|logic||category theory||type theory|
|true||terminal object/(-2)-truncated object||h-level 0-type/unit type|
|false||initial object||empty type|
|proposition||(-1)-truncated object||h-proposition, mere proposition|
|cut rule||composition of classifying morphisms / pullback of display maps||substitution|
|cut elimination for implication||counit for hom-tensor adjunction||beta reduction|
|introduction rule for implication||unit for hom-tensor adjunction||eta conversion|
|logical conjunction||product||product type|
|disjunction||coproduct ((-1)-truncation of)||sum type (bracket type of)|
|implication||internal hom||function type|
|negation||internal hom into initial object||function type into empty type|
|universal quantification||dependent product||dependent product type|
|existential quantification||dependent sum ((-1)-truncation of)||dependent sum type (bracket type of)|
|equivalence||path space object||identity type|
|equivalence class||quotient||quotient type|
|induction||colimit||inductive type, W-type, M-type|
|higher induction||higher colimit||higher inductive type|
|completely presented set||discrete object/0-truncated object||h-level 2-type/preset/h-set|
|set||internal 0-groupoid||Bishop set/setoid|
|universe||object classifier||type of types|
|modality||closure operator, (idemponent) 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|
Following (Moggi91, Benton-Bierman-de Paiva 95, Kobayashi 97) modal type theory is specifically understood as being a type theory equipped with (co-)monads on its type system, representing the intended modalities. Since monads in computer science embody a notion of computation, some authors also speak of computational type theory here (Benton-Bierman-de Paiva 95, Fairtlough-Mendler 02).
The starting point for Moggi’s work is an explicit semantic distinction between computations and values. If is an object which interprets the values of a particular type, then is the object which models computation of that type . For a wide variety of notions of computation, the unary operator turns out to have the categorical structure of a strong monad on an underlying cartesian closed category of values. On a purely intuitive level and particularly if one thinks about non-termination, there is certainly something appealing about the idea that a computation of type represents the possibility of a value of type .
When the underlying type theory is homotopy type theory these modalities are a “higher” generalization of traditional modalities, with “higher” in the sense of higher category theory: they have categorical semantics in (∞,1)-categories given by (∞,1)-monads. See (Shulman 12, HoTTBook, section 7.7) for definition of such higher modalities, and see at reflective subuniverse.
See for instance (Benton-Bierman-de Paiva, section 3.2), (Kobayashi), (Gabbay-Nanevski, section 8), (Gaubault-Larrecq, Goubault, section 5.1), (Park-Harper, section 2.6), (Shulman) as examples of the first, and (Moggi, def. 4.7), (Awodey-Birkedal, section 4.2) as examples of the second.
As a special case of the modalities-are-monads relation, a Grothendieck topology on the site underlying a presheaf topos is equivalently encoded in the sheafification monad induced by the sheaf topos geometric embedding. More generally, any geometric subtopos is equivalently represented by a left-exact idempotent monad.
When restricted to act on (-1)-truncated objects (i.e. subterminal objects or more generally monomorphisms), this becomes a universal closure operator. When internalized as an operation on the subobject classifier, this becomes the corresponding Lawvere-Tierney operator. This modal perspective on sheafification was maybe first made explicit by Bill Lawvere:
A Grothendieck ‘topology’ appears most naturally as a modal operator of the nature ‘it is locally the case that’ (Lawvere).
More discussion along these lines is in (Goldblatt), where this kind of modality is called a geometric modality.
For higher toposes, it is no longer true in general that a subtopos is determined by its behavior on the -truncated objects, but we can still regard the entire sheafification monad as a higher modality in the internal homotopy type theory.
By adding to homotopy type theory three (higher) modalities that encode discrete types and codiscrete types and hence implicitly a non-(co)-discrete notion of cohesion one obtained cohesive homotopy type theory. Adding furthermore modalities for infinitesimal (co)discreteness yields differential homotopy type theory.
M. Fairlough, Michael Mendler, Propositional lax logic, Information and computation 137(1):1-33 (1997)
Satoshi Kobayashi, Monad as modality, Theoretical Computer Science, Volume 175, Issue 1, 30 (1997), Pages 29–74
See also the type-theoretic generalization of adjoint logic, as discussed in
The modal systems “CL” and “PLL” in (Benton-Bierman-Paiva) and (Fairlough-Mendler), respectively, turn out to be equivalent to the geometric modality of Goldblatt. The system CS4 in (Kobayashi) yields a constructive version of S4 modal logic.
Explicit mentioning of type theory equipped with such a monad as modal type theory or computational type theory is in
Proceedings of the Tenth Workshop on Logical and Semantic Frameworks, with Applications (LSFA 2015)
A survey of the field of modal type theory is in the collections
M. Fairtlough, M. Mendler, Eugenio Moggi (eds.), Modalities in Type Theory, Mathematical Structures in Computer Science, Vol. 11, No. 4, (2001)
Valeria de Paiva, Rajeev Goré, Michael Mendler, Modalities in constructive logics and type theories, Special issue of the Journal of Logic and Computation, editorial pp. 439-446, Vol. 14, No. 4, Oxford University Press, (2004) (pdf)
Modal type theory with an eye towards homotopy type theory is discussed in
Formalization of modalities in homotopy type theory is discussed
and specifically in cohesive homotopy type theory in
Urs Schreiber, Michael Shulman, Quantum gauge field theory in Cohesive homotopy type theory, in Ross Duncan, Prakash Panangaden (eds.) Proceedings 9th Workshop on Quantum Physics and Logic, Brussels, Belgium, 10-12 October 2012 (arXiv:1408.0054)
Monadic modal type theory with idempotent monads/monadic reflection is discussed in
Oleg Kiselyov and Chung-chieh Shan, Embedded Probabilistic Programming. Working conference on domain-specific languages, (2009) (pdf)
Giuseppe Primiero, A multi-modal dependent type theory (pdf)
Jean Goubault-Larrecq, Éric Goubault, On the geometry of intuitionistic S4 proofs, Homology, homotopy and applications vol 5(2) (2003)
A list of related references is also kept at