This entry is about scales in algebra and linear logic. For scales in geometry and physics, see length scale.
symmetric monoidal (∞,1)-category of spectra
The idea of a scale comes from Peter Freyd, in his attempt to give an algebraic (in the sense of universal algebra) description of the real interval, which he believed to be more fundamental than the real numbers themselves in analysis, and of concepts from analysis such as Lipschitz continuity, limits, differentiation and differential equations. Freyd also discovered that scales are models of multiplicative-additive linear logic with a midpoint operation.
A scale is a minor scale satisfying the scale identities:
for all and in ,
for all and in ,
Every scale with is trivial.
As a scale is a closed midpoint algebra, a scale has a partial order . For all and in , if and only if .
A subset of a scale is an ideal if , and if and only if and . An ideal is a zoom-invariant ideal if it is closed under -zooming.
A subset of a scale is a -face if , and if and only if and .
Every -face is a zoom-invariant ideal.
Given an element in scale , a principal -face is the subset of all in such that for all large in .
A Jacobson radical of a scale is the set of all in such that for all in , . is semi-simple if is trivial.
The proof of the Linear Representation Theorem in section 8 of Algebraic Real Analysis by Peter Freyd requires the use of excluded middle through its implicit definition of the quasiorder from the algebraically defined partial order . In particular, that every scale is a *-autonomous category and thus a model for linear logic and that every equational axiom added to the theory of minor scales is either a consequence of the scale identity for scales or is inconsistent with the theory of minor scales are classical results, as certain lemmas used in the proofs have only been derived from the scale identities through the Linear Representation Theorem. The same is true of the definition of simple scales in section 10, of the algebraic construction of the standard interval from simple scales in section 11, and various results involving absolute retracts in section 25. Since quasiorders can be constructed from partial orders in any inequality space, these results hold if the scales have a tight apartness relation, but it is unknown if these results still hold for general scales in constructive mathematics.
The unit interval with , , , , , , and is an example of a scale.
The set of truth values in Girard’s linear logic is a scale.
Last revised on June 3, 2021 at 16:11:05. See the history of this page for a list of all contributions to it.