On deformations of pasting diagrams. II. (Q2927671)

Continuing the second author's Part I [ibid. 22, 24--53 (2009; Zbl 1167.18001)], this paper presents a deformation theory for pasting arbitrary diagrams of \(k\)-linear categories, \(k\)-linear functors, and natural transformations with a field \(k\), generalizing classical results of \textit{M. Gerstenhaber} [Ann. Math. (2) 79, No.1, 59--103 (1964; Zbl 0123.03101); ibid. 84, 1--19 (1966; Zbl 0147.28903)] and \textit{M. Gerstenhaber} and \textit{S. D. Schack} [Trans. Am. Math. Soc. 279, 1--50 (1983; Zbl 0544.18005)] concerning infinitesimal deformations of associative algebras and poset-indexed diagrams of associative algebras. In [Zbl 1167.18001], the result that obstructions are cocycles was established merely for the simplest cases, while this paper succeeds in establishing it in general. The heart of the proof is based upon the first author's doctoral dissertation, in which ``edges of polygons are labeled with arrow-valued operations and corresponding deformation terms in such a way that a labeled polygon will simulataneously encode a coherence condition, a related cocycle condition, the formula for a direct-summand of an obstruction, and the condition on that direct-summand corresponding to the requirement that the next deformation term cobound the obstruction, depending on which indices of deformation terms are included in a summation''.