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.