Let [Fscr ] be the category of functors from the category of finite-dimensional [ ]2-vector
spaces to [ ]2-vector spaces. The concept of &∇tilde;-nilpotence in the category [Fscr ]
is used to define a ‘dimension’ for the category of analytic functors which has good
properties. In particular, the paper shows that the tensor product F [otimes ] G of analytic
functors which are respectively &∇tilde;s and
&∇tilde;t nilpotent is &∇tilde;s+t − 1-nilpotent.
The notion of &∇tilde;-nilpotence is extended to define a dimension in the category
of unstable modules over the mod 2 Steenrod algebra, which is shown to coincide
with the transcendence degree of an unstable Noetherian algebra over the Steenrod algebra.