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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.

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  • $\begingroup$ This maybe relates to something that has been irritating me lately; if Ens is the (superlarge) category of large sets, then Cat(Set) is somehow simultaneously the internal category of the categories internal to the internal category Set, and also a subcategory of Cat(Ens), $\endgroup$
    – user13113
    Commented Oct 29, 2015 at 18:43
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    $\begingroup$ I don't think it is reasonable to try to define large category as colimit of diagram of small category as you will need large diagram to do this hence your already need to know large category. A good way of internalizing the notion of large category are indexed categories, you then have notion of "locally small category" indexed categories and of "small" indexed category which corresponds to category object. $\endgroup$
    – Simon Henry
    Commented Oct 30, 2015 at 11:22

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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.

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