nLab initiality conjecture



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

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
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



The initiality conjecture in type theory states that the term model of a type theory should be an initial object in the category of models of that type theory. Initiality guarantees that the relation between type theory and category theory works as expected, hence that formal syntactical proofs in type theory match theorems in categories that interpret these type theories.

A careful proof of initiality for the special case of the calculus of constructions was given in Streicher 91. Since then, initiality for more complex type theories (such as Martin-Löf dependent type theory) has often been treated as established, as a straightforward extension of Streicher’s result, but never written up carefully for a larger theory.

Around 2010, various researchers (notably Voevodsky 15, 16, 17) raised the question of whether these extensions really were sufficiently straightforward to consider them established without further proof. Since then, views on the status of initiality have varied within the field; but the issue has been, at least, a frustrating unresolved point.

A proof of the initiality conjecture for a full-featured Martin-Löf type theory is given/announced in de Boer 20, Brunerie-Lumsdaine 20.

(text adapted from Brunerie-Lumsdaine 20)


A proof of the initiality conjecture for Martin-Löf dependent type theory is implicit in the proof of its generalized algebraic semantics (for more on this see Uemura 2019/21 below), due to:

Proof of the initiality conjecture for the calculus of constructions:

  • Thomas Streicher, Chapter 4 of: Semantics of type theory – Correctness, completeness and independence results, Progress in Theoretical Computer Science, Birkhäuser Boston, Inc., Boston, MA, 1991, xii+298 pp., (ISBN:0-8176-3594-7, doi:10.1007/978-1-4612-0433-6)

Relevance of proof of more general versions of the conjecture was amplified in:

Early status reports on the full proof appeared in:

A full proof of the initiality conjecture for full Martin-Löf type theory, formalized in Agda, is given/announced in:

and in

most of those who believed initiality to remain unresolved have been convinced by Taichi Uemura’s doctoral thesis, which gives a more detailed alternative to Cartmell’s 1978 proof for a more structured class of theories called second-order generalized algebraic theories — a class that includes homotopy type theory, cubical type theory, and many other type theories. [Paraphrased from a comment by Jon Sterling, Jul 19, 2022]

Last revised on July 22, 2022 at 11:34:17. See the history of this page for a list of all contributions to it.