nLab
selection monad

Contents

Context

Category theory

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(-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
coinductionlimitcoinductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
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

Contents

Idea

In a cartesian closed category, given an object SS, the selection monad, also known as the select monad, is the endofunctor J S(X)[[X,S],X]J_S(X) \mapsto [[X, S], X]. It is a strong monad.

There is a monad homomorphism from the selection monad to the continuation monad for SS, K S(X)=(XS)SK_S(X) = (X \to S) \to S, which sends ϵJ S(X)\epsilon \in J_S(X) to ϵ¯K S(X)\bar{\epsilon} \in K_S(X), where ϵ¯(p)=p(ϵ(p))\bar{\epsilon}(p) = p(\epsilon(p)).

If we understand the continuation monad as mapping an object to the generalized quantifiers over it, with SS a generalized truth value, a selection function for a generalized quantifier is an element of its preimage under the monad morphism.

For instance, a selection functional for the supremum functional sup:(XS)Ssup: (X \to S) \to S, when it exists, applied to a function, p:XSp: X \to S, gives a point in XX at which pp attains its maximum value.

Due to the resemblance of an algebra, J S(A)AJ_S(A) \to A, to Peirce's law in logic, ((pq)p)p((p \Rightarrow q) \Rightarrow p) \Rightarrow p , J SJ_S is also called the Peirce monad in (Escardó-Oliva 2012).

Properties

There is a distributive law TJ SJ STT J_S \Rightarrow J_S T for every strong monad TT (Fiore 2019).

References

Last revised on December 30, 2020 at 12:02:41. See the history of this page for a list of all contributions to it.