I'm trying to take familiarity with homotopy theory and I have the following questions. Let $\mathcal{C}$ be a small category, and let $F\: : \:\mathcal{C}\to \mathcal{M}$ that take values in a (simplicial) model category.
For any simplicial presheaves $K\: : \: \mathcal{C}^{op}\to sSet$, I denote with
$$
K\otimes\_{\mathcal{C}} F
$$
the functor tensor product. Let \* be the constant simplicial presheaf, it is easy to note that
$$
\*\otimes\_{\mathcal{C}} F=colim F
$$
Now choose a cofibrant replacement P of \* in the projective model structure of simplicial presheaves. If we assume that $F$ is pointwise cofibrant, then the homotopy colimit is given by
$$
P\otimes\_{\mathcal{C}} F
$$
A good candidate for the cofibrant replacement for $\*$ is the nerve functor $N(-/ \mathcal{C})$. Another strategy to compute the homotopy colimit is given by the bar construction $B(K, \mathcal{C}, F)$ (see for example www.math.harvard.edu/~eriehl/hocolimits.pdf or the book *categorical homotopy theory*). In particular it is possible to show that there is a natural isomorphism 
$$
B(\*, \mathcal{C}, F)\to N(-/ \mathcal{C})\otimes\_{\mathcal{C}} F
$$
Thus in some sense the bar construction contains the data of a cofibrant replacement for the simplicial presheaf $\*$.
Here are my questions:



> 
> 1) Is it possible to generalize the above facts for a general simplicial presheaf $K\: : \: \mathcal{C}^{op}\to sSet$, i.e there exists a cofibrant replacament QK of K and a natural isomorphism
>  $$
> B(K, \mathcal{C}, F)\to QK\otimes\_{\mathcal{C}} F?
> $$
> 
> 
> 2) If not, it is possible to find at least a weak equivalence?
> 
> 
> 


Another formulation of this question is as follows: since $K\otimes\_{\mathcal{C}} F$ may be intepretated as a weighted colimit, does the bar construction compute the *homotopy* weighted colimits?



> 
> 3)(i don'think make sense) If $\mathcal{M}$ is not a simplicial category, it is possible to find a relation between the homotopy weighted colimit and the Bar construction?
> 
> 
> 



homotopy tensor product of functors and bar construction
Yes. (Of course for these constructions to be homotopically well-behaved you need $F$ to be levelwise cofibrant.) In fact, assuming that such $QK$ exists we can express it by an explicit formula which then proves that it indeed exists. All we need to know is that tensoring over $\newcommand{\C}{\mathcal{C}}\C$ with a functor represented by $x \in \mathcal{C}$ is just evaluation at $x$. Hence $QK\_x = QK \otimes\_\C \C(x,-) = B(K, \C, \C(x,-))$ and that's how we construct $QK$.


Now the formula for $B(K, \C, F)$ holds for general $F$.


$$QK \otimes\_\C F = B(K, \C, \C(-,-)) \otimes\_\C F \\ = |\coprod\_{y\_0, \ldots, y\_m} \C(-, y\_0) \times \C(y\_0, y\_1) \times \ldots \times \C(y\_{m-1}, y\_m) \times K\_{y\_m}| \otimes\_\C F \\ = |\coprod\_{y\_0, \ldots, y\_m} F \otimes\_\C \C(-, y\_0) \times \C(y\_0, y\_1) \times \ldots \times \C(y\_{m-1}, y\_m) \times K\_{y\_m}| \\ = |\coprod\_{y\_0, \ldots, y\_m} F y\_0 \times \C(y\_0, y\_1) \times \ldots \times \C(y\_{m-1}, y\_m) \times K\_{y\_m}| = B(K, \C, F)$$


Moreover, $QK$ is projectively cofibrant. Indeed, we can explicitly write it as a cell complex in the projective model structure on diagrams $\C^{\mathrm{op}} \to \mathsf{sSet}$. For each $m$ there is a functor $G\_m K \colon \C^{\mathrm{op}} \to \mathsf{Set}$ where $G\_m K\_x$ is the set


$$\{ (y, z) \mid y \colon [m] \to x \downarrow \C, z \in K\_{y\_m}, (y,z) \text{ is a non-deg. simplex of } N(x \downarrow \C) \times K\_{y\_m} \}$$


and a pushout square


$$
\begin{array}{ccccccccc}
G\_m K \times \partial\Delta[m] & \xrightarrow{} & \mathrm{Sk}^{m-1} QK & 
\newline
 \downarrow & & \downarrow & & \newline
G\_m K \times \Delta[m] & \xrightarrow[]{} & \mathrm{Sk}^m QK .& 
\end{array}
$$


It remains to see that the map $QK \to K$ is a weak equivalence, but this follows from the fact that $\C(x,-)$ is projectively cofibrant since that map coincides with $K \otimes\_\C B(\C(-,-),\C,\C(x,-)) \to K \otimes\_\C \C(x,-)$ by a computation similar to the one above.


