|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-level 1-type/h-prop|
|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 monad||modal type theory, monad (in computer science)|
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.
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.
A survey of the field of modal type theory is in the collections
A general relation between modalities in type theory and monads (in computer science) was noted in
This was observed to systematically yield constructive modal logic in (independently)
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
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)
Formalization of modalities in homotopy type theory is discussed also around def. 1.11 of
Giuseppe Primiero, A multi-modal dependent type theory (pdf)
A list of related references is also kept at