basic constructions:
strong axioms
further
A regular cardinal is is a cardinal number that is ‘closed under union’. The category of sets bounded by a regular cardinal has several nice properties, making it a universe that is handy for some purposes but falls short of being a Grothendieck universe. Unlike Grothendieck universes (which are based on inaccessible cardinals rather than regular cardinals), it is easy to prove that (even uncountable) regular cardinals exist.
An infinite cardinal $\kappa$ is a regular cardinal if it satisfies the following equivalent properties:
no set (in a material set theory) of cardinality $\kappa$ is the union of fewer than $\kappa$ sets of cardinality less than $\kappa$.
no set (in a structural set theory) of cardinality $\kappa$ is the disjoint union of fewer than $\kappa$ sets of cardinality less than $\kappa$.
given a function $P \to X$ (regarded as a family of sets $\{P_x\}_{x\in X}$) such that ${|X|} \lt \kappa$ and ${|P_x|} \lt \kappa$ for all $x \in X$, then ${|P|} \lt \kappa$.
the category $\Set_{\lt\kappa}$ of sets of cardinality $\lt\kappa$ has all colimits (or just all coproducts) of size $\lt\kappa$.
the cofinality of $\kappa$ is equal to $\kappa$.
A cardinal that is not regular is called singular.
Traditionally, one requires regular cardinals to be infinite. This clause may be removed, in which case $0$, $1$, and $2$ are all regular cardinals.
Other modifications of the definition which are equivalent for infinite cardinals may include some of $0$, $1$, and $2$ but not all. For instance, if we regard an indexed disjoint union $\sum_{i\in\lambda} \alpha_i$ as a binary operation taking as input $\lambda$ and a $\lambda$-indexed family $\alpha$, then closure under this binary operation as in the above definition also entails closure under the ternary version $\sum_{i\in\lambda} \sum_{j\in \mu_i} \alpha_{i,j}$, and so on. The unary version is simply the identity operation, whereas the nullary version will always output the singleton set $1$. (This can be seen by thinking in terms of trees of uniform finite height, or remembering that a dependent sum includes a binary cartesian product as a special case, so a nullary dependent sum should at least include a nullary product.) Thus, from this perspective, $2$ is a regular cardinal, but $0$ and $1$ are not. In applications for which this perspective is the relevant one, such as familial regularity and exactness, one may more precisely be interested in an arity class rather than a regular cardinal.
We may rule out all three finite regular cardinals by additionally generalising from indexed disjoint unions to finitary disjoint unions.
Then in terms of $Set_{\lt\kappa}$, the (potential) conditions on a (possibly finite) regular cardinal are as follows:
These are all variations on the theme of closure under disjoint unions.
Clauses (2–4) hold of all infinite cardinals, while clauses (2&3) together force $\kappa$ to be greater than any finite cardinal. However, if we require only clauses (1&2), then $2$ is a regular cardinal.
Thinking of a regular cardinal as a cardinal number makes the most sense using the axiom of choice. Otherwise, we probably want to think of it as a collection of cardinals, or equivalently think of it as the category $Set_{\lt\kappa}$.
From this perspective, a regular cardinal is a full subcategory of $Set$ that is closed under taking quotient objects and satisfies the condition on $Set_{\lt\kappa}$ above. We can then recover $\kappa$ as the smallest cardinal number greater than every cardinal in $Set_{\lt\kappa}$, if we accept the axiom of choice.
Note that if we require only conditions (1&2) on $Set_{\lt\kappa}$, then (even classically), $\{1\}$ is an acceptable (and finite) regular collection of cardinals, even though it is not actually of the form $Set_{\lt\kappa}$ for any cardinal number $\kappa$.
In the absence of the axiom of choice, it is not clear that there exist arbitrarily large regular cardinals. Thus in weaker foundations, regular cardinals (or “regular sets of cardinals”) can be regarded as a large cardinal property.
At least if “regular cardinal” has its classical meaning of a particular ordinal, then the statement that there exist arbitrarily large regular cardinals is independent of ZF; in fact it is consistent with ZF that all uncountable cardinals are singular. A foundational axiom which is related to the existence of regular cardinals (but considers them as sets with various closure properties, rather than cardinal numbers) is the regular extension axiom.
Assuming the axiom of choice, the successor of any infinite cardinal, such as $\aleph_0$, is a regular cardinal.
In the case of $\aleph_0$, this means that a countable union of countable sets is countable. Note that this implies that there exist arbitarily large regular cardinals: for any cardinal $\lambda$ there is a greater regular cardinal, namely $\lambda^+$.
