Hey,

Suppose I want to establish a theory of the category $C$ (vector spaces or whatever), but what I really have is $D$, some precisely known category.  This is to say, I know all the axioms of $D$, but I only have an intuition for $C$ and I want to develop a theory of $C$.  I would normally have diagrams in $C$ by mapping cpos or Domains into $C$.  But instead, I want to do it with $D$.  What I do is define the largest category, $J$, of Domains in $D$.  I do this by defining a dcpo with objects as elements in $D$ and relations as arrows in $D$.  Then any functor will map the domains in $J$ to diagrams in $C$.

It seems like I am just inserting a category $D$ in the normal diagram functor $J \rightarrow C$ resulting in $J \rightarrow D \rightarrow C$ which seems to miss the point of the exercise.  The point of the exercise, I think, is to try to do a lot of category theory when you have to live in some category $D$.

We start by saying that we "have access to" all diagrams in $D$.  Further, we say that we have access to none of the morphisms in any other category.  So if we want to talk about a category $C$, it will have to be in terms of diagrams in $D$.  Next, we intuit the existence of a category $C$  (I am using this restricted language to reflect the notion that we do don't have access to $C$).  Next, we consider endofuntors of $D$, but we really see them as diagrams in $D$ indexed by the domains we constructed in $D$ by $J$.  These endofunctors are meant to mimic functors from $D$ to $C$.  We are pretending to have access to $C$, by attempting a construction of $C$ in $D$.

Sorry that this is so unclear, especially the idea of having an "intuition of C" and "attempting a construction of".  I think that this is an expression of a Topos, and so I have some questions.  Firstly, what kind of minimum structure do we need in $D$ to really start doing some work?  Second, if we really want to say that we only have access to $D$, then we cannot present $D$ as a set of morphisms and a set of objects because that would imply we are actually in SET, not $D$.  Is there any way to start working only in $D$?  This goes back to the first question (although thinking about this too much is a bit of a morass).

It's sort of like the inverse function theorem, and that is why it is so strong.  If you have $n$ functions vanishing at the origin of $k^n$ and want to know if they give a local coordinate system, you ask if their differentials are independent at the origin.  Or equivalently if their differentials generate the cotangent space at the origin.  So in a [not necessarily noetherian, thanks Georges!] local ring $(\mathcal{O},\mathfrak{m})$, Nakayama's lemma says you can detect that elements of the maximal ideal generate that ideal, hence act sort of like coordinate functions, just by knowing their differentials, i.e. their residues in the Zariski cotangent space $\mathfrak{m}/\mathfrak{m}^2$, generate that linear space.

Those versions of the lemma you linked to are almost unrecognizable forms of this simple statement, but that's the way abstract math goes as we know. But the idea is the same, you have a hypotheses about a truncated version of your statement, and you get out the fuller version.  The Jacobson radical stuff is there to disguise the fact that it doesn't say much unless you are in a local setting.  I.e. in a local ring the Jacobson radical is pretty big and you get a better result.  In a polynomial ring with tiny Jacobson radical you get nothing.
