** **

**Abstract:**

The purpose of this talk is to explain the
basic definition

and properties of d-dimensional iterated
integrals for d=1,2,3, in particular the case d=2.

(d=1): These are Chen's iterated integrals.
I will explain how iterated integration

of a connection on a vector bundle gives the
holonomy.

In the special case that I study, these
holonomies are chain maps between chain complexes.

(d=2): 2-dimensional iterated integrals give
``chain homotopies.'' This is the main point of the lecture.

(d=3): in dimension 3 we get a degree 2
mapping between chain complexes whose boundary

is a chain homotopy. I.e., this is a
``higher homotopy.''

These integration formulae convert

Riemannian geometry into category theory and
homological algebra.

It is a long story of which I am telling
only one part.