nLab classical modality

Idea

In as far as a bireflective subcategory inclusion

$$\mathcal{C} \xhookrightarrow{\phantom{-} \iota \phantom{-}} \mathcal{B} \xrightarrow{\phantom{-} \beta \phantom{-}} \mathcal{C} \,, \;\;\;\;\;\;\; \beta \dashv \iota \dashv \beta$$

is understood as an inclusion (via forming of zero object-bundles) of a category $\mathcal{C}$ of “classical” objects into a category $\mathcal{B}$ of bundles of infinitesimal fibers over these (as in infinitesimal cohesion), it makes sense to address the corresponding bireflecting idempotent Frobenius monad

$$\natural \,\coloneqq\, \iota \circ \beta \;\colon\; \mathcal{B} \longrightarrow \mathcal{B}$$

as the modal operator which projects out the “mode of being classical”, hence the classical modality.

For example, interpreting a tangent $\infty$-topos $T \mathbf{H}$ (generally an $\infty$-topos of parameterized module spectra) as the home of a kind of higher quantum information theory (as discussed at motivic quantization and quantum circuits via dependent linear types) then its infinitesimal cohesion $\iota \,\colon\, \mathbf{H} \hookrightarrow T \mathbf{H}$ characterizes exactly the “classical objects” in the standard sense of classical vs quantum mechanics.

A modal homotopy type theoretic formulation of the classical modality $\natural \,\colon\, T \mathbf{H} \to T \mathbf{H}$ in the internal language of such tangent $\infty$-toposes is proposed in Riley, Finster & Licata (2021), its combination with a linear multiplicative conjunction $\otimes$ (and the required bunched logic) has been developed by Riley (2022) and the interpretation as a “classical modality” in the sense of classical/quantum mechanics is discussed in Myers et al. (2023).

Related concepts

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

  $(\esh \dashv \flat \dashv \sharp )$

  • discrete object, codiscrete object, concrete object

  • points-to-pieces transform

  • structures in cohesion

• dR-shape modality$\dashv$ dR-flat modality

  $\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

• classical modality

tangent cohesion

• differential cohomology diagram

differential cohesion

• (reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)

  $(\Re \dashv \Im \dashv \&)$

  • reduced object, coreduced object, formally smooth object

  • formally étale map

  • structures in differential cohesion

graded differential cohesion

• fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality

  $(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$

singular cohesion

$$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$$

References

A system of inference rules for a classical modality added to homotopy type theory has been proposed in

• Mitchell Riley, Eric Finster, Daniel R. Licata, Synthetic Spectra via a Monadic and Comonadic Modality [arXiv:2102.04099]

and combined with a linear multiplicative conjunction and the necessary bunched logic in

• Mitchell Riley, A Bunched Homotopy Type Theory for Synthetic Stable Homotopy Theory, PhD Thesis (2022) [doi:10.14418/wes01.3.139, ir:3269, pdf]

The role as a “classical modality” (and introducing this terminology) making the resulting linear homotopy type theory a formal quantum language is indicated in

• David Jaz Myers, Mitchell Riley, Hisham Sati, Urs Schreiber: Quantum Certification via Linear Homotopy Types

with further exposition in:

• Urs Schreiber: Quantum Data Types via Linear Homotopy Type Theory, talk at Workshop on Quantum Software @ QTML 2022, Naples (Nov 2022) [slides:pdf]