nLab
bosonic modality
Redirected from "bosonic".
Contents
Context
Modalities, Closure and Reflection
Super-Geometry
Contents
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.
fermions ⇉ ⊣ ⇝ bosons
\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
infinitesimal cohesion
tangent cohesion
differential cohesion
graded differential cohesion
singular cohesion
id ⊣ id ∨ ∨ fermionic ⇉ ⊣ ⇝ bosonic ⊥ ⊥ bosonic ⇝ ⊣ R h rheonomic ∨ ∨ reduced ℜ ⊣ ℑ infinitesimal ⊥ ⊥ infinitesimal ℑ ⊣ & étale ∨ ∨ cohesive ʃ ⊣ ♭ discrete ⊥ ⊥ discrete ♭ ⊣ ♯ continuous ∨ ∨ ∅ ⊣ *
\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
}
Last revised on July 23, 2022 at 18:45:51.
See the history of this page for a list of all contributions to it.