computational trinitarianism



Type theory

Category theory

Constructivism, Realizability, Computability



Under the identifications

  1. propositions as types

  2. programs as proofs

  3. relation between type theory and category theory

the following notions are equivalent:

  1. A proof of a proposition. (In logic.)

  2. A program with output some type. (In type theory and computer science.)

  3. A generalized element of an object. (In category theory.)

This is referred to as “computational trinitarianism” in (Harper), where also an exposition is given.

The central dogma of computational trinitarianism holds that Logic, Languages, and Categories are but three manifestations of one divine notion of computation. There is no preferred route to enlightenment: each aspect provides insights that comprise the experience of computation in our lives.

Computational trinitarianism entails that any concept arising in one aspect should have meaning from the perspective of the other two. If you arrive at an insight that has importance for logic, languages, and categories, then you may feel sure that you have elucidated an essential concept of computation–you have made an enduring scientific discovery. (Harper)

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


In the introduction of

  • Paul-André Melliès, Functorial boxes in string diagrams, Procceding of Computer Science Logic 2006 in Szeged, Hungary. 2006 (article)

the insight is recalled to have surfaced in the 1970s, with an early appearance in print being the monograph

  • Joachim Lambek, Phil Scott, Introduction to Higher Order Categorical Logic, Cambridge Studies in Advanced Mathematics Vol. 7. Cambridge University Press, 1986.

See also at History of categorical semantics of linear type theory for more on this.

A exposition of the relation between the three concepts is in

An exposition with emphasis on linear logic/quantum logic and the relation to physics is in

A discussion in the context of homotopy type theory is in

For further references see at programs as proofs, propositions as types, and relation between category theory and type theory.

Textbooks on the foundations of mathematics and foundations of programming language which connect via the common theme of type theory/categorical logic include the following:

Last revised on November 10, 2018 at 08:15:54. See the history of this page for a list of all contributions to it.