κ-ary regular and exact categories
arity class: unary, finitary, infinitary
regularity
regular category = unary regular
coherent category = finitary regular
geometric category = infinitary regular
exactness
exact category = unary exact
An arity class is a class of cardinalities which is suitable to be the collection of arities? for the operations in an algebraic theory.
An arity class is a class $\kappa$ of small cardinalities such that
$1\in\kappa$.
$\kappa$ is closed under indexed sums: if $\lambda\in\kappa$ and $\alpha: \lambda \to\kappa$, then $\sum_{i\in \lambda} \alpha(i)$ is also in $\kappa$.
$\kappa$ is closed under indexed decompositions: if $\lambda\in\kappa$ and $\sum_{i\in \lambda} \alpha(i)\in \kappa$, then each $\alpha(i)$ is also in $\kappa$.
A set or family is called $\kappa$-small if its cardinality belongs to $\kappa$. A theory or other object with a collection of “operations” whose inputs are all $\kappa$-small is called $\kappa$-ary.
By induction, the second condition implies closure under iterated indexed sums, in the sense that for any $n\ge 2$, we have
is in $\kappa$ if all the $\lambda$‘s are. The first condition may be regarded as the case $n=0$ of this (the case $n=1$ being just “$\lambda\in\kappa$ iff $\lambda\in\kappa$”).
An alternative, more category-theoretic, way to state the second and third conditions is that for any function $f:I\to J$, if ${|J|}\in\kappa$, then ${|I|}\in\kappa$ if and only if all fibers of $f$ are in $\kappa$.
The set $\{1\}$ is an arity class. A $\{1\}$-ary object is called unary.
The set $\{0,1\}$ is an arity class. A$\{0,1\}$-ary object is called subunary.
The set $\omega = \mathbb{N} = \{0,1,2,3\dots\}$ is an arity class. An $\omega$-ary object is called finitary.
For any regular cardinal $\kappa$, the set of all cardinalities strictly less than $\kappa$ is an arity class, which we abusively denote also by $\kappa$. The previous example $\omega$ is a special case of this, as is $\{0,1\}$ if we consider $2$ to be a regular cardinal.
In particular, if $\kappa$ is the “size of the universe” — e.g., an inaccessible cardinal for which we have chosen to call sets of cardinality $\lt\kappa$ small, or literally the proper-class cardinality of the universe, depending on how one thinks of it —, then it is an arity class. In this case we call $\kappa$-ary objects infinitary or $\infty$-ary.
In classical mathematics, these examples in fact exhaust all arity classes. Classically, if $\lambda$ is any cardinal number strictly greater than $1$, then for any cardinal numbers $\mu\le \nu$, we can write $\nu$ as a $\lambda$-indexed sum containing $\mu$. Hence, if an arity class contains any cardinality $\gt 1$, it must be down-closed, and a down-closed arity class must arise from a regular cardinal.
In constructive mathematics, however, not every arity class besides $\{1\}$ must be downward-closed, and not every downward-closed arity class must arise from a regular cardinal. Arguably, however, in constructive mathematics one should consider downward-closed arity classes instead of regular cardinals.