  Since Quillen proved his famous equivalences of homotopy categories in 1969,
much work has been done towards classifying the rational homotopy types of
simply connected topological places. The majority of this work has focused on
rational homotopy types with the same cohomology algebra. The models in this
case were differential graded algebras and acted similarly to differential
forms. These models were then used together with some deformation theory to
describe a moduli space for all rational homotopy types with a given cohomology
algebra. Indeed, this theory has been very well developed. However, there is
another case to consider. That is, the collection of rational homotopy types
with the same homotopy Lie algebra (same homotopy groups and Whitehead product
structure). This case, arguably, is closer to the heart of homotopy theory, as
it fixes the homotopy groups themselves and how they interact with each other.
However, the Lie case has received less attention and is less developed than
its cohomology counterpart. The main purpose of this paper is to completely
develop the theory for rational homotopy types of simply-connected topological
spaces with the same homotopy Lie algebra. It will include some foundations of
the theory as well as some new work. Often, previously-known results will be
streamlined, reworded, or reproven to make them directly relevant to the
results of this paper. By the end of the paper, deformation theory will be
developed and the moduli space for rational homotopy types with a fixed
homotopy Lie algebra will be defined and justified.
