Higher Holonomy via a Simplicial Viewpoint

TL;DR

This paper develops a novel simplicial set framework for higher holonomy using A-infinity-categories, introducing fiberwise integrals and an analog of Stokes's theorem to extend classical concepts.

Contribution

It introduces a new simplicial approach to higher holonomy, utilizing fiberwise integrals and A-infinity-categories, extending Chen's iterated integrals.

Findings

Constructed fiberwise integrals on simplicial sets

Defined an iterated integral analogous to Chen's

Proved an analog of Stokes's theorem for fiberwise integrals

Abstract

In this paper, we construct an analogy of holonomy of connection to simplicial sets using A-infinity-categories. To construct it, we develop fiberwise integrals on simplicial sets and define an iterated integral on simplicial sets. It is an analogy to Chen's iterated integral. We also prove an analogy of Stokes's theorem for fiberwise integrals.