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

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



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)


Proof of the initiality conjecture for the calculus of constructions is due to:

  • 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:

Last revised on June 24, 2022 at 04:31:22. See the history of this page for a list of all contributions to it.