Consider the usual model structure on Cat (category of small categories).
Which are the fibrations of the injective model structure on the category of cosimplicial categories $Fun( \Delta ,Cat )$?


Actually, I wonder how can I compute homotopy limits of cosimplicial categories (considering them as diagrams in Cat)...


Edit: I'm most interested in understanding how to compute the homotopy limit of a cosimplicial diagram in $Cat$


Thank you in advance



Homotopy limit of a cosimplicial category
I will answer the title question about computing homotopy limits. Let $A : \mathcal{C} \to \mathbf{Cat}$ be a small (strict) diagram. By expanding the definitions of the cobar construction, one eventually discovers that $\operatorname{holim} A$ can be computed as the end
$$\int\_{[n] : \mathbf{\Delta}} \left[ \mathbf{I} [n], \prod\_{(c\_0, \ldots, c\_n)} \mathcal{C} (c\_n, c\_{n-1}) \times \cdots \times \mathcal{C} (c\_1, c\_0), A c\_0 \right]$$
where $\mathbf{I} [n]$ is the contractible groupoid with $n + 1$ objects. In more familiar terms, this is just the hom-category of all morphisms between a certain pair of cosimplicial categories. 


Here is a description of an object in $\operatorname{holim} A$:


* For each object $c$ in $\mathcal{C}$, we have an object $a\_c$ in $A c$.
* For each morphism $f : c\_1 \to c\_0$ in $\mathcal{C}$, we have an isomorphism $\mu\_f : f (a\_{c\_1}) \to a\_{c\_0}$ in $A c\_0$; and $\mu\_{\mathrm{id}\_c} = \mathrm{id}\_{a\_c}$ for all objects $c$ in $\mathcal{C}$.
* For each composable pair $f\_1 : c\_2 \to c\_1, f\_0 : c\_1 \to c\_0$ in $\mathcal{C}$, we have a commutative triangle in $A c\_0$,
$$\begin{array}{ccc}
f\_0 (f\_1 (a\_{c\_2})) & \rightarrow & f\_0 (a\_{c\_1}) \\
& \searrow & \downarrow \\
&& a\_{c\_0}
\end{array}$$
i.e. $\mu\_{f\_0 \circ f\_1} = \mu\_{f\_0} \circ f\_0 (\mu\_{f\_1})$.
* For each composable triple in $\mathcal{C}$, we have a commutative tetrahedron in $A c\_0$.
* etc.


In fact, the coherence conditions above degree 2 are automatic if all the triangles commute. Thus we see that an object in $\operatorname{holim} A$ is the same thing as a pseudocone over $A$ (where we regard $A$ as a pseudofunctor with the canonical coherence data). Unsurprisingly, the morphisms are the same, so $\operatorname{holim} A$ (as constructed above) is the pseudolimit of $A$.


