  The aim of this thesis is to analyze part of the results presented by R.E.
Borcherds in his article on renormalization and quantum field theories. To
facilitate this, the first chapter presents the basic facts on category theory,
sheaf theory and the theory of distributions on a manifold. In the particular
case of distributions on manifolds we differentiate between the H-distribution
(introduced by H\"ormander) and the D-distributions (described in the book of
Dieudonn{\'e}) and extend the product of distributions defined by H\"ormander
only for H-distibution, to an action from H-distributions over D-distributions.
  Each one of these facts will be used along the thesis in order to exhibit
more detailed descriptions and explanations for the results. In Chapter 2 the
concepts of spacetime, fields, Lagrangians, propagators and Feynman measure are
defined. The notion of Feynman measure is introduced with a more detailed
explanation of the Gaussian condition and its relation with the cut propagator.
In the last chapter we define the renormalization group and immediately
characterize it using the results from Chapter 1. The algebraic structure of
this group is described in detail. We present a descending filtration of the
renormalization group by normal subgroups, compute the corresponding quotients
and commutators, and show that any element of the renormalization group can be
written as an infinite product of elements each one living in one of the
aforementioned subgroups. Lastly we show that an action from the
renormalization group over the set of Feynman measures is well defined and
prove that this action is transitive over the Feynman measures associated with
a given cut local propagator.
