  Lifting theorems form an important collection of tools in showing that Galois
representations are associated to automorphic forms. (Key examples in dimension
n>2 are the lifting theorems of Clozel, Harris and Taylor and of Geraghty.) All
present lifting theorems for n>2 dimensional representations have a certain
rather technical hypothesis---the residual image must be `big'. The aim of this
paper is to demystify this condition somewhat.
  For a fixed integer n, and a prime l larger than a constant depending on n,
we show that n dimensional mod l representations which fail to be big must be
of one of three kinds: they either fail to be absolutely irreducible, are
induced from representations of larger fields, or can be written as a tensor
product including a factor which is the reduction of an Artin representation in
characteristic zero. Hopefully this characterization will make the bigness
condition more comprehensible, at least for large l.
