2-cosemisimplicial objects in a 2-category, permutohedra and deformations of pseudo\-functors (Q1886436)

This is the continuation of the deformation theory for \(K\)-linear pseudofunctors developed in a preceding paper [\textit{J. Elgueta}, Adv. Math. 182, 204--277 (2004; Zbl 1046.18003)]. In an arbitrary 2-category a notion of 2-cosimplicial object (it is a 2-dimensional version of the classical cosemisimplicial objects in a category) is introduced. The coherence question makes appear the permutohedra (like for Milgram in the context of iterated loop spaces). Then one has a general method to obtain usual cochain complexes of \(K\)-modules in the special case of the 2-category \({\mathcal C}at\) of small \(K\)-linear categories (i.e., categories enriched over the monoidal category of \(K\)-vector spaces). This method permits to obtain the deformation complex of a pseudofunctor \(F\) from a 2-cosimplicial object associated to \(F\). Then the obstructions to the integrability of a deformation of \(F\) correspond to cocycles in the third cohomology of the deformation complex of \(F\).