nLab
cartesian theory

Context

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
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive 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, (idemponent) 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

A theory in cartesian logic.

Definition

References

Cartesian theories were introduced under different names in the early seventies by John Isbell, Peter Freyd and Michel Coste (cf. Johnstone 1979). A standard source is Johnstone (2002).

  • Peter Freyd, Cartesian Logic , Theor. Comp. Sci. 278 (2002) pp.3-21.

  • Peter Johnstone, A Syntactic Approach to Diers’ Localizable Categories , pp.466-478 in Springer LNM 753 Heidelberg 1979.

  • Peter Johnstone, Sketches of an Elephant II , Oxford UP 2002. (Around D1.3.4 p.833)

Revised on June 6, 2017 15:33:07 by Thomas Holder (82.113.121.174)