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\begin{document}

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\section*{solid topos}

\begin{quote}%
under construction


\end{quote}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohesive_toposes}{}\paragraph*{{Cohesive Toposes}}\label{cohesive_toposes}

[[!include cohesive infinity-toposes - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{super_formal_smooth_sets}{Super formal smooth sets}\dotfill \pageref*{super_formal_smooth_sets} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The concept of \emph{[[solid topos]]} is meant to be a characterization of \emph{[[gros toposes]]} which further refines that of [[elastic toposes]] and [[cohesive toposes]]. The idea is that we may consistently regard the [[objects]] of such toposes as [[generalized spaces]] for some flavor of [[geometry]], and that the axioms on the topos determine aspects of the geometric nature of these [[generalized spaces]], as follows:

\begin{tabular}{l|l}
$\phantom{A}$[[gros topos]]$\phantom{A}$&$\phantom{A}$[[generalized spaces]] obey\ldots{}$\phantom{A}$\\
\hline 
$\phantom{A}$[[cohesion]]&$\phantom{A}$principles of [[differential topology]]$\phantom{A}$\\
$\phantom{A}$[[elastic topos&elasticity]]\\
$\phantom{A}$[[solid topos&solidity]]$\phantom{A}$\\
\end{tabular}

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{SuperDifferentialCohesion}\hypertarget{SuperDifferentialCohesion}{}
\textbf{([[solid topos]])}

Let $\mathbf{H}_{bos}$ be an [[elastic topos]] (\href{geometry+of+physics+--+categories+and+toposes#DifferentialCohesion}{this Def.}) over a [[cohesive topos]] $\mathbf{H}_{red}$ (\href{geometry+of+physics+--+categories+and+toposes#CohesiveTopos}{this Def.}). Then a \emph{solid topos} or \emph{super-differentially cohesive topos} over $\mathbf{H}_{bos}$ is a [[sheaf topos]] $\mathbf{H}$, which is

\begin{enumerate}%
\item a [[cohesive topos]] over [[Set]] (\href{geometry+of+physics+--+categories+and+toposes#CohesiveTopos}{this Def.}),


\item an [[elastic topos]] over $\mathbf{H}_{red}$ (\href{geometry+of+physics+--+categories+and+toposes#DifferentialCohesion}{this Def.}).


\item equipped with a [[adjoint quadruple|quadruple]] of [[adjoint functors]] (\href{geometry+of+physics+--+categories+and+toposes#AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}{this Def.}) to $\mathbf{H}_{bos}$ of the form

\begin{displaymath}
\mathbf{H}_{bos}
    \itexarray{
      \overset{\phantom{A} even \phantom{A} }{\longleftarrow}
      \\
      \overset{\phantom{AA} \iota_{sup} \phantom{AA} }{\hookrightarrow}
      \\
      \overset{\phantom{AA} \Pi_{sup} \phantom{AA} }{\longleftarrow}
      \\
      \overset{\phantom{AA} Disc_{sup} \phantom{AA} }{\hookrightarrow}
    }
  \mathbf{H}
\end{displaymath}
hence with $\iota_{sup}$ and $Disc_{sup}$ being [[fully faithful functors]].



\end{enumerate}
\end{defn}
\begin{lemma}
\label{ProgressionOfSubcategoriesOfSolidTopos}\hypertarget{ProgressionOfSubcategoriesOfSolidTopos}{}
\textbf{(progression of ([[coreflective subcategory|co-]])[[reflective subcategories]] of [[solid topos]])}

Let $\mathbf{H}$ be a [[solid topos]] (Def. \ref{SuperDifferentialCohesion}) over an [[elastic topos]] $\mathbf{H}_{red}$:

