On Deformations of Pasting Diagrams

TL;DR

This paper develops a deformation theory for 2-categorical pasting diagrams, extending existing algebraic deformation frameworks to more general, non-commutative, and non-composable cases within $k$-linear categories.

Contribution

It adapts Power's work to describe deformations of general pasting diagrams, introducing a bicategorical analog of homotopy G-algebras and proving standard deformation results.

Findings

Established a deformation theory for $k$-linear category diagrams.

Constructed a bicategorical analog of homotopy G-algebras.

Proved standard deformation results for these diagrams.

Abstract

We adapt the work of Power to describe general, not-necessarily composable, not-necessarily commutative 2-categorical pasting diagrams and their composable and commutative parts. We provide a deformation theory for pasting diagrams valued in $k$-linear categories, paralleling that provided for diagrams of algebras by Gerstenhaber and Schack, proving the standard results. Along the way, the construction gives rise to a bicategorical analog of the homotopy G-algebras of Gerstenhaber and Voronov.