nLab arity class

Idea

An arity class is a class of cardinalities which is suitable to be the collection of arities for the operations in an algebraic theory.

Definition

An arity class is a class $\kappa$ of small cardinalities such that

1. $1\in\kappa$.

2. $\kappa$ is closed under indexed sums: if $\lambda\in\kappa$ and $\alpha: \lambda \to\kappa$, then $\sum_{i\in \lambda} \alpha(i)$ is also in $\kappa$.

3. $\kappa$ 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.

Remark

By induction, the second condition implies closure under iterated indexed sums, in the sense that for any $n\ge 2$, we have

$$\sum_{i_1\in\lambda_1} \; \sum_{i_2\in\lambda_2(i_1)} \cdots \sum_{i_{n-1} \in\lambda_{n-1}(i_1,\dots,i_{n-2})} \lambda_n(i_1,\dots,i_{n-1})$$

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 $\lambda\in\kappa$”).

Remark

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 $\kappa$.

Examples

• 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.

Related pages

• algebraic theory

• ∞-ary exact category, ∞-ary site

• arity space

References

• Michael Shulman, “Exact completions and small sheaves”. Theory and Applications of Categories, Vol. 27, 2012, No. 7, pp 97-173. Free online