nLab formal (infinity,1)-category theory



(,1)(\infty,1)-Category theory

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




Formal (,1)(\infty,1)-category theory is to ( , 1 ) (\infty,1) -category theory what formal category theory is to 1-category theory, that is: a synthetic approach to (,1)(\infty,1)-categories standing in relation to synthetic homotopy theory as (,1)(\infty, 1)-category theory stands to homotopy theory. Consequently, formal (,1)(\infty,1)-category theory has also been called synthetic (,1)(\infty,1)-category theory.


There have been two main styles of approaches to formal (,1)(\infty,1)-category theory: categorical, which are (,2)(\infty,2)-categorical in nature; and type theoretic, which propose deductive systems for reasoning about (,1)(\infty,1)-categories.

Category theoretic approaches

Type theoretic approaches


Category theoretic approaches

With (small) (∞,1)-categories modeled as quasi-categories, their homotopy 2-category was considered first in

and then developed further (in the generality of homotopy 2-categories of ∞-cosmoi) in:

In an (∞,1)-category theoretic version of proarrow equipments:

Type theoretic approaches

For synthetic (infinity, 1)-category theory in cubical type theory via bicubical sets:

  • Matthew Weaver, Daniel Licata, A Constructive Model of Directed Univalence in Bicubical Sets, in: Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science. LICS ’20, Association for Computing Machinery (2020) 915–928 [doi:10.1145/3373718.3394794]

Exposition in

  • Matthew Weaver, A Constructive Model of Directed Univalence in Bicubical Sets, talk at HoTTEST (April 2020) [pdf, video]

For synthetic (infinity, 1)-category theory in simplicial type theory:

A talk on synthetic (infinity,1)-category theory in simplicial type theory and infinity-cosmos theory:

Last revised on October 14, 2023 at 06:21:06. See the history of this page for a list of all contributions to it.