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\begin{document}

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\section*{geometric function object}

\hypertarget{content}{}\section*{{Content}}\label{content}

\noindent\hyperlink{content}{Content}\dotfill \pageref*{content} \linebreak
\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{overcategories_and_groupoidification}{over-categories and groupoidification}\dotfill \pageref*{overcategories_and_groupoidification} \linebreak
\noindent\hyperlink{remarks}{remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{undercategories_of_quantities}{under-categories of $\infty$-quantities}\dotfill \pageref*{undercategories_of_quantities} \linebreak
\noindent\hyperlink{quasicoherent_sheaves}{quasicoherent sheaves}\dotfill \pageref*{quasicoherent_sheaves} \linebreak


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

This entry list details on concrete constructions for examples of [[geometric function theory|geometric function theories]], or closely related structures.

Recall the notion of \emph{geometric function object} from [[geometric function theory]]:

Given an [[(∞,1)-topos]] $\mathbf{H}$ of [[∞-stack]]s -- in the simplest case just [[Top]] or [[∞-Grpd]] -- a \emph{geometric function theory} is some kind of assignment

\begin{displaymath}
C : \mathbf{H} \to (\infty,1)Cat
\end{displaymath}
such that for $X \in \mathbf{H}$ the object $C(X)$ behaves to some useful extent like a collection of ``functions on $X$''.

More concretely, this will usually be taken to mean that $C$ satisfies properties of the following kind:

\begin{itemize}%
\item \textbf{existence of pull-push} -- For every morphism $f : A \to B$ in $\mathbf{H}$ there is naturally (functorially) an [[adjunction]] $f_* : C(A) \stackrel{\leftarrow}{\to} C(B) : f^*$ with $f_*$ playing the role of push-forward of functions along $f$ and $f^*$ playing the role of pullback of functions along $f$;


\item \textbf{respect for composition of spans} -- Pull-push through [[span]]s should be functorial: if

\begin{displaymath}
\itexarray{
    &&&& Y_1 \times_{X_2} Y_2
    \\
    &&& {}^{p_1}\swarrow && \searrow^{p_2}
    \\
    && Y_1 &&&& Y_2
    \\
    & {}^t\swarrow && \searrow^u && {}^v\swarrow && \searrow^w
    \\
    X_1 &&&& X_2 &&&& X_4
  }
\end{displaymath}
is a composite of two [[span]]s, then the pull-push through both spans seperately should be equivalent to that through the total span

\begin{displaymath}
w_* v^* u_* t^* 
  \simeq
  {w}_* {p_2}_* p_1^* t^* 
  \,.
\end{displaymath}
Of course this just means that the two ways to pull-push through the pullback diamond

\begin{displaymath}
\itexarray{
    &&&& Y_1 \times_{X_2} Y_2
    \\
    &&& {}^{p_1}\swarrow && \searrow^{p_2}
    \\
    && Y_1 &&&& Y_2
    \\
    & && \searrow^u && {}^v\swarrow && 
    \\
    &&&& X_2 &&&& 
  }
\end{displaymath}
should coincide.


\item \textbf{respect for fiber products} -- With respect to some suitable [[tensor product]] of geometric functions one has for each ([[homotopy pullback|homotopy]]) [[fiber product]] $X \times_Z Y$ in $\mathbf{H}$ that

\begin{displaymath}
C(X \times_Z Y) \simeq C(X) \otimes_{C(Z)} C(Y)
 \,.
\end{displaymath}


\end{itemize}
\hypertarget{overcategories_and_groupoidification}{}\subsection*{{over-categories and groupoidification}}\label{overcategories_and_groupoidification}

This first example is rather minimalistic and may feel a bit tautological, as compared to more involved constructions as discussed below. It does nevertheless have interesting applications and, due to its structural simplicity, should serve as a good model on which to study the structural aspects of geometric function theory.

So consider here the assignment

\begin{displaymath}
C := \mathbf{H}/(-) : \mathbf{H} \to (\infty,1)Cat
\end{displaymath}
that sends each object $X \in \mathbf{H}$ to its [[over category]] $\mathbf{H}/X$.

Checking that this assignment does satisfy a good deal of the properties of a geometric function object amounts to recalling the properties of [[over category|over categories]].

