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The abstract of the paper Set Theory for Category Theory (arXiv:0810.1279 [math.CT]) by Michael Shulman opens thus:

Questions of set-theoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made can have noticeable effects on what categorical constructions are permissible.

It appears to me that these issues arise when category theory is formalized within the context of some set theory, but as far as I know it is also possible to take category theory itself as the foundation of mathematics, with no reference to any underlying set theory. If this approach is taken, is there any good reason to distinguishing between small sets and large sets, and between small and large categories?

Forget about good reason, is there even any sense in distinguishing between small and large categories if category theory is taken as the foundation of mathematics? Take, for instance, the following definition of small and large categories from Wikipedia:

A category C is called small if both ob(C) and hom(C) are actually sets and not proper classes, and large otherwise.

If there is no underlying set theory in which category theory is formulated, does this definition even make sense? And even if you can somehow imbue it with sense, is it worthwhile to do so?

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  • $\begingroup$ I imagine that the answer is that in a categorical foundation a set is an object of $\bf Set$, which may or may not be assumed to satisfy some properties. So what would that mean? I guess the existence of a functor from the objects/morphisms of $\bf C$ into $\bf Set$ which acts in some reasonable way. But of course, I will wait for someone more familiar with categorical foundations to speak up. $\endgroup$ – Asaf Karagila Jun 24 '18 at 17:44
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Category theory wants to contain set theory (Set is an important category! Also, sets are basically discrete categories), so all of the usual reasons compel you to pay attention to size issues. They can even show up without appealing to sets; e.g. there can't be a category of all categories.

A more category-theoretic flavoring of size issues is:

  • A small category is a category object in Set
  • A locally small category is a Set-enriched category

so even if there weren't size issues, smallness would still be an important topic in category theory, even if it were just a special case of enriched category theory and internal category theory.

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  • $\begingroup$ Thanks. According to nLab, "Bill Lawvere proposed to found mathematics on ETCC (for ‘ Elementary Theory of the Category of Categories’), a first-order axiomatisation of the category of categories. This has not been very successful, but his other proposal, a first-order axiomatisation of the category of sets, works well." Does Lawvere's axiomatization of the category of sets provide both sets and classes? If so, category theory can be based on the set theory established by this axiomatization, right? $\endgroup$ – Evan Aad Jun 24 '18 at 18:04
  • $\begingroup$ @EvanAad Every first-order theory has classes. Classes are just (represented by) first-order formulas with a single free variable. You can talk about the class of all squares in the first-order (algebraic even) theory of groups. $\endgroup$ – Derek Elkins Jun 24 '18 at 19:54
  • $\begingroup$ @EvanAad: Generally, one additionally posits some sort of universe; in ZFC this would be a large cardinal axiom. In ETCS, I think it would be a universe, in the sense of topos theory. If you wanted to mimic the overly restrictive foundations consisting of ZFC without a large cardinal axiom, I imagine you would mimic NBG and axiomatize something like "A category with finite limits generated by an ETCS object", with the idea that this category's objects are the classes. I don't know if there is any interest in such a thing. $\endgroup$ – Hurkyl Jun 24 '18 at 19:54
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Most foundations consist of two components: a logic (i.e. a formal system of reasoning), and an axiomatization (i.e. a description in the formal language of that system of reasoning) of a system of basic mathematical objects in which most if not all of informal mathematics can be encoded.

The most common meaning of the phrase a foundation mathematics is a foundation whose logic is given by first-order logic. In this usual sense then, the distinctions between different foundations is a distinction regarding what the system of basic mathematical objects looks like.

When set theory is considered as a foundation of mathematics, the system of mathematical objects is the class of sets equipped with a membership relation $\in$ satisfying certain axioms (e.g. ZFC).

When category theory is considered as a foundation of mathematics, the system of mathematical objects is a category, i.e. a pair of classes consisting respectively of the objects of the category and the morphisms of the category, equipped with certain auxiliary functions between classes (domain, codomain, identity, partially-defined composition) satisfying certain axioms.

Two points are in order.

First, classes are present in both foundations. In fact, classes are inescapable when using first-order logic where class is simply an equivalence class of formulas (i.e. statements) under logical equivalence. In other words, a class in first-order logic is a collection of objects specified by a formula in first-order logic (the extension of the formula).

Second, the distinction between classes and sets, hence between large and small categories in the set-theoretical setting, appears vacuous: classes and sets are different kinds of things. This is because the correct distinction to drawn is not between classes and sets, but between proper classes and small classes. A class is small if it is the class of elements of some set (formally, the class $\phi(x)$ is small if $\exists x:\phi(y)\leftrightarrow y\in x$ is provable). Otherwise the class is proper.

Russel's paradox then shows that there exist proper classes. What this indicates is a limitation of first-order logic, because set theory is supposed to be an axiomatization of how we like to manipulate collections, and it turns out that not only do the collections described by first-order logic fail to directly admit these operations, but that even if we tried to indirectly implement the manipulations of collections we would like to do, we would not be able to manipulate all collections this way.

The previous two paragraphs have an analog in categorical foundations. First, observe that the class of elements of a set $X$ is the same as the class of functions from a singleton $\{*\}\to X$. Therefore, we can redefine a class to be small if it is (in bijection with) the class of morphisms to an object $X$ from some fixed terminal object.

Second, it is not too difficult to show by a diagonalization argument that if a category is such that, for some class containing all morphisms of the category, every object of the category admits a power by that class, then between any two objects there exists at most one morphism. Consequently, if we are to have a good category theory, we need a distinction between small classes for which a category can be complete (have all limits indexed by small diagrams, i.e. admit all small constructions) without being trivial.

Finally, what makes a notion of smallness good for the purposes of category theory is being what is sometimes called an arity class, which roughly says that a family of collections indexed by a small collection consists of small collections if and only if its disjoint union is a small collection. Important example of small classes in this sense are the finite classes, and, in set theory, the classes of elements of regular cardinals.

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