2-cosemisimplicial objects in a 2-category, permutohedra and deformations of pseudofunctors

In this paper we take up again the deformation theory for $K$-linear pseudofunctors initiated in a previous work (Adv. Math. 182 (2004) 204-277). We start by introducing a notion of a 2-cosemisimplicial object in an arbitrary 2-category and analyzing the corresponding coherence question, where the permutohedra make their appearence. We then describe a general method to obtain cochain complexes of K-modules from (enhanced) 2-cosemisimplicial objects in the 2-category ${\bf Cat}_K$ of small $K$-linear categories and prove that the deformation complex introduced in the above mentioned work can be obtained by this method from a 2-cosemisimplicial object that can be associated to the pseudofunctor. Finally, using a generalization to the context of $K$-linear categories of the deviation calculus introduced by Markl and Stasheff for $K$-modules (J. Algebra 170 (1994) 122), it is shown that the obstructions to the integrability of an $n^{th}$-order deformation of a pseudofunctor indeed correspond to cocycles in the third cohomology group, a question which remained open in our previous work.