regular category = unary regular
coherent category = finitary regular
geometric category = infinitary regular
is closed under indexed sums: if and , then is also in .
is closed under indexed decompositions: if and , then each is also in .
By induction, the second condition implies closure under iterated indexed sums, in the sense that for any , we have
is in if all the ‘s are. The first condition may be regarded as the case of this (the case being just “ iff ”).
An alternative, more category-theoretic, way to state the second and third conditions is that for any function , if , then if and only if all fibers of are in .
The set is an arity class. A -ary object is called unary.
The set is an arity class. A-ary object is called subunary.
The set is an arity class. An -ary object is called finitary.
For any regular cardinal , the set of all cardinalities strictly less than is an arity class, which we abusively denote also by . The previous example is a special case of this, as is if we consider to be a regular cardinal.
In particular, if is the “size of the universe” — e.g., an inaccessible cardinal for which we have chosen to call sets of cardinality 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 -ary objects infinitary or -ary.
In classical mathematics, these examples in fact exhaust all arity classes. Classically, if is any cardinal number strictly greater than , then for any cardinal numbers , we can write as a -indexed sum containing . Hence, if an arity class contains any cardinality , 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 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.