On a local topos/local (∞,1)-topos $\mathbf{H}$, hence with extra fully faithful right adjoint $coDisc$ to the global section geometric morphism $(Disc \dashv \Gamma)$, is canonically induced the idempotent comonad $\flat \coloneqq Disc\circ \Gamma$. This modality sends for instance pointed connected objects $\mathbf{B}G$ to coefficients $\flat \mathbf{B}G$ 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 $\sharp \coloneqq coDisc \circ \Gamma$. If $\mathbf{H}$ is in addition a cohesive (∞,1)-topos then it is also the right adjoint in an adjoint modality with the shape modality $\int$.
We assume a dependent type theory with crisp term judgments $a::A$ in addition to the usual (cohesive) type and term judgments $A \; \mathrm{type}$ and $a:A$, as well as context judgments $\Xi \vert \Gamma \; \mathrm{ctx}$ where $\Xi$ is a list of crisp term judgments, and $\Gamma$ is a list of cohesive term judgments. A crisp type is a type in the context $\Xi \vert ()$, 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:
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.
The codiscrete objects in a local topos are the modal types for the sharp modality.
The discrete objects in a local topos are the modal types for the flat modality.
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
The terminology of the flat-modality in the above sense was introduced – in the language of $(\infty,1)$-toposes and as part of the axioms on “cohesive $(\infty,1)$-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:
Last revised on November 15, 2022 at 20:36:14. See the history of this page for a list of all contributions to it.