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 Elgueta (Adv. Math. 182 (2004) 204–277). We start by introducing a notion of 2-cosemisimplicial object in an arbitrary 2-category and analyzing the corresponding coherence question, where the permutohedra make their appearance. We then describe a general method to obtain usual cochain complexes of K-modules from (enhanced) 2-cosemisimplicial objects in the 2-category CatK of small K-linear categories and prove that the deformation complex image introduced in Elgueta (to appear) can be obtained by this method from a 2-cosemisimplicial object that can be associated to image. Finally, using this 2-cosemisimplicial object of image and 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 nth-order deformation of image indeed correspond to cocycles in the third cohomology group image, a question which remained open in Elgueta (Adv. Math. 182 (2004) 204–277).