internalization of the concept of large and small category I have been poking around the internet and nlab looking at the concept of large and small categories.  My original focus was locally presentable categories of categories and I was thinking of finding categories like Hilb and Group and Set as colimits over diagrams of compact objects.  This lead to trying to define "largeness" and "smallness" in terms of colimits, ie, large categories as colimits of diagrams of small categories.
There is plenty of work done on internalizing the concept of "category".  We even know that a "small category" is one that is internal to set.  Is it possible to internalize how the concept of smallness is related to largeness?  I had one thought on the matter, which was to have some ambient category that could host SET, and so all the small categories are internal there.  We want to define largeness as maps from internal categories in other categories into this.  That is a stupid attempt, but its all I have.  Is it even possible to have a category which contains large categories and small categories?
I just don't see anywhere an attempt to internalize the concepts of how largeness relates to smallness.
 A: I am not sure exactly what sort of thing you are looking for, but you would probably find the work on algebraic set theory relevant and possibly interesting.
Algebraic set theory studies elementary set theory through category-theoretic methods. A central idea is an abstract notion of smallness which is axiomatized in a category-theoretic fashion. I recommend Steve Awodey's overview article A Brief Introduction to Algebraic Set Theory. Briefly, we set things up by postulating (I am quoting from Section 1.2 of Steve's paper):


*

*a Heyting category $\mathcal{C}$ of "classes"

*a subcategory $\mathcal{S} \hookrightarrow \mathcal{C}$ of "sets"

*a "powerclass" functor $P : \mathcal{C} \to \mathcal{C}$ of subsets

*a "universe" $U$ which is a free algebra for $P$.


The classes in $\mathcal{C}$ admit the interpretation of first-order logic; the sets $\mathcal{S}$ capture an abstract notion of “smallness” of some classes; the powerclass $P(C)$ of a class $C$ is the class of all subsets $A \rightarrowtail C$; and this restriction on $P$ to subsets (as opposed to subclasses) permits the assumption of a universe $U$ which, as a free algebra, has an isomorphism $i : P(U) \cong U$. 
There are many models of this setup apart from the canonical Zermelo-Fraenkel set theory.
