# nLab Heyting scale

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

The idea of a Heyting scale comes from Peter Freyd.

## Definition

### In terms of Heyting implication

A Heyting scale or chromatic scale is a scale $M$ with a Heyting implication operation $(-)\rightarrow(-):M \times M \to M$ such that

• $(\bot \rightarrow \bot)^\bullet = \bot$

• for all $a$ in $M$, $a \wedge (a \rightarrow \bot) = \bot$

• for all $a$ and $b$ in $M$, $((a \wedge b) \rightarrow \bot)^\bullet = (a \rightarrow \bot)^\bullet \wedge (b \rightarrow \bot)^\bullet$

## Properties

Every Heyting scale with $\bot = \top$ is trivial.

## Examples

The set of truth values in Girard’s linear logic is a Heyting scale.

## References

• Peter Freyd, Algebraic real analysis, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 (tac:20-10)

Last revised on June 2, 2021 at 15:14:21. See the history of this page for a list of all contributions to it.