nLab 2-type theory

Contents

Context

Type theory

natural deduction metalanguage, practical foundations

judgement
hypothetical judgement, sequent
- antecedents $\vdash$ consequent, succedents

type theory (dependent, intensional, observational type theory, homotopy type theory)

calculus of constructions

syntax object language

theory, axiom
proposition/type (propositions as types)
definition/proof/program (proofs as programs)
theorem

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logic	set theory (internal logic of)	category theory	type theory
proposition	set	object	type
predicate	family of sets	display morphism	dependent type
proof	element	generalized element	term/program
cut rule		composition of classifying morphisms / pullback of display maps	substitution
introduction rule for implication		counit for hom-tensor adjunction	lambda
elimination rule for implication		unit for hom-tensor adjunction	application
cut elimination for implication		one of the zigzag identities for hom-tensor adjunction	beta reduction
identity elimination for implication		the other zigzag identity for hom-tensor adjunction	eta conversion
true	singleton	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	empty set	initial object	empty type
proposition, truth value	subsingleton	subterminal object/(-1)-truncated object	h-proposition, mere proposition
logical conjunction	cartesian product	product	product type
disjunction	disjoint union (support of)	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	function set (into subsingleton)	internal hom (into subterminal object)	function type (into h-proposition)
negation	function set into empty set	internal hom into initial object	function type into empty type
universal quantification	indexed cartesian product (of family of subsingletons)	dependent product (of family of subterminal objects)	dependent product type (of family of h-propositions)
existential quantification	indexed disjoint union (support of)	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
logical equivalence	bijection set	object of isomorphisms	equivalence type
	support set	support object/(-1)-truncation	propositional truncation/bracket type
		n-image of morphism into terminal object/n-truncation	n-truncation modality
equality	diagonal function/diagonal subset/diagonal relation	path space object	identity type/path type
completely presented set	set	discrete object/0-truncated object	h-level 2-type/set/h-set
set	set with equivalence relation	internal 0-groupoid	Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
	equivalence class/quotient set	quotient	quotient type
induction		colimit	inductive type, W-type, M-type
higher induction		higher colimit	higher inductive type
-		0-truncated higher colimit	quotient inductive type
coinduction		limit	coinductive type
	preset		type without identity types
	set of truth values	subobject classifier	type of propositions
domain of discourse	universe	object classifier	type universe
modality		closure operator, (idempotent) monad	modal type theory, monad (in computer science)
linear logic		(symmetric, closed) monoidal category	linear type theory/quantum computation
proof net		string diagram	quantum circuit
(absence of) contraction rule		(absence of) diagonal	no-cloning theorem
		synthetic mathematics	domain specific embedded programming language

homotopy levels

semantics

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Idea

As type theory has categorical semantics in 1-categories, 2-type theory has semantics in 2-categories. In particular, a 2-type theory can have semantics in the 2-category Cat, which would mean that contexts and closed types are modelled as categories.

There are, potentially, many different kinds of “2-type theory” with different uses and semantics. 2-type theory is closely related to (and sometimes the same as) directed type theory.

Applications

The “mode theories” in some general approaches to modal type theory and adjoint type theory are a form of 2-type theory, where the 2-cells represent a general form of “structural rules” acting on modal judgments.
The 2-cells in 2-type theory can also be used to model rewriting, e.g. the process of $\beta$ -reduction.

Related concepts

type theory, logic
2-type theory, 2-logic

directed homotopy type theory
(∞,1)-type theory, (∞,1)-logic

References

On 2-type theory:

R.A.G. Seely, Modeling computations: a 2-categorical framework, LICS 1987 (pdf)
Richard Garner, Two-dimensional models of type theory, Mathematical structures in computer science 19.4 (2009): 687-736 (doi:10.1017/S0960129509007646, pdf)
Tom Hirschowitz, Cartesian closed 2-categories and permutation equivalence in higher-order rewriting. Logical Methods in Computer Science, Logical Methods in Computer Science Association, 2013, 9 (3), pp.10. (pdf)
Mike Shulman, 2-categorical logic
Philip Saville, Cartesian closed bicategories: type theory and coherence. PhD thesis, 2020. pdf.
A type system with semantics in virtual equipments:

Max S. New and Daniel R. Licata. 2023. A Formal Logic for Formal Category Theory. In Foundations of Software Science and Computation Structures - 26th International Conference, FoSSaCS 2023. DOI
Andrea Laretto, Fosco Loregian, and Niccolò Veltri. 2024. Directed equality with dinaturality. ArXiv

Application to adjoint logic and modal type theory:

Daniel Licata, Dependently Typed Programming with Domain-Specific Logics PhD Thesis (2011) (pdf) - chapter 7
Dan Licata, Mike Shulman, Adjoint logic with a 2-category of modes, in Logical Foundations of Computer Science 2016 (pdf, slides)
Daniel Licata, Mike Shulman, Mitchell Riley, A Fibrational Framework for Substructural and Modal Logics (extended version), in Proceedings of 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017) (doi: 10.4230/LIPIcs.FSCD.2017.25, pdf)