The Zigzag Hochschild Complex

TL;DR

This paper introduces a new higher Hochschild complex with an iterated integral map, modeling differential forms on bigons and relating to gerbes with 2-group structures, advancing the mathematical understanding of higher holonomy.

Contribution

It defines a novel curved zigzag Hochschild complex that models local 2-holonomy of gerbes with 2-group structures, connecting higher algebraic structures to differential geometry.

Findings

Constructs a higher Hochschild complex with iterated integral map.

Models local 2-holonomy for gerbes with 2-group structure.

Provides foundational work for future research in higher geometric structures.

Abstract

In this paper, a new higher Hochschild Complex is defined with an Iterated Integral map to locally model differential forms on the space of bigons on $M$. In particular, given the local data for a gerbe with structure 2-group given by a crossed module of matrix-groups, there is an element in our curved zigzag Hochschild complex associated to the local 2-holonomy given by such a gerbe. This paper introduces an initial construction central to the author's PhD Thesis.