# Characterizing categories by size

Usually one distinguishes five classes of categories by size, and there are examples for all of them:

• finite categories
• locally finite categories
• small categories
• locally small categories
• large categories

Strictly speaking there are two more classes characterized by sizes:

• discrete categories: $|\text{Hom}(X,X)|=1, |\text{Hom}(X,Y)|=0$ for $X\neq Y$

• indiscrete categories: $|\text{Hom}(X,Y)|=1$ for all $X,Y$

I wonder up to which point a more detailed classification of categories by size would make sense and/or for which sizes of categories there are examples resp. no examples.

Let the "size spectrum" of a category $\mathcal{C}$ be the functions $K_1, K_2$ from the class of cardinal numbers to itself (including the - improper - cardinality $\aleph_\infty$ of a proper class) which assign to each cardinality $\kappa$ the size of the class

$$O_1(\kappa) := \lbrace X \in \text{Obj}(\mathcal{C}) : |\text{Hom}(X,X)| = \kappa \rbrace$$ i.e. $K_1(\kappa) = |O_1(\kappa)|$ ("the number of objects with $\kappa$ morphisms to itself"), resp. the size of the class

$$O_2(\kappa) := \lbrace (X,Y) \in \text{Obj}(\mathcal{C})^2 : X \neq Y\ \text{and}\ |\text{Hom}(X,Y)| = \kappa \rbrace$$ i.e. $K_2(\kappa) = |O_2(\kappa)|$ ("the number of pairs of distinct objects with $\kappa$ morphisms from the first to the second").

With this definitions we have:

• all categories: $K_1(0) = 0$

• connected categories: $K_2(0) = 0$

• finite categories: $\Sigma_\kappa K_1(\kappa) < \aleph_0$, $K_1(\kappa) = K_2(\kappa) = 0$ for $\kappa \geq \aleph_0$

• locally finite categories: $K_1(\kappa) = K_2(\kappa) = 0$ for $\kappa \geq \aleph_0$

• small categories: $\Sigma_\kappa K_1(\kappa) < \aleph_\infty$, $K_1(\aleph_\infty) = K_2(\aleph_\infty) = 0$

• locally small categories: $K_1(\aleph_\infty) = K_2(\aleph_\infty) = 0$

• discrete categories: $K_1(\kappa) = 0$ for $\kappa \neq 1$, $K_2(\kappa) = 0$ for $\kappa > 0$

• indiscrete categories: $K_1(\kappa) = K_2(\kappa) = 0$ for $\kappa \neq 1$

The characterizations have basically the form $K_1(\kappa) = 0$, $K_2(\kappa) = 0$ for $\kappa$ in specific subclasses $\mathbb{K}_1$, $\mathbb{K}_2$ of the cardinal numbers.

I wonder whether there are "interesting" and simply definable subclasses $\mathbb{K}_1$, $\mathbb{K}_2$ of the cardinal numbers such that there are no categories with $K_1(\kappa) = 0$ for $\kappa \in \mathbb{K}_1$ and/or $K_2(\kappa) = 0$ for $\kappa \in \mathbb{K}_2$.

This is an interesting idea, which I can comment almost nothing about in full generality. But regarding your singled-out question at the end, the answer is no-ish. Specifically, if $|S|=\kappa$ and $\cdot$ is a monoid structure on $S$, then there's a category with one object whose self-maps are $S$ with composition $\cdot$, so that $K_1(\kappa)=1$. But can we find such a monoid structure? It's known that there's a group of every cardinality iff the axiom of choice holds, but I don't know analogous facts about monoid structures. The proof for groups relies fundamentally on the freeness of a group's self-action, so the proof itself doesn't generalize. But defining a monoid structure on an arbitrary set feels quite choicy, so maybe the same fact will hold.
Once we have a monoid $S$ of size $\kappa,$ we can also set up a category $\mathcal{C}$ on objects $\star,\square$ with $\mathcal{C}(a,a)\cong S$ for $a$ either object. Then for every $s\in S$ put $sf$ into $\mathcal{C}(\star,\square)$ and define $s_1\circ s_2f\circ s_3=s_1s_2s_3 f$. Note for this we need to pick canonical isomorphisms between $\mathcal{C}(\star,\star)$, $\mathcal{C}(\square,\square),$ and $S$. Let $\mathcal{C}(\square,\star)=\{g\}$ with $s_1\circ g\circ s_2=g.$ Finally define $g\circ sf=s:\star\to\star$ and similarly for $sf\circ g$. Associativity of composition follows from that of $S$, so this finishes the definition of $\mathcal{C}$, which gives an example with $K_2(\kappa)=1$.
We can extend these examples via disjoint union of categories to give $K_1,K_2$ any cardinal or $\aleph_\infty$ whenever there's a monoid structure on $\kappa$. So this doesn't look like a useful angle from which to analyze your functions.
Update: I asked about the existence of monoid structures on arbitrary sets, and Zhen Lin pointed out there's an easy one that doesn't need choice: on a set with at least two elements, pick a $0$ and a $1$ and define the product of any pair of elements neither of which is $1$ to be $0$. So indeed the $K_i$ take on every value at every $\kappa.$