\begin{displaymath}
Set
    \itexarray{
      \overset{\phantom{A} \Pi_{red} \phantom{A} }{\longleftarrow}
      \\
      \overset{\phantom{A} Disc_{red} \phantom{A} }{\hookrightarrow}
      \\
      \overset{\phantom{A} \Gamma_{red} \phantom{A} }{\longleftarrow}
      \\
      \overset{\phantom{A} coDisc_{red} \phantom{A} }{\hookrightarrow}
    }
  \mathbf{H}_{red}
    \itexarray{
      \overset{\phantom{AA} \iota_{inf} \phantom{AA} }{\hookrightarrow}
      \\
      \overset{\phantom{AA} \Pi_{inf} \phantom{AA} }{\longleftarrow}
      \\
      \overset{\phantom{AA} Disc_{inf} \phantom{AA} }{\hookrightarrow}
      \\
      \overset{\phantom{AA} \Gamma_{inf} \phantom{AA} }{\longleftarrow}
      \\
      \phantom{A}
      \\
      \phantom{A \atop A}
    }
  \mathbf{H}_{bos}
    \itexarray{
      \overset{\phantom{AA} even \phantom{AA} }{\longleftarrow}
      \\
      \overset{\phantom{AA} \iota_{sup} \phantom{AA} }{\hookrightarrow}
      \\
      \overset{\phantom{AA} \Pi_{sup} \phantom{AA} }{\longleftarrow}
      \\
      \overset{\phantom{AA} Disc_{sup} \phantom{AA} }{\hookrightarrow}
      \\
      \overset{\phantom{AA} \Gamma_{sup} \phantom{AA} }{\longleftarrow}
      \\
      \phantom{A}
      \\
      \phantom{A \atop A}
      \\
      \phantom{A \atop A}
    }
  \mathbf{H}
\end{displaymath}
Then these adjoint functors arrange and decompose as shown in the following [[diagram]]:

Here the composite [[adjoint quadruple]]

\begin{displaymath}
Set
    \itexarray{
      \overset{\Pi \simeq \Pi_{red}\Pi_{inf} \Pi_{sup} }{\longleftarrow}
      \\
      \overset{Disc = Disc_{sup} Disc_{inf} Disc_{red}}{\hookrightarrow}
      \\
      \overset{\Gamma = \Gamma_{sup} \Gamma_{inf} \Gamma_{red} }{\longleftarrow}
      \\
      \overset{\phantom{AA} coDisc \phantom{AA} }{\hookrightarrow}
    }
  \mathbf{H}
\end{displaymath}
exhibits the [[cohesion]] of $\mathbf{H}$ over [[Set]], and the composite adjoint quadruple

\begin{displaymath}
\mathbf{H}_{red}
    \itexarray{
      \overset{\iota_{sup} \iota_{inf}}{\hookrightarrow}
      \\
      \overset{\Pi_{inf} \Pi_{sup} }{\longleftarrow}
      \\
      \overset{Disc_{inf} Disc_{red}}{\hookrightarrow}
      \\
      \overset{ \Gamma_{sup} }{\longleftarrow}
    }
  \mathbf{H}
\end{displaymath}
exhibits the [[elastic topos|elasticity]] of $\mathbf{H}$ over $\mathbf{H}_{red}$.

\end{lemma}
\begin{proof}
As in the proof of \href{geometry+of+physics+--+categories+and+toposes#ProgressionOfSubcategoriesOfElasticTopos}{this Prop.}, this is immediate by the essential uniqueness of adjoints (\href{geometry+of+physics+--+categories+and+toposes#UniquenessOfAdjoints}{this Prop.}) and of the [[global section]]-[[geometric morphism]] (\href{geometry+of+physics+--+categories+and+toposes#GlobalSectionsGeometricMorphism}{this Example}).

\end{proof}
\begin{defn}
\label{SuperDiffCohesiveModalities}\hypertarget{SuperDiffCohesiveModalities}{}
\textbf{([[adjoint modalities]] on [[solid topos]])}

Given a [[solid topos]] $\mathbf{H}$ over $\mathbf{H}_{bos}$ (Def. \ref{SuperDifferentialCohesion}), [[composition]] of the functors in Lemma \ref{ProgressionOfSubcategoriesOfSolidTopos} yields, via \href{geometry+of+physics+--+categories+and+toposes#ModalOperatorsEquivalentToReflectiveSubcategories}{this Prop.}, the following [[adjoint modalities]] (\href{geometry+of+physics+--+categories+and+toposes#AdjointModality}{this Def.})