So an [[object]] in $C(X)$ is a morphism $\psi : \Psi \to X$ in $\mathbf{H}$. A [[morphism]] $(\psi,\Psi) \to (\psi',\Psi')$ is a diagram

\begin{displaymath}
\itexarray{
    \Psi &&\to&& \Psi'
    \\
    & {}_\psi\searrow && \swarrow_{\psi'}
    \\
    && X
  }
\end{displaymath}
in $\mathbf{H}$.

For $f : X \to Y$ a morphism in $\mathbf{H}$ the push-forward functor

\begin{displaymath}
f_* : C(X) \to C(Y)
\end{displaymath}
is simply given by postcomposition with $f$:

\begin{displaymath}
f_* \;\;:\;\; 
  \left(
    \itexarray{
      \Psi
      \\
      \downarrow^\psi
      \\
      X
    }
  \right)
  \;\;
  \mapsto
  \;\;
  \left(
    \itexarray{
      \Psi
      \\
      \downarrow^\psi
      \\
      X
      \\
      \downarrow^f
      \\
      Y
    }
  \right)
  \,.
\end{displaymath}
The pullback functor

\begin{displaymath}
f^* : C(Y) \to C(X)
\end{displaymath}
is literally given by the ([[homotopy pullback|homotopy]]) [[pullback]]

\begin{displaymath}
\itexarray{
    f^* \Psi &\to& \Psi
    \\
    \downarrow^{f^* \psi} && \downarrow^\psi
    \\
    X &\stackrel{f}{\to}& Y
  }
\end{displaymath}
of a morphism $\psi : \Psi \to Y$ along $f$.

A quick way to check that pushforward $f_*$ and pullback $f^*$ defined this form a pair of [[adjoint functor]]s is to notice the hom-isomorphism

\begin{displaymath}
Hom_{\mathbf{H}/X}(\Psi, f^* \Phi)
  \simeq
  Hom_{\mathbf{H}/X}(f_* \Psi, \Phi)
\end{displaymath}
which is established by the essential uniqueness of the universal morphism into the pullback

\begin{displaymath}
\itexarray{
    \Psi &&\to&&
    \\
    & \searrow^{\exists ! \bar k} & && \downarrow^{k}
    \\
    \downarrow && f^* \Phi &\to& \Phi
    \\
    &\searrow& \downarrow^{f^* \psi} && \downarrow^\psi
    \\
    && X &\stackrel{f}{\to}& Y
  }
\end{displaymath}
Here the outer diagram exhibits a morphism $k : f_* \Psi \to \Phi$. The universal property of the [[pullback]] says that this essentially uniquely corresponds to the [[adjunct]] morphism $\bar k : \Psi \to f^* \Phi$.

The fact that the pull-push respects composition of spans is a direct consequence of the way [[pullback]] diagrams compose under pasting: recall that in a diagram

\begin{displaymath}
\itexarray{
    A &\to& B &\to& C
    \\
    \downarrow && \downarrow && \downarrow
    \\
    D &\to& E &\to& F
  }
\end{displaymath}
for which the left square is a pullback, the total rectangle is a pullback precisely if the right square is, too.

Apply this to the pull-push of an object $\left(\itexarray{ \Psi \\ \downarrow^{\psi} \\ Y_1}\right) \in C(Y_1)$ through a pullback diamond (see the introduction above)

\begin{displaymath}
\itexarray{
      &&&& Y_1 \times_{X_2} Y_2
      \\
      &&& {}^{p_1}\swarrow && \searrow^{p_2}
      \\
      && Y_1 &&&& Y_2
      \\
      & && \searrow^u && {}^v\swarrow && 
      \\
      &&&& X_2 &&&& 
    }
    \,.
\end{displaymath}
This is described by the diagram

\begin{displaymath}
\itexarray{
      && p_1^* \Psi
      \\
      & {}^{q_1}\swarrow && \searrow^{p_1^* \psi}
      \\
      \Psi&&&& Y_1 \times_{X_2} Y_2
      \\
      &\searrow^f && {}^{p_1}\swarrow && \searrow^{p_2}
      \\
      && Y_1 &&&& Y_2
      \\
      & && \searrow^u && {}^v\swarrow && 
      \\
      &&&& X_2 &&&& 
    }
    \,.
\end{displaymath}
By the above definitions, the push-pull operation $v^* u_*$ is encoded in the pullback property of the total outer rectangle. On the other hand, the pull-push operation ${p_2}_* p_1^*$ is determined by the pullback property of the upper square. By the above fact both properties are equivalent. This means that indeed

\begin{displaymath}
v^* u_* \simeq {p_2}_* p_1^*
\end{displaymath}
and hence that the pull-push operations defined by over-categories are compatible with composition of [[span]]s.

