Size: Small, Large, Locally Small

Every category met so far has been described with the deliberately vague word collection: a collection of objects, a collection of arrows. The vagueness has a reason. For most of the categories that matter ($\mathbf{Set}$, $\mathbf{Grp}$, $\mathbf{Top}$), the objects are too numerous to form a set, and a theory that ignored the difference would inherit the paradoxes that set theory spent the early twentieth century escaping. The distinction matters concretely at the adjoint functor theorems, where a size hypothesis is what makes a construction go through.

Set-theoretic assumptions

Leinster's approach to foundations is to list the properties the category of sets is expected to have, rather than to build sets from a membership relation. The inventory, each item a fact about arrows:¹

• Initial and terminal objects. $\varnothing$ admits exactly one function to any set; a one-element set $1$ admits exactly one function from any set. Elements are recovered as arrows: a function $1\to B$ is precisely an element of $B$.
• Products and sums. Any family $(A_i{)}_{i\in I}$ has a product ${\prod }_i{A}_i$ (families of elements) and a sum ${\sum }_i{A}_i$ (disjoint union).
• Function sets. For sets $A,B$ there is a set $A^B$ of functions $B\to A$; the arithmetic notation is earned, since $A^{B+C}\cong A^B\times A^C$ and $(A^B{)}^C\cong A^{B\times C}$.
• Subsets and the two-element set. Subsets of $A$ correspond one-to-one with functions $A\to 2$; the powerset is $\mathcal{P}(A)=2^A$.
• Equalizers and quotients. $\{a\in A\mid f(a)=g(a)\}$ exists for any parallel pair, and any equivalence relation has a quotient $A/\sim$ whose projection has a universal factorization property.
• Natural numbers. $\mathbb{N}$ with $0$ and successor supports unique recursive definitions.
• Choice. Every surjection has a section.

None of these is a theorem; with no prior definition of set there is nothing to prove them from. They are the interface that the rest of mathematics programs against, and each clause is a statement about the category$\mathbf{Set}$ rather than about membership.

Cardinality and Cantor's theorem

Comparison of sizes is defined through arrows as well. Write $|A|\le |B|$ when an injection $A\to B$ exists; the expression $|A|$ has no standalone meaning. Identities and composites of injections make $\le$ reflexive and transitive, and the substance is the antisymmetry-up-to-isomorphism:

We write $|A|=|B|$ (same cardinality) for $A\cong B$, and $|A|<|B|$ for $|A|\le |B|$ with $|A|\ne |B|$. The central theorem:

Immediately, there is no biggest set: powerset always climbs strictly. And with one more step, no set-indexed family of sets is exhaustive:

Small, large, locally small

The word for a collection that may be too big to be a set is class; the usage here is informal but disciplined. All sets are classes; a class is small if it is a set and large otherwise. The proposition above says precisely that the class of isomorphism classes of sets is large. The crucial asymmetry:⁴

The classification of the standard examples:

Local smallness is why the hom-functors can honestly take values in $\mathbf{Set}$: for a locally small $\mathcal{C}$, each $\mathcal{C}(A,B)$ is an object of $\mathbf{Set}$, and the Yoneda machinery built on hom-functors quietly assumes it. Many authors fold local smallness into the definition of category for exactly this reason.

Essential smallness

Smallness is not invariant under equivalence: a category equivalent to a small one may have a proper class of objects, all redundantly isomorphic. The robust notion quotients that redundancy:

The category of finite sets has a proper class of objects, but every finite set is isomorphic to some $\{1,\dots ,n\}$, so it is equivalent to a small category with objects indexed by $\mathbb{N}$. In an essentially small category the class of isomorphism classes of objects is small (equivalences induce bijections on isomorphism classes), and this gives the clean impossibility results:

The same strategy — show the category has at least as many non-isomorphic objects as there are sets — handles the algebraic cases. For ${\mathbf{Vect}}_k$: given any set-indexed family $(V_i)$ of vector spaces, let $S=\mathcal{P}({\sum }_{i}U(V_i))$ where $U$ is the forgetful functor; the free space $F(S)$ contains $S$ as a basis, so $|U(V_i)|<|S|\le |UF(S)|$ for every $i$, and $F(S)\ncong V_i$. Neither $\mathbf{Grp}$, $\mathbf{Ab}$, $\mathbf{Ring}$, nor $\mathbf{Top}$ is essentially small, by the same free-object argument.

