What is a noetherian category? What is a noetherian category?
I'm a little bit familiar with category theory, but I've no idea what this could be.
Do you know what it is good for or examples?
 A: From http://ncatlab.org/nlab/show/noetherian+category :
A category $C$ is noetherian if  the class of objects of its skeleton is a set and every object in $C$ is a noetherian object.
For example you can take the category of finite dimensional vector spaces.
A: OK, well a good start would be to google "noetherian category" and strike upon the nLab entry (which is the very first hit) which explains that a Noetherian category is an essentially small category in which the objects are all Noetherian objects.
The "essentially small" part is unfamiliar to me, and a bit of a surprise since I suspected that the class of Noetherian modules for any ring would be a Noetherian category. Since such a category seems to frequently not be essentially small, I guess one would have to make additional restrictions. For example, the category of submodules of a Noetherian module over a simple Artinian ring would work, since there are only finitely many isotypes of submodules of such a module.

While I was editing I see that Boris has suggested what looks like a very nice example of finite dimensional vector spaces, which is a really helpful illustration of "essentially small".

Hrm, well reading more at the link, and some help from Zhen Lin in the comments, I've learned that it is true that the category of Noetherian modules over a ring with identity form a Noetherian category. (I hoped as much!)