Finally, there is a simple observation on the [[cartesian product]] on [[over category|over categories]]:

for

\begin{displaymath}
\itexarray{
     && Y
     \\
     && \downarrow^g
     \\
     X &\stackrel{f}{\to}& Z
  }
\end{displaymath}
a diagram in $\mathbf{H}$, notice that the objects in the [[fiber product]] of [[over category|over categories]]

\begin{displaymath}
(\mathbf{H}/X) \times_{\mathbf{H}/Y} \mathbf{H}/Y
\end{displaymath}
are those pairs $\psi : \Psi \to X$ and $\phi : \Phi \to Y$ such that we get a (homotopy) commutative diagram

\begin{displaymath}
\itexarray{
     \Psi \simeq \Phi &\stackrel{\phi}{\to}& Y
     \\
     \downarrow^\psi && \downarrow^g
     \\
     X &\stackrel{f}{\to}& Z
  }
  \,.
\end{displaymath}
Again by the universal property of the pullback this is the same as maps

\begin{displaymath}
(\Psi \simeq \Phi) \to X \times_Z Y
\end{displaymath}
which are precisely the objects of $C(X \times_Z Y)$. So we get

\begin{displaymath}
C(X \times_Z Y) \simeq C(X) \times_{C(Z)} C(Y)
\end{displaymath}
\hypertarget{remarks}{}\subsubsection*{{remarks}}\label{remarks}

\begin{itemize}%
\item This is -- more or less implicitly -- the notion of geometric ∞-functions that underlies [[John Baez]]` notion of [[groupoidification]] as well as the generalized sections that appear at \href{http://ncatlab.org/schreiber/show/Nonabelian+cocycles+and+their+sigma+model+QFTs}{these sigma-model notes}.


\item The definition seems to be disturbingly non-linearized, but this should be viewed in light of the possible nature of the $X$s considered here. If $X = E$ is, for instance, the [[groupoid]] incarnation of the total space of the [[vector bundle]] associated to a $G$-[[principal bundle]], then a choice of groupoid over $E$ picks a bunch of vectors in that bundle, hence picks a ``distributional section'' of that bundle.



\end{itemize}
\hypertarget{undercategories_of_quantities}{}\subsection*{{under-categories of $\infty$-quantities}}\label{undercategories_of_quantities}

By essentially simply applying [[Isbell duality]] for the case that the underlying [[site]] is [[CartSp]] to the above example one obtains the following example.

Tentative.

Recall the notion of [[∞-quantity]]. Notice that by the discussion at [[models for ∞-stack (∞,1)-toposes]] every object $A \in \mathbf{H}$ may be modeled as a [[simplicial presheaf]]. Let $C^\infty(-)$ be the map that sends simplicial presheaves to cosimplicial copresheaves as described at [[∞-quantity]].

Then consider the assignment

\begin{displaymath}
C(-) : \mathbf{H} \to (\infty,1)Cat
\end{displaymath}
that sends every $X$ to the $(\infty,1)$-category of cosimplicial copresheaves to the [[under category]]

\begin{displaymath}
C(X) = C^\infty(X)/CoSCoSh
\end{displaymath}
or

\begin{displaymath}
C(X) = C^\infty_{loc}(X)/CoSCoSh
  \,.
\end{displaymath}
From the discussion at [[∞-quantity]] and [[Lie-∞ algebroid representation]] we see that we can think of objects in $C(X)$ defines this way as representations of the [[Lie-∞ algebroid]] of $X$.

Now pullback is left adjoint and push-forward is right adjoint.

\hypertarget{quasicoherent_sheaves}{}\subsection*{{quasicoherent sheaves}}\label{quasicoherent_sheaves}

The choice $C(X) =$ the [[stable (∞,1)-category]] of [[quasicoherent sheaf|quasicoherent sheaves]] on a [[derived stack]] $X$ is discussed at

\begin{itemize}%
\item [[geometric ∞-function theory]].

\end{itemize}
[[!redirects examples for geometric function objects]] [[!redirects example for geometric function objects]] [[!redirects geometric function objects]]



\end{document}