nLab preset

Presets

Context

Foundations

Category theory

(,1)(\infty,1)-Category theory

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition, truth value(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/h-set
setinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
set of truth valuessubobject classifiertype of propositions
universeobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels

semantics

Presets

Idea

A preset is a set without an equality relation. Conversely, a set may be defined as a preset XX equipped with an equality relation (technically an equivalence prerelation on XX).

In his seminal work The Foundations of Constructive Analysis (1967), Errett Bishop explained what you must do to define a set (see also Bishop set) in three steps:

  1. You must state what one must do to construct an element of the set;
  2. Given two elements constructed as in (1), you must state what one must do to prove that the elements are equal;
  3. You must prove that the relation defined in (2) is reflexive, symmetric, and transitive (which can be phrased in similar ‘what one must do’ terms, but that's kind of wordy).

If you only do (1), then you don't have a set, according to Bishop; you only have a preset.

A given preset may define many different sets, depending on the equality relation. For example, the set Q +Q^+ of positive rational numbers may be defined using the same preset as the set Z +×Z +Z^+ \times Z^+ of ordered pairs of positive integers, but the equality relation is different; two pairs (a,b)(a,b) and (c,d)(c,d) of positive integers are equal iff a=ca = c and b=db = d, while two positive rational numbers a/ba/b and c/dc/d are equal iff ad=bca d = b c. (Of course, these definitions require that one already has the set Z +Z^+ of positive integers, including its equality relation, and the operation of multiplication on it.)

Prefunctions and prerelations

As functions go between sets, so prefunctions go between presets. (Bishop used the term ‘operation’ instead of ‘prefunction’, but ‘operation’ has many other meanings.) Even if XX and YY are sets, a prefunction from XX to YY is not the same as a function from XX to YY, because a prefunction need not preserve equality; that is, we may have a=ba = b without f(a)=f(b)f(a) = f(b). Conversely, we may define a function as a prefunction (between sets) that preserves equality; such a prefunction is said to be extensional.

For example, consider the identity prefunction on the underlying preset of both Q +Q^+ and Z +×Z +Z^+ \times Z^+, as defined above. From Z +×Z +Z^+ \times Z^+ to Q +Q^+, this is a function, since a/b=c/da/b = c/d if (a,b)=(c,d)(a,b) = (c,d). But from Q +Q^+ to Z +×Z +Z^+ \times Z^+, it is not a function, since (for example) 2/4=3/62/4 = 3/6 but (2,4)(3,6)(2,4) \neq (3,6). A related example is the operation of taking the numerator of a (positive) rational number; from Q +Q^+ to Z +Z^+, we may view this as a prefunction but not as a function, although it is a function on Z +×Z +Z^+ \times Z^+.

In general, the prefunctions from XX to YY form a preset, since there is no way to compare them for equality. (Of course, it is still impredicative, at least in the classical sense, to form this preset.) However, if YY is a set, then these prefunctions do form a set, with f=gf = g defined to mean that f(a)=g(a)f(a) = g(a) for every aa in XX. If XX is also a set, then the function set from XX to YY is a subset of this set of prefunctions.

Composition of prefunctions is also possible, but likewise does not preserve equality.

A (say binary) prerelation between XX and YY may be thought of as a prefunction from X×YX \times Y to truth values. Even if one is too predicative to allow a (pre)set of truth values, still one may have a notion of prerelation, by fiat if nothing else. Note that one can compare prerelations for equality; R=SR = S means that a Rba \sim_R b if and only if a Sba \sim_S b. (In other words, a preset of truth values becomes a set under the biconditional, so we can compare functions to it.) We define a relation between sets to be a prerelation that respects equality.

Many properties of relations can also be predicated of prerelations, but not all. In particular, prerelations may be reflexive, symmetric, and transitive, so we have a notion of equivalence prerelation, which completes the definition of sets in terms of presets. A prerelation may also be entire, but it makes no sense to ask if it is functional unless YY is a set. In that case, there is a correspondence between prefunctions and functional entire prerelations as usual. In general, however, there is no way to define the prerelation corresponding to a given prefunction (which would be a sort of pre-graph). In other words, the idea that functions are certain relations (namely the functional entire ones) does not extended to prefunctions and prerelations unless YY is a set.

In general, prerelations are

Presets, types, sets, and setoids

Presets do not have equality. However, the types in many type theories come equipped with some kind of identity relation, such as an equivalence relation or an identity type. Thus, strictly speaking, the types in the type theories are not presets.

The sets defined by Bishop as a preset with an equivalence relation do not have quotient sets, so there is still a distinction between Bishop sets and the quotient set of an object with an equivalence relation: this is the difference between a set and a setoid. Some people call a set with no quotient sets a “preset”, however, this use of “preset” is at odds with preset as something that does not have equality.

Formalisation

In category theory

One may hope to formalise the category of presets and prefunctions between presets, but unfortunately, presets and prefunctions do not form a category. Instead, they only form a magmoid.

In type theory

It is possible to develop type-theoretic foundations in which presets are not equipped with identity relations (only metamathematical identity or interconvertibilty judgements); see preset for some discussion. Unlike the case for types with identity relations, the presentation axiom is not provable in the base theory, although it is provable in the impredicative version (where identity relations can be defined, following Leibniz's definition of equality).

In logic

If an untyped first-order logic does not have equality of propositions or equality of predicates, then the domain of discourse is a preset instead of a set. This remains true for typed first-order logic with multiple types, which are presets if each type does not have local equality.

The sorts in Michael Makkai's FOLDS are presets. FOLDS is very different from the other foundations considered above, since it is based strictly on prerelations and has no notion of prefunction. As far as I can tell, it therefore does not prove the presentation axiom.

Applications

To make the principle of equivalence hold automatically, a category should have only a preset of objects and only its hom-sets as sets. Then a category whose set of objects is a set may be called a strict category, which is really a special case of a strict ∞-category. Alternatively, one may keep sets as sets but adopt preclasses; then a small category is strict but a large category is not.

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition, truth value(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/h-set
setinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
set of truth valuessubobject classifiertype of propositions
universeobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

Last revised on May 18, 2022 at 11:33:06. See the history of this page for a list of all contributions to it.