$$.

• The final object in D=C(k) is the zero chain complex, so *=0 and Map(*,F(-))=0. It would appear then that your question is trivial. Commented Jul 16, 2019 at 0:58
• Fixed, thanks. In my case this is $k$. The reference to Goerss Jardine isn't really correct now, but anyway, just take $D=C(k)$ :) Commented Jul 16, 2019 at 7:00

• Thank you. Yet I'm wondering : as a functor from $C$ to $\mathsf{sSet}$, $|F|$ does not seem to preserve cofibrations or anything like this, so how do we know that it actually has a left derived functor ? Commented Jul 17, 2019 at 9:15
• Ok I was confused, thanks. I have one last problem : why would $|F|\circ Q$ perserve weak equivalences ? Commented Jul 17, 2019 at 15:58