On Iterated Integral on Simplicial Sets

TL;DR

This paper investigates the properties of a simplicial analogue of Chen's iterated integral, compares it with the classical integral, and explores their relationship with rational homotopy theory and infinity-categories.

Contribution

It establishes the elementary properties of simplicial iterated integrals and clarifies their connection to Chen's integrals and rational homotopy theory.

Findings

Simplicial iterated integral's properties are characterized.

A relationship between iterated integral and homotopy pullback is described.

Comparison with Chen's iterated integral and implications for rational homotopy groups are provided.

Abstract

A simplicial analogy of Chen's iterated integral was introduced in another paper. However, its properties were hardly investigated in the paper. In particular, no mention is made of whether it coincides with Chen's iterated integral as a special case. In this paper, we answer this question. This paper consists of two main parts. One of them is the research of elementary properties of simplicial iterated integral. In particular, we describe a relationship between iterated integral and homotopy pullback. The other one is a comparison of Chen's iterated integral and simplicial iterated integral. At the end of this paper, we observe Chen's theorem and Hain's theorem which connects rational homotopy groups and (co)homologies of a smooth manifold, and give an infinity-categorical viewpoint.