Under the axiom of choice, the successor of a cardinal number is the Hartogs number (see there): if $\lambda$ is the cardinality of $X$, then $\lambda^+$ is the order type of the well-ordered set $\aleph(X)$. If $(\lambda_\alpha) = (0 = \lambda_0 \lt \lambda_1 \lt \ldots)$ is a set of ordinals with least upper bound $\lambda^+$, and supposing, working toward a contradiction, that this set has cardinality $\leq \lambda$, then the corresponding initial segments $X_\alpha$ of $\aleph(X)$ provide a partition
and by construction of $\aleph(X)$, each $X_{\alpha+1} \setminus X_\alpha$ has cardinality $\leq \lambda$. Thus the union would have at most $\lambda \cdot \lambda = \lambda$ elements, which is less than the cardinality $\lambda^+$ of $\aleph(X)$, contradiction.
$\aleph_\omega = \bigcup_{n\in \mathbb{N}} \aleph_n$ is singular, more or less by definition, since $\aleph_n\lt\aleph_\omega$ and ${|\mathbb{N}|} = \aleph_0 \lt\aleph_\omega$.
More generally, any limit cardinal that can be “written down by hand” should be singular, since if it were regular then it would be weakly inaccessible, and the existence of weakly inaccessible cardinals cannot be proven in ZFC (if $ZFC$ is consistent). We say ‘should’ rather than ‘must’, since there are exceptions, but they are sort of cheating: one (definable with Choice) is ‘the smallest limit cardinal that is regular if and only if some weakly innaccessible cardinal exists’.
Assuming the consistency (with ZFC) of ‘there is a proper class of strongly compact cardinals’, it is consistent with $ZF$ that every uncountable cardinal is singular (and in fact every infinite well-orderable cardinal has cofinality $\aleph_0$), a result due to Moti Gitik?. (Of course this conclusion is inconsistent with $ZFC$, in which many uncountable cardinals, starting with $\aleph_1$, are regular.)
Regular cardinals $\lambda$ are used in the definitions of $\lambda$-filtered colimits, $\lambda$-presentable objects, $\lambda$-accessible categories, locally $\lambda$-presentable categories, $\lambda$-ind-completion, and many notions derived from these, e.g., $\lambda$-combinatorial model categories.
Then notions make sense for all cardinals, not necessarily regular. However, the relevant concepts reduce to those for regular cardinals.
The relevance of regular cardinals for these concepts was already pointed out by Gabriel and Ulmer in their original treatise on locally presentable categories, where on page 2 we read:
Sei $\alpha$ eine Kardinalzahl, wobei $3 \le \alpha \lt \infty$. Eine geordnete Menge $(N,\le)$ heisst $\alpha$-gerichtet, wenn es für jede Familie $(\nu_i)_{i\in I}$ in $N$ mit $Kard(I) \lt \alpha$ ein $\mu$ gibt derart, dass $\nu_i\le\mu$. Sei $\beta$ die kleinste reguläre Kardinalzahl $\ge \alpha$. Dann ist jede $\alpha$-gerichtete Menge auch $\beta$-gerichtet. Wir setzen deshalb im folgenden zusatzlich voraus, dass $\alpha$ regulär ist (vgl. §0).
where the meaning of $\alpha\lt\infty$ is explained on page 13:
Ausserdem bezeichnen wir mit $\infty$ die kleinste Kardinalzahl, die nicht mehr zu $U$ gehört (die also in unserer Sprache keine Menge ist).
Dieser Arbeit liegt die Mengenlehre von Zermelo-Fraenkel und ein fest gewähltes
Universum $U$ zugrunde. Wir setzen dabei voraus, dass $U$ die Menge $\mathbf{N}$ der natürlichen Zahlen enthält.
If $\lambda$ is not a regular cardinal, then a category has $\lambda$-filtered colimits if and only if it has $\lambda^+$-filtered colimits, and $\lambda^+$ is always a regular cardinal (assuming the axiom of choice). In this case, a category is locally $\lambda$-presentable if and only if it is locally $\lambda^+$-presentable.
Reference: Proposition 5.6 in Rezk2021.
Peter Gabriel, Friedrich Ulmer, Lokal präsentierbare Kategorien, Lecture Notes in Mathematics 221, Springer 1971 (doi:10.1007/bfb0059396, MR0327863).
Charles Rezk, Generalizing accessible ∞-categories, 2021 (pdf).