Where the distinction matters

Three concrete places where ignoring the distinction produces error or nonsense.

• Self-membership. If the objects of $\mathbf{Set}$ formed a set $W$, then $W$ would be a candidate element of itself, and Cantor's theorem applied to $W$ against the family of all sets yields a set bigger than every set — a contradiction of exactly the Russell type. The objects of $\mathbf{Set}$ form a proper class, and Cantor's theorem speaks only about sets.
• Hom-functors need targets. $\mathcal{C}(A,-)$ is a functor into $\mathbf{Set}$ only when $\mathcal{C}$ is locally small. For a category with large hom-classes the representable machinery must be rebuilt against a bigger universe.
• Limits over large diagrams. A category with products indexed by a proper class collapses: a classical argument shows any such category is a preorder. The completeness notions of the limits module are therefore always small completeness — all limits over small diagrams — and the solution-set condition in the adjoint functor theorem exists precisely to tame a large construction by a small amount of data.

Some treatments avoid classes by positing a hierarchy of universes: each universe is a set large enough to contain all ordinary mathematics, with every universe an element of a bigger one. Small then means in the chosen universe, and $\mathbf{Set}$ becomes a legitimate object one level up. The bookkeeping is different; the discipline it enforces is the same.⁶

Historical remarks

Leinster closes his interlude with intellectual history that explains why category theory treats what are the elements really? as a malformed question.⁷

Cantor created set theory in the late nineteenth century and proved, among much else, that infinity comes in different sizes. The axiomatizers who followed (Zermelo, Fraenkel, and others, arriving at ZFC) took sets and membership as the primitive notions. That choice has consequences that sit oddly with mathematical practice:

• Everything is a set. In ZFC, $\sqrt{2}$ is a set, $\log$ is a set, and the number $4$ is officially $\{\varnothing ,\{\varnothing \},\{\varnothing ,\{\varnothing \}\},\{\varnothing ,\{\varnothing \},\{\varnothing ,\{\varnothing \}\}\}\}$: an encoding, like a file format, that nobody intends literally.
• Types are erased. Because everything is a set, the question is $\mathbb{Q}\in \sqrt{2}$? is grammatical, and even has an answer, one that depends on the encoding conventions and means nothing. Working mathematics has an instinctive type discipline (a number is not a function is not a group), and membership-based foundations discard it.
• Membership is global. Any two sets can be intersected, so does the cyclic group of order 10 meet $\mathbb{Z}$? is a legitimate ZFC question. Its answer is again encoding-dependent noise.

The categorical alternative takes sets and functions (with composition) as primitive, exactly the inventory of assumptions listed above. One popular categorical axiomatization can be summarized in ten clauses (composition is associative and unital; there are terminal and empty sets; functions are determined by their effect on elements; products, function sets, inverse images, powersets-via-$2$, natural numbers, and sections of surjections exist). Nothing in the list mentions a general notion of category; the axioms speak directly about sets and functions, so there is no circularity in founding categories on sets so axiomatized. The approach preserves types (asking whether an element of a group equals an open set of a space is ungrammatical, as it should be), and every meaningful theorem of Cantor's theory survives.

Category theory gets inside an object by probing it with arrows, never by inspecting members. An element of $A$ is an arrow $1\to A$; a subset of $A$ is an arrow $A\to 2$; a variable element is any arrow into $A$. The membership relation is not denied; it is reconstructed as composition, at which point it stops being primitive and becomes one construction among many. The next module specifies every basic construction of mathematics — products, quotients, function spaces, free objects — purely by the pattern of arrows around it.