\begin{displaymath}
\rightrightarrows
  \;\dashv\;
  \rightsquigarrow
  \;\dashv\;
  Rh
  \;\;\colon\;\;
  \mathbf{H}
    \itexarray{
      \overset{ \rightrightarrows \;\coloneqq\; \iota_{sup} \circ even  }{\longleftarrow}
      \\
      \overset{\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup}  }{\longrightarrow}
      \\
      \overset{ Rh \;\coloneqq\; Disc_{sup}  \circ \Pi_{sup} }{\longleftarrow}
    }
  \mathbf{H}
  \,.
\end{displaymath}
Since $\iota_{sup}$ and $Disc_{sup}$ are [[fully faithful functors]] by assumption, these are ([[comodal operator|co-]])[[modal operators]] (\href{geometry+of+physics+--+categories+and+toposes#ModalOperator}{this Def.}) on the [[cohesive topos]], by \href{geometry+of+physics+--+categories+and+toposes#ModalOperatorsEquivalentToReflectiveSubcategories}{this Prop.}.

We pronounce these as follows:

\begin{tabular}{l|l|l}
$\phantom{A}$ [[fermionic modality]] $\phantom{A}$&$\phantom{A}$ [[bosonic modality]] $\phantom{A}$&$\phantom{A}$ [[rheonomy modality]] $\phantom{A}$\\
\hline 
$\phantom{A}$ $\rightrightarrows \;\coloneqq\; \iota_{sup} \circ even$ $\phantom{A}$&$\phantom{A}$ $\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup}$ $\phantom{A}$&$\phantom{A}$ $Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup}$ $\phantom{A}$\\
\end{tabular}

and we refer to the corresponding [[modal objects]] (\href{geometry+of+physics+--+categories+and+toposes#ModalObjects}{this Def.}) as follows:

\begin{itemize}%
\item a $\rightsquigarrow$-[[comodal object]]

\begin{displaymath}
\overset{\rightsquigarrow}{X}
    \underoverset{\simeq}{\epsilon^\rightsquigarrow_X}{\longrightarrow}
  X
\end{displaymath}
is called a \emph{[[bosonic object]]};


\item a $Rh$-[[modal object]]

\begin{displaymath}
X
    \underoverset{\simeq}{ \eta^{Rh}_X}{\longrightarrow}
  Rh X
\end{displaymath}
is called a \emph{rheonomic object};



\end{itemize}
\end{defn}
\begin{prop}
\label{ProgressionOfModalitiesOnElasticTopos}\hypertarget{ProgressionOfModalitiesOnElasticTopos}{}
\textbf{(progression of [[adjoint modalities]] on [[solid topos]])}

Let $\mathbf{H}$ be a [[solid topos]] (Def. \ref{SuperDifferentialCohesion}) and consider the [[adjoint modalities]] which it inherits

\begin{enumerate}%
\item for being a [[cohesive topos]], from \href{geometry+of+physics+--+categories+and+toposes#CohesiveModalities}{this Def.},


\item for being an [[elastic topos]], from \href{geometry+of+physics+--+categories+and+toposes#DiffCohesiveModalities}{this Def.},


\item for being a [[solid topos]], from Def. \ref{SuperDiffCohesiveModalities}:



\end{enumerate}
\begin{tabular}{l|l|l}
$\phantom{A}$ [[shape modality]] $\phantom{A}$&$\phantom{A}$ [[flat modality]] $\phantom{A}$&$\phantom{A}$ [[sharp modality]] $\phantom{A}$\\
\hline 
$\phantom{A}$ $&#643; \;\coloneqq\; Disc \Pi$ $\phantom{A}$&$\phantom{A}$ $\flat \;\coloneqq\; Disc \circ \Gamma$ $\phantom{A}$&$\phantom{A}$ $\sharp \;\coloneqq\; coDisc \circ \Gamma$ $\phantom{A}$\\
&&\\
$\phantom{A}$ \textbf{[[reduction modality]]} $\phantom{A}$&$\phantom{A}$ \textbf{[[infinitesimal shape modality]]} $\phantom{A}$&$\phantom{A}$ \textbf{[[infinitesimal flat modality]]} $\phantom{A}$\\
$\phantom{A}$ $\Re \;\coloneqq\; \iota_{sup} \iota_{inf} \circ \Pi_{inf}\Pi_{sup}$ $\phantom{A}$&$\phantom{A}$ $\Im \;\coloneqq\; Disc_{sup} Disc_{inf} \circ \Pi_{inf} \Pi_{sup}$ $\phantom{A}$&$\phantom{A}$ $\& \;\coloneqq\; Disc_{sup} Disc_{inf} \circ \Gamma_{inf}\Gamma_{sup}$ $\phantom{A}$\\
&&\\
$\phantom{A}$ \textbf{[[fermionic modality]]} $\phantom{A}$&$\phantom{A}$ \textbf{[[bosonic modality]]} $\phantom{A}$&$\phantom{A}$ \textbf{[[rheonomy modality]]} $\phantom{A}$\\
$\phantom{A}$ $\rightrightarrows \;\coloneqq\; \iota_{sup} \circ even$ $\phantom{A}$&$\phantom{A}$ $\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup}$ $\phantom{A}$&$\phantom{A}$ $Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup}$ $\phantom{A}$\\
\end{tabular}

