On Deformations of Pasting Diagrams, II

TL;DR

This paper extends the deformation theory of pasting diagrams of k-linear categories by proving that all obstructions are cocycles in general, using polygon representations and trivial deformations.

Contribution

It provides a general proof that all obstructions are cocycles for pasting diagrams, expanding on previous elementary cases.

Findings

Obstructions are cocycles in general pasting diagrams.

Polygon representations rigorously encode deformation formulas.

Deformations with some cells fixed trivially are analyzed.

Abstract

We continue the development of the infinitesimal deformation theory of pasting diagrams of k-linear categories begun in Yetter, D.N. "On Deformations of Pasting Diagrams", Theory and Applications of Categories 22 (2009) 24-53. In that paper, the standard result that all obstructions are cocycles was established only for the elementary, composition-free parts of pasting diagrams. In the present work we give a proof for pasting diagrams in general. As tools we use (1) the method developed by Shrestha, in his Kansas State University doctoral dissertation, of representing formulas for obstructions, along with the corresponding cocycle and cobounding conditions by suitably labeled polygons, giving a rigorous exposition of the previously heuristic method, and (2) deformations of pasting diagrams in which some cells are required to be deformed trivially.