Small objects in categories I would like to pick out small objects from a category. I would like to find such a notion which
Dream 1. Picks out the schemes of finite type over $k$ from the category of $k$-schemes. Or at least picks out something relevant.
Dream 2. Picks out the top spaces homotopic to finite CW-complexes from the category of topological spaces (equipped with the Quillen model structure). (or something relevant)
Question: What notions exist for "small" objects (other than the compact objects)?
References or comments are welcomed.:)
Thanks!
 A: The subcategory of spaces equivalent to finite CW-complexes is the smallest one containing a point and closed under finite homotopy colimits. (In fact, this is the universal homotopy theory generated under finite hocolims by a single object). 
Someone more patient than I could might fill in the details to make the following sentence both (i) sensical and (ii) correct: If $G$ is finite, the subcategory of $G$-spaces equivalent to a finite $G$-CW-complex is the smallest containing a point and closed under finite 'enriched' (weighted is probably the right word) homotopy colimits. (Here I'm enriched in $G$-spaces instead of spaces like above). 
The category of schemes is probably too ugly of a place to try and make a similar statement, but perhaps one of the many colimit-populated enlargements would allow you to make the statement that some reasonable enlargement of the category of schemes of finite type over $k$ is generated by a point under some allowed collection of colimits.
(By the way, in the first two examples, honest compact objects are obtained by idempotent completion, i.e. adding retracts.)
A: I'm not really sure what you're asking, so this may not qualify as an answer. But it is too long to be a comment. When I see the word compact, I think of it as $\aleph_0$-small (a.k.a. $\aleph_0$-presentable). Of course, there is also the notion of $\lambda$-small (a.k.a. $\lambda$-presentable or $\lambda$-compact) for any ordinal $\lambda$. Is this what you're asking about? If so, I can give you a number of references to learn more about this.
One small point of confusion is that for many years people studying triangulated categories used the words compact and small interchangeably. With Amnon Neeman's book Triangulated Categories this seems to have changed. That book is a great source to learn about notions related to smallness, e.g. the notion of a $\lambda$-well-generated triangulated category. 
I don't think the theory of small objects is intrinsically bound to that of model categories, but model category theorists have worked out much of the theory of smallness because they need it (e.g. for the small object argument). I learned everything I know about smallness from Mark Hovey's book Model Categories and I think the nLab article is very good also. For question 1, a model category theorist might consider the category of motivic spaces. This category is compactly generated, as is proven in the appendix to Po Hu's article S-modules in the category of schemes
Question 2 seems like it should be asked as a separate question, because I think it's doubtful that the answer will be very related to a general discussion of smallness. However, it does remind me to mention $\Delta$-generated spaces, which form a combinatorial model category Quillen equivalent to the usual model category of topological spaces (i.e. all objects are small). Dan Dugger has a survey about this topic, and I also like the treatment given in this paper by Fajstrup and Rosicky.
