nLab
arity class

Arity classes

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 κ\kappa of small cardinalities such that

  1. 1κ1\in\kappa.

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

  3. κ\kappa is closed under indexed decompositions: if λκ\lambda\in\kappa and iλα(i)κ\sum_{i\in \lambda} \alpha(i)\in \kappa, then each α(i)\alpha(i) is also in κ\kappa.

A set or family is called κ\kappa-small if its cardinality belongs to κ\kappa. A theory or other object with a collection of “operations” whose inputs are all κ\kappa-small is called κ\kappa-ary.

Remark

By induction, the second condition implies closure under iterated indexed sums, in the sense that for any n2n\ge 2, we have

i 1λ 1 i 2λ 2(i 1) i n1λ n1(i 1,,i n2)λ n(i 1,,i n1)\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 κ\kappa if all the λ\lambda’s are. The first condition may be regarded as the case n=0n=0 of this (the case n=1n=1 being just ”λκ\lambda\in\kappa iff λκ\lambda\in\kappa”).

Remark

An alternative, more category-theoretic, way to state the second and third conditions is that for any function f:IJf:I\to J, if Jκ{|J|}\in\kappa, then Iκ{|I|}\in\kappa if and only if all fibers of ff are in κ\kappa.

Examples

  • The set {1}\{1\} is an arity class. A {1}\{1\}-ary object is called unary.

  • The set {0,1}\{0,1\} is an arity class. A{0,1}\{0,1\}-ary object is called subunary.

  • The set ω=={0,1,2,3}\omega = \mathbb{N} = \{0,1,2,3\dots\} is an arity class. An ω\omega-ary object is called finitary.

  • For any regular cardinal κ\kappa, the set of all cardinalities strictly less than κ\kappa is an arity class, which we abusively denote also by κ\kappa. The previous example ω\omega is a special case of this, as is {0,1}\{0,1\} if we consider 22 to be a regular cardinal.

  • In particular, if κ\kappa is the “size of the universe” — e.g., an inaccessible cardinal for which we have chosen to call sets of cardinality <κ\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 κ\kappa-ary objects infinitary or \infty-ary.

In classical mathematics, these examples in fact exhaust all arity classes. Classically, if λ\lambda is any cardinal number strictly greater than 11, then for any cardinal numbers μν\mu\le \nu, we can write ν\nu as a λ\lambda-indexed sum containing μ\mu. Hence, if an arity class contains any cardinality >1\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 may arise from a regular cardinal. Arguably, however, in constructive mathematics one should consider arity classes instead of regular cardinals.

Revised on June 19, 2013 03:32:31 by Toby Bartels (98.23.135.113)