nLab
internal set theory

Contents

Context

2-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
-0-truncated higher colimitquotient 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

Just as one could define an internal logic, such as first order logic, higher order logic, or geometric logic in a (1,1)-category with sufficient structure, in a (2,1)-category, one should be able to define an internal set theory.

Defining a set theory internal to a (1,1)-category is not possible, unless one is talking about material set theories such as ZFC, because any groupoid or category internal to an ambient (1,1)-category is necessarily a strict groupoid or category, which violates the principle of equivalence and is thus evil. Instead, in order to construct an internal set theory, an internal notion of weak category is needed.

The basic ideas of the internal set theory induced by a given category C are:

  • the objects AA of CC are regarded as collections of things of a given type AA

  • the morphisms ABA\rightarrow B of CC are regarded as terms of type BB containing a free variable of type AA (i.e. in a context x:Ax:A)

  • A faithful morphism F:ABF:A \rightarrow B is regarded as a set family of sets by thinking of it as the discrete collection of all those things of type BB for which the type AA is inhabited. BB is regarded as an index set. If BB is a terminal object ** in CC, then AA and FF are equivalent as discrete objects and are regarded as sets.

    • The hom-set of morphisms in the weak category of discrete objects of AA and faithful morphisms are regarded as the collection of functions between two sets X:A*X:A \rightarrow * and Y:A*Y:A \rightarrow *. In particular, functions have set-theoretic equality.
  • Set theoretic operations are implemented by universal constructions on discrete objects.

and so on.

  • A dependent type over an object AA of CC may be interpreted as a morphism BAB \rightarrow A whose “fibres” represent the types B(x)B(x) for x:Ax:A. This morphism might be restricted to be a display map or a fibration.

See also

Created on February 28, 2021 at 11:53:05. See the history of this page for a list of all contributions to it.