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\begin{document}

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\section*{coshape of an (infinity,1)-topos}

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

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory}

[[!include (infinity,1)-topos - 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{in_terms_of_universe_enlargements}{In terms of Universe Enlargements}\dotfill \pageref*{in_terms_of_universe_enlargements} \linebreak
\noindent\hyperlink{enlarging_the_category_of_toposes}{Enlarging the category of toposes}\dotfill \pageref*{enlarging_the_category_of_toposes} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

Just as the [[shape of an (∞,1)-topos]] is the functor $\infty Gpd\to \infty Gpd$ which it \emph{corepresents} (after identifying ∞-groupoids with their presheaf (∞,1)-topoi), so the \emph{coshape} of an (∞,1)-topos is the functor $\infty Gpd^{op}\to \infty Gpd$ which it \emph{represents}.

Unlike the shape, which is only (co)representable (by an ∞-groupoid) when the topos is [[locally ∞-connected (∞,1)-topos|locally ∞-connected]], the coshape is \emph{always} representable, albeit possibly by a \emph{large} ∞-groupoid---specifically the ∞-groupoid of [[point of a topos|points]] of the (∞,1)-topos in question.

From here on, this page uses the [[implicit ∞-category theory convention]].

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

\begin{udefn}
For $\mathbf{H}$ a [[(∞,1)-topos|topos]], we say its \textbf{co-shape} $\Gamma \mathbf{H}$ is the functor $\Gamma(\mathbf{H}) \colon Gpd^{op}\to Gpd$ defined by

\begin{displaymath}
\Gamma(\mathbf{H})(A) =
  Topos(PSh(A), \mathbf{H})
\end{displaymath}
\end{udefn}
Let $Pt(\mathbf{H}) = Topos(*,\mathbf{H})$ denote the (possibly large) groupoid of \emph{points} of $\mathbf{H}$, where $*$ denotes the terminal topos $Gpd$.

\begin{uprop}
The coshape $\Gamma(\mathbf{H})$ is represented by $Pt(\mathbf{H})$, i.e. for any (small) groupoid $A$ we have

\begin{displaymath}
\Gamma(\mathbf{H})(A) \simeq GPD(A, Pt(\mathbf{H})).
\end{displaymath}
\end{uprop}
\begin{proof}
Recall that colimits in $Topos$ are calculated via \emph{limits} on the level of underlying categories. In particular, the [[copower]] of $\mathbf{K}$ by a groupoid $A$ is the topos $\mathbf{K}^A$.

Thus, in even more particular, the functor $Psh$ preserves copowers, since each groupoid is the copower of the terminal object by itself. Therefore, we have $Topos( Psh(A), \mathbf{H} ) \simeq GPD(A, Topos( *, \mathbf{H} )) = GPD(A,Pt(\mathbf{H}))$, as desired.

\end{proof}
\hypertarget{in_terms_of_universe_enlargements}{}\subsection*{{In terms of Universe Enlargements}}\label{in_terms_of_universe_enlargements}

Another way of phrasing the above argument is as follows. For the same reason cited in the proof, the embedding $Psh$ preserves all small colimits. Therefore, since $\Gamma(\mathbf{H}) \colon Gpd^{op}\to Gpd$ is the composite of this embedding with the representable functor $Topos(-,\mathbf{H})$, it must also preserves all small limits in $Gpd^{op}$ (i.e. small colimits in $Gpd$).

Therefore, we can regard it as an object of the category $Cts(Gpd^{op},Gpd)$ of small-limit-preserving functors, also known as the [[very large (∞,1)-sheaf (∞,1)-topos]] on [[?Gpd|Gpd]] (and also the $\kappa$-[[ind-object in an (∞,1)-category|ind-objects]] of $Gpd$, for $\kappa$ the cardinality of the universe). However, by the general theory of [[universe enlargement]] (generalized to $(\infty,1)$-categories), this category is equivalent to $GPD$, and the equivalence gives the representability theorem above.

\hypertarget{enlarging_the_category_of_toposes}{}\subsubsection*{{Enlarging the category of toposes}}\label{enlarging_the_category_of_toposes}

Instead of being content with a ``large-representability'' result as above, we might wish that the coshape would actually give us a right adjoint to the embedding $Psh$. For this to be possible, we would need to enlarge $Gpd$ to $GPD$, but if we also enlarged $Topos$ to its naive enlargement $TOPOS$, we would face the same problem ``one universe higher.''

Thus, to get ``better behavior'' we can instead replace $Topos$ by its locally presentable enlargement $\Uparrow Topos$, also called the [[very large (∞,1)-sheaf (∞,1)-topos]] on $Topos$. We can then say:

\begin{uprop}
Coshape [[Yoneda extension|Yoneda-extends]] to a pair of [[adjoint (∞,1)-functor|adjoint functors]]s

\begin{displaymath}
GPD
  \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{Codisc}{\to}}
  \Uparrow Topos.
\end{displaymath}
\end{uprop}
\begin{proof}
By [[Higher Topos Theory|HTT, lemma 6.3.5.21]] we have a [[(∞,1)-functor|functor]]

\begin{displaymath}
\Uparrow Topos
  \to 
  \Uparrow Grpd = GRPD
\end{displaymath}
that preserves $\mathcal{U}$-small colimits and finite limits and is given by sending

\begin{displaymath}
F : Topos^{op} \to Grpd
\end{displaymath}
to the composite

\begin{displaymath}
Grpd
   \stackrel{PSh(-)}{\to}
   (Topos/Grpd)_{et}^{op}
   \stackrel{}{\to}
   Topos^{op}
   \stackrel{F}{\to}
   Grpd
  \,,
\end{displaymath}
where the first step is forming [[(∞,1)-presheaf (∞,1)-topos|presheaf toposes]] which sit by their terminal [[global section]] [[(∞,1)-geometric morphism|geometric morphisms]] over [[∞Grpd|Grpd]], and the second step is the evident projection.

Applied to a [[representable functor|representable]] $F = Topos(-,\mathbf{H})$ this composite is hence $A \mapsto \Gamma(\mathbf{H})(A)$.

\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[shape of an (∞,1)-topos]]


\item \textbf{coshape of an $(\infty,1)$-topos}



\end{itemize}
[[!redirects coshape of an (∞,1)-topos]]



\end{document}