nLab subsingleton

Redirected from "subterminal".
Note: subterminal object and subsingleton both redirect for "subterminal".

Contents

Definition

A subsingleton generally refers to a subset (of some ambient set AA) having at most one element. That is, it is a subset BB of AA such that any two elements of BB are equal.

Of course, classically any subsingleton is either empty or a singleton, but constructively this need not hold. In a topos, the “object of subsingletons in AA” is the partial map classifier for AA, often denoted A A_\bot.

Remark

Sometimes a slightly different convention is used: There what we call subsingletons are called subterminals, and a subset BB of AA is a subsingleton if and only if there exists an element aAa \in A such that every element of BB is equal to aa. With this nomenclature, any subsingleton is a subterminal, but the converse doesn’t hold in general. (See flabby sheaf for a class of examples where the converse does hold.)

 Generating subsingletons from propositions

Given a proposition PP and a singleton {*}\{*\}, one can construct the subsingleton

{p{*}|(p=*)P}\{p \in \{*\} \vert (p = *) \wedge P\}

This means that one can use the set-theoretic operations to define the BHK interpretation of logic: Given two propositions PP and QQ, purely PP or QQ is given by an element of the binary disjoint union

{p{*}|(p=*)P}{p{*}|(p=*)Q}\{p \in \{*\} \vert (p = *) \wedge P\} \uplus \{p \in \{*\} \vert (p = *) \wedge Q\}

and the pure existential quantifier of a predicate P(x)P(x) on a set AA is given by an element of the indexed disjoint union

xA{p{*}|(p=*)P(x)}\biguplus_{x \in A} \{p \in \{*\} \vert (p = *) \wedge P(x)\}

Relation to the principles of omniscience

Suppose that AA is a subsingleton. Then the existential quantifier x.xA\exists x.x \in A which says that AA is an inhabited set is equivalent to the existential quantifier x:A.(λt.1)(x)=1\exists x:A.(\lambda t.1)(x) = 1, where λt.1\lambda t.1 is the constant function from AA to the boolean domain which takes elements of AA to the boolean 1bool1 \in \mathrm{bool}. Then, one can show that various principles of omniscience are equivalent to weak versions of the internal excluded middle, via the special case of the constant function λt.1\lambda t.1:

See also

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

logicset theory (internal logic of)category theorytype theory
propositionsetobjecttype
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject 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 September 25, 2024 at 21:38:07. See the history of this page for a list of all contributions to it.