nLab computer science



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
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/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


Category theory



Computer science studies programs and languages to express them, as well as the operation, application and design of computers and computer networks. This includes aspects relating to concurrency, semantics of programming languages, and aspects of mathematical logic.

From the nPOV, computer science is part of the computational trinity, together with logic and category theory.

Some (theoretical) computer scientists


A discussion of foundations of programming languages is in

A suggestion for a classification of structures arising in computer science is in

An old discussion on the n-cat café can be found here. The discussion revolved around

  • Joseph Goguen, A categorical manifesto, Mathematical Structures in Computer Science 1 (1991), 49-67.

for which also see A Categorical Manifesto.

Other aspects are treated in

Logical Methods in Computer Science is an open access journal for papers on Theoretical Computer Science and other areas of Computer Science in which logical methods play a large part.

category: software

Last revised on June 6, 2022 at 02:29:16. See the history of this page for a list of all contributions to it.