Weak 2-cosemisimplicial objects in a 2-category and deformations of a pseudofunctor
نویسنده
چکیده
In this paper we take up again the deformation theory for K-linear pseudofunctors initiated in [4] with a two-fold purpose. Firstly, we introduce the notion of weak 2-cosemisimplicial object in a 2-category and show that the deformation complex X•(F) introduced in [4] can be obtained from one such object in the 2-category CatK of small K-linear categories. In doing this, we describe a family of graphs, conjecturally the 1-skeleta of a new family of convex polytops we call the cosemisimplihedra, and related to the higher-order cosemisimplicial identities. Secondly, using this construction and a generalization to the context of K-linear categories of the deviation calculus introduced by Markl and Stasheff for K-modules [8], we prove that the obstructions to the integrability of an n-order deformation of F indeed correspond to cocycles in the third cohomology group H(X(F)), a question which remained open in [4].
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