nLab bosonic modality

Idea

On super smooth infinity-groupoids there is a modal operator $\stackrel{\rightsquigarrow}{(-)}$ which projects onto the bosonic components of the supergeometry. On formal dual superalgebras this is given by passing to the body. In terms of physical fields this is the projection onto boson fields, which are hence the modal types of $\rightsquigarrow$, and so it makes sense to speak of the bosonic modality.

This has a left adjoint $\rightrightarrows$ (which on superalgebras passes to the even-graded sub-algebra) and hence together these form an adjoint modality which may be thought of as characterizing the supergeometry. See at super smooth infinity-groupoid – Cohesion. With $\rightrightarrows$ being opposite to $\rightsquigarrow$ thereby, it makes sense to call it the fermionic modality.

Notice that the fermionic currents in physics (e.g. the electron density current) are indeed fermionic bilinears, i.e. are indeed in the even subalgebras of the underlying superalgebra.

$$\array{ fermions & \rightrightarrows &\stackrel{}{\dashv}& \rightsquigarrow & bosons }$$

The modal objects for $\rightsquigarrow$ are the bosonic objects.

The right adjoint of the bosonic modality is the rheonomy modality.

See also

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

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