nLab
modal type theory

Idea

Modal type theory is a flavor of type theory with type formation rules for modalities, as in modal logic. A survey of the field is in (de Paiva-Goré-Mendler).

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. See (Shulman) for remarks on such higher modalities.

Properties

Relation to monads

At least in many cases, modalities in type theory are identified with monads or comonads on the underlying type universe, or on the subuniverse of propositions.

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.

Examples

Geometric modality – Grothendieck topology

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 $\mathrm{PSh}(C)\to \mathrm{Sh}(C)\hookrightarrow \mathrm{PSh}(C)$ 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 $(-1)$-truncated objects, but we can still regard the entire sheafification monad as a higher modality in the internal homotopy type theory.

Closure modality

The canonical monad on a local topos gives rise to a closure modality on propositions in its internal language, as described in (Awodey-Birkedal).

Cohesive and differential modality

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.

Related concepts

• closure operator

References

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)

and

• 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)

and

• Valeria de Paiva, Brigitte Pientka (eds.) Intuitionistic Modal Logic and Applications (IMLA 2008), . Inf. Comput. 209(12): 1435-1436 (2011) (web)

The historically first modal type theory, the interpretation of sheafification as a modality in the internal language of a Grothendieck topos goes back to the notion of Lawvere-Tierney operator

• Bill Lawvere, Quantifiers and sheaves, Actes, Congrès intern, math., 1970. Tome 1, p. 329 à 334 (pdf)

reviewed in

• Robert Goldblatt, Grothendieck topology as geometric modality, Mathematical Logic Quarterly, Volume 27, Issue 31-35, pages 495–529, (1981)

A general relation between modalities in type theory and monads (in computer science) was noted in

• Eugenio Moggi, Notions of computation and monads Information And Computation, 93(1), 1991. (pdf)

This was observed to systematically yield constructive modal logic in (independently)

• P.N. Benton , G.M. Bierman , Valeria de Paiva, Computational Types from a Logical Perspective I (1995) (web)

and

• M. Fairlough, M. Mendler, Propositional lax logic, Information and computation 137(1):1-33 (1997)

and

• Satoshi Kobayashi, Monad as modality, Theoretical Computer Science, Volume 175, Issue 1, 30 (1997), Pages 29–74

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.

Modal type theory with an eye towards homotopy type theory is discussed in

• UF-IAS-2012, Modal type theory

Monadic modal type theory with idempotent monads/monadic reflection is discussed in

• Andrzej Filinski, Representing Layered Monads, POPL 1999. (pdf)

• Andrzej Filinski, On the Relations between Monadic Semantics, TCS 375:1-3, 2007. (pdf)

• Andrzej Filinski, Monads in Action, POPL 2010. (pdf)

• Oleg Kiselyov and Chung-chieh Shan, Embedded Probabilistic Programming. Working conference on domain-specific languages, (2009) (pdf)

• Mike Shulman, Higher modalities (pdf)

Formalization of modalities in homotopy type theory is discussed also around def. 1.11 of

• Egbert Rijke, Bas Spitters, Sets in homotopy type theory (arXiv:1305.3835)

See also

• Frank Pfenning, Towards modal type theory (2000) (pdf)

• Frank Pfenning, Intensionality, Extensionality, and Proof Irrelevance in Modal Type Theory, Pages 221–230 of: Symposium on Logic in Computer Science (2001) (web)

• Aleksandar Nanevski, Frank Pfenning, Brigitte Pientka, Contextual Model Type Theory (2005) (web, slides)

• Giuseppe Primiero, A multi-modal dependent type theory (pdf)

• Murdoch Gabbay, Aleksandar Nanevski, Denotation of contextual modal type theory (CMTT): syntax and metaprogramming (pdf)

A modality in the internal language of a local topos is discussed in section 4.2 of

• Steve Awodey, Lars Birkedal, Elementary axioms for local maps of toposes, Journal of Pure and Applied Algebra, 177(3):215-230, (2003) (ps, pdf )

• Jean Goubault-Larrecq, Éric Goubault, On the geometry of intuitionistic S4 proofs, Homology, homotopy and applications vol 5(2) (2003)

• Sungwoo Park, Robert Harper, A modal language for Effects (2004) (web)

• Dan Licata, Robert Harper, A Monadic Formalization of ML5 (arXiv:1009.2793)

A list of related references is also kept at

• Valeria de Paiva, Intuitionistic Modal Logic Selected Publications (web)