Then these arrange into the following progression, via the [[preorder]] on modalities from \href{geometry+of+physics+--+categories+and+toposes#PreorderOnModalities}{this Def.}:

\begin{displaymath}
\itexarray{    
    id &\dashv& id
    \\
    \vee && \vee
    \\
    \rightrightarrows &\dashv& \rightsquigarrow &\dashv& Rh
    \\
    && \vee && \vee
    \\
    && \Re &\dashv& \Im &\dashv& \&
    \\
    && && \vee && \vee
    \\
    && && &#643; &\dashv& \flat &\dashv& \sharp
    \\
    && && && \vee && \vee
    \\
    && && && \emptyset &\dashv& \ast
  }
\end{displaymath}
where we are displaying, for completeness, also the [[adjoint modalities]] at the [[bottom]] $\emptyset \dashv \ast$ and the [[top]] $id \dashv id$ (\href{geometry+of+physics+--+categories+and+toposes#InitialAndFinalAdjointModality}{this Example}).

\end{prop}
\begin{proof}
By \href{geometry+of+physics+--+categories+and+toposes#ProgressionOfModalitiesOnElasticTopos}{this Prop.}, it just remains to show that for all [[objects]] $X \in \mathbf{H}$

\begin{enumerate}%
\item $\Im X$ is an $Rh$-[[modal object]], hence $Rh \Im X \simeq X$,


\item $\Re X$ is a [[bosonic object]], hence $\overset{\rightsquigarrow}{\Re X} \simeq \Re X$.



\end{enumerate}
The proof is directly analogous to the proof of \href{geometry+of+physics+--+categories+and+toposes#ProgressionOfModalitiesOnElasticTopos}{that Prop.}, now using the decompositions from Lemma \ref{ProgressionOfSubcategoriesOfSolidTopos}:

\begin{displaymath}
\begin{aligned}
    Rh \Im 
    & =
    Disc_{sup} 
    \underset{
      \simeq id
    }{
    \underbrace{
      \Pi_{sup} \;\; Disc_{sup}
    } 
    }
    Disc_{inf} \Pi_{inf} \Pi_{sup}
    \\
    & \simeq 
    Disc_{sup} Disc_{inf} \Pi_{inf} \Pi_{sup}
    \\
    & = \Im
  \end{aligned}
\end{displaymath}
and

\begin{displaymath}
\begin{aligned}
    \rightsquigarrow \Re
    & =
    \iota_{sup} 
    \underset{\simeq id}{\underbrace{
    \Pi_{sup} \;\; \iota_{sup}}}
    \iota_{inf} \Pi_{inf}\Pi_{sub}
    \\
    & \simeq
    \iota_{sup} \iota_{inf} \Pi_{inf} \Pi_{sub}    
    \\
    & \simeq \Re
  \end{aligned}
\end{displaymath}
\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{super_formal_smooth_sets}{}\subsubsection*{{Super formal smooth sets}}\label{super_formal_smooth_sets}

The [[sheaf topos]] of [[super formal smooth sets]] is solid over that of [[formal smooth sets]], which is [[elastic topos|elastic]] over that of [[smooth sets]], which is [[cohesive topos|cohesive]] over [[Set]].

See \href{geometry+of+physics+--+supergeometry#SuperSmoothSetsSystemOfAdjunctions}{this Prop} at \emph{[[geometry of physics -- supergeometry]]}.

[[!redirects super-differentially cohesive topos]]

[[!redirects solid model topos]] [[!redirects solid model toposes]]

[[!redirects ∞-solid site]] [[!redirects ∞-solid sites]]



\end{document}