Contents
Context
Modalities, Closure and Reflection
Contents
Definition
On a local topos/local (∞,1)-topos , hence with extra fully faithful right adjoint to the global section geometric morphism , is canonically induced the idempotent comonad . This modality sends for instance pointed connected objects to coefficients for flat principal ∞-connections, and may therefore be referred to as the flat modality. It is itself the left adjoint in an adjoint modality with the sharp modality . If is in addition a cohesive (∞,1)-topos then it is also the right adjoint in an adjoint modality with the shape modality .
In type theory
We assume a dependent type theory with crisp term judgments in addition to the usual (cohesive) type and term judgments and , as well as context judgments where is a list of crisp term judgments, and is a list of cohesive term judgments. A crisp type is a type in the context , where is the empty list of cohesive term judgments. In addition, we also assume the dependent type theory has typal equality and judgmental equality.
From here, there are two different notions of the flat modality which could be defined in the type theory, the strict flat modality, which uses judgmental equality in the computation rule and uniqueness rule, and the weak flat modality, which uses typal equality in the computation rule and uniqueness rule. When applied to a type they result in strict flat types and weak flat types respectively. The natural deduction rules for strict and weak flat types are provided as follows:
- Formation rule for weak and strict flat types:
- Introduction rule for weak and strict flat types:
- Elimination rule for weak and strict flat types:
- Computation rule for weak and strict flat types respectively:
Weak flat modalities are primarily used in cohesive objective type theories, while strict flat modalities are typically used in cohesive type theories with judgmental equality, such as cohesive Martin-Löf type theory (cohesive homotopy type theory or cohesive higher observational type theory.
Properties
Relation to discrete and codiscrete objects
cohesion
infinitesimal cohesion
tangent cohesion
differential cohesion
graded differential cohesion
singular cohesion
References
The terminology of the flat-modality in the above sense was introduced – in the language of -toposes and as part of the axioms on “cohesive -toposes” – in:
See also the references at local topos.
Early discussion in view of homotopy type theory and as part of a set of axioms for cohesive homotopy type theory is in
The dedicated type theory formulation with “crisp” types, as part of the formulation of real cohesive homotopy type theory, is due to: