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\begin{document}

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\section*{Sierpinski topos}

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

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{presentation_and_homotopy_type_theory}{Presentation and Homotopy type theory}\dotfill \pageref*{presentation_and_homotopy_type_theory} \linebreak
\noindent\hyperlink{Cohesion}{Connectedness, locality, cohesion}\dotfill \pageref*{Cohesion} \linebreak
\noindent\hyperlink{cohesive_structures}{Cohesive structures}\dotfill \pageref*{cohesive_structures} \linebreak
\noindent\hyperlink{DifferentialCohesion}{Differential cohesion}\dotfill \pageref*{DifferentialCohesion} \linebreak
\noindent\hyperlink{AsAClassifyingTopos}{As a classifying topos}\dotfill \pageref*{AsAClassifyingTopos} \linebreak
\noindent\hyperlink{cohomology}{Cohomology}\dotfill \pageref*{cohomology} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{Sierpinski [[topos]]} is the [[arrow category]] of [[Set]].

Equivalently, this is the [[category of presheaves]] over the [[interval category]] $\Delta[1] := \mathbf{2} = \{0 \to 1\}$, or equivalently the [[category of sheaves]] over the [[Sierpinski space]] $Sierp$

\begin{displaymath}
Sh(Sierp) \simeq PSh(\Delta[1]) \simeq Set^{\Delta[1]}
  \,.
\end{displaymath}
Yet another description is that it is the [[Freyd cover]] of \textbf{Set}.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
Similarly, the \textbf{Sierpinski [[(∞,1)-topos]]} is the [[arrow (∞,1)-category]] $\infty Grpd^{\Delta[1]}$ of [[∞Grpd]].

Equivalently this is the [[(∞,1)-category of (∞,1)-presheaves]] on $\Delta[1]$ and equivalently the [[(∞,1)-category of (∞,1)-sheaves]] on $Sierp$:

\begin{displaymath}
Sh_{(\infty,1)}(Sierp)
  \simeq
  PSh_{(\infty,1)}(\Delta[1])
  \simeq
  \infty Grpd^{\Delta[1]}
  \,.
\end{displaymath}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{presentation_and_homotopy_type_theory}{}\subsubsection*{{Presentation and Homotopy type theory}}\label{presentation_and_homotopy_type_theory}

Being a [[(∞,1)-category of (∞,1)-functors]], the Sierpinski [[(∞,1)-topos]] is [[presentable (∞,1)-category|presented]] by any of the [[model structure on simplicial presheaves]] $[\Delta[1], sSet]$.

Specifically the [[Reedy model structure]] of [[simplicial presheaves]] on the [[interval category]] $[\Delta[1], sSet]_{Reedy}$ provides a [[univalence|univalent]] model for [[homotopy type theory]] in the Sierpinski $(\infty,1)$-topos (\hyperlink{Shulman}{Shulman})

\hypertarget{Cohesion}{}\subsubsection*{{Connectedness, locality, cohesion}}\label{Cohesion}

We discuss the connectedness, locality and cohesion of the Sierpinski topos. We do so relative to an arbitrary [[base topos]]/[[base (∞,1)-topos]] $\mathbf{H}$, hence regard the [[global section geometric morphism]]

\begin{displaymath}
\mathbf{H}^I \to \mathbf{H}
  \,.
\end{displaymath}
\begin{prop}
\label{}\hypertarget{}{}
The Sierpinski topos is a \emph{[[cohesive topos]].}

The Sierpinski $(\infty,1)$-topos is a [[cohesive (∞,1)-topos]].

\begin{displaymath}
(\Pi \dashv \Disc \dashv \Gamma \dashv coDisc)
  : 
  \mathbf{H}^I
  \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\leftarrow}}}}
  \mathbf{H}
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
For the first statement, see the detailed discussion at \emph{[[cohesive topos]]} \emph{\href{cohesive+topos#FamiliesOfSets}{here}}.

For the second statement, see the discussion at \emph{[[cohesive (∞,1)-topos]]} \emph{\href{cohesive+%28infinity%2C1%29-topos#CohesiveDiagramToposes}{here}}.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
The fact that the Sierpienski $(\infty,1)$-topos is, therefore, in particular

\begin{enumerate}%
\item a [[locally ∞-connected (∞,1)-topos]];


\item an [[∞-connected (∞,1)-topos]];


\item a [[local (∞,1)-topos]]



\end{enumerate}
all follow directly from the fact that it is the image, under [[localic reflection]], of the [[Sierpinski space]] (hence that it is [[n-localic (∞,1)-topos|0-localic]], its [[n-truncated|(-1)-truncation]] being the [[frame]] of opens of the Sierpinski space).

That space $Sierp$, in turn,

\begin{enumerate}%
\item is a [[contractible topological space]];


\item a [[locally contractible topological space]].


\item has a [[focal point]]



\end{enumerate}
which implies the corresponding three properties of the Sierpinski $\infty$-topos above.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
By the discussion at \emph{[[cohesive (∞,1)-topos]]} every such may be thought of as a \emph{fat point}, the abstract \emph{cohesive blob}. In this case, this fat point \emph{is} the Sierpinski space. This space can be thought of as being the abstract ``point with open neighbourhood''.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Accordingly, the objects of the Sierpinski $(\infty,1)$-topos may be thought of as [[∞-groupoids]] (relative to $\mathbf{H}$) equipped with the notion of [[cohesive (∞,1)-topos|cohesion]] modeled on this: they are [[bundles]] $[P \to X]$ of [[∞-groupoids]] whose [[fibers]] are regarded as being \href{cohesive+%28infinity%2C1%29-topos+--+structures#Homotopy}{geometrically contractible}, in that

\begin{displaymath}
\Pi([P \to X]) \simeq X
\end{displaymath}
and so in particular

\begin{displaymath}
\Pi([Q \to *]) \simeq *
  \,.
\end{displaymath}
Hence these objects are [[discrete ∞-groupoids]] $X$, to each of whose points $x : * \to X$ may be attached a contractible cohesive blob with inner structure given by the $\infty$-groupoid $P_x := P \times_X \{x\}$.

Accordingly, the \emph{underlying} $\infty$-groupoid of such a bundle $[P \to X]$ is the union

\begin{displaymath}
\Gamma([P \to X]) \simeq P
\end{displaymath}
of the discrete base space and the inner structure of the fibers.

The [[discrete object]] in the Sierpinski $(\infty,1)$-topos on an object $X \in \mathbf{H}$ is the bundle

\begin{displaymath}
Disc(X) \simeq [X \stackrel{id}{\to} X]
\end{displaymath}
which is $X$ with ``no cohesive blobs attached''.

Finally the [[codiscrete object]] in the Sierpinski $(\infty,1)$-topos on an object $X \in \mathbf{H}$ is

\begin{displaymath}
coDisc(X) \simeq [X \to *]
  \,,
\end{displaymath}
the structure where all of $X$ is regarded as one single contractible cohesive ball.

The $(\Pi \dashv Disc)$-[[unit |adjunction unit]]

\begin{displaymath}
i :  id \to Disc \Pi
\end{displaymath}
on $[P \to X]$ is

\begin{displaymath}
\itexarray{
    \mathllap{[}P &\to& X\mathrlap{]}
    \\
    \downarrow && \downarrow
    \\
    \mathllap{[}X &\to& X\mathrlap{]}
  }
  \,.
\end{displaymath}
The $(Disc  \dashv \Gamma)$-counit $Disc \Gamma  \to id$ on $[P \to X]$ is

\begin{displaymath}
\itexarray{
    \mathllap{[}P &\to& P\mathrlap{]}
    \\
    \downarrow && \downarrow
    \\
    \mathllap{[}P &\to& X\mathrlap{]}
  }
  \,.
\end{displaymath}
Hence the canonical [[natural transformation]]

\begin{displaymath}
\itexarray{
    \Gamma && \to && \Pi
    \\
     & {}_{\mathllap{\Gamma(i)}}\searrow & & \nearrow_{\mathrlap{\simeq}}
     \\
     && \Gamma Disc \Pi
  }
\end{displaymath}
from ``points to pieces'' is on $[P \to X]$ simply the morphism $P \to X$ itself

\begin{displaymath}
(\Gamma \to \Pi)([P \to X]) = (P \to X)
  \,.
\end{displaymath}
Therefore

\begin{enumerate}%
\item the [[full sub-(∞,1)-category]] on those objects in $\mathbf{H}^I$ for which ``\href{cohesive%20%28infinity,1%29-topos#PiecesHavePoints}{pieces have points}'', hence those for which $\Gamma \to \Pi$ is an [[effective epimorphism in an (∞,1)-category|effective epimorphism]], is the $(\infty,1)$-category of effective epimorphisms in the ambient $(\infty,1)$-topos, hence the $(\infty,1)$-category of [[groupoid object in an (∞,1)-category|groupoid objects]] in the ambient $(\infty,1)$-topos;


\item the full sub-$(\infty,1)$-category on the objects with ``\href{cohesive+topos#ObjectsWithOnePointPerCohesivePiece}{one point per piece}'' is the ambient $(\infty,1)$-topos itself.



\end{enumerate}
\end{remark}
\hypertarget{cohesive_structures}{}\subsubsection*{{Cohesive structures}}\label{cohesive_structures}

We unwind what some of the canonical [[cohesive (infinity,1)-topos -- structures|structures in a cohesive (∞,1)-topos]] are when realized in the Sierpinski $(\infty,1)$-topos.

A group object $\mathbf{B}[\hat G \to G]$ in $\mathbf{H}^I$ is a morphism in $\mathbf{H}$ of the form $= \mathbf{B}\hat G \to \mathbf{B}G$.

The corresponding flat coefficient object $\mathbf{\flat} \mathbf{B}[\hat G \to G] \to \mathbf{B}[\hat G \to G]$ is

\begin{displaymath}
\itexarray{
    \mathbf{\hat G} &\to& \mathbf{B} \hat G
    \\
    \downarrow && \downarrow
    \\
    \mathbf{\hat G} &\to& \mathbf{G}
  }
  \,.
\end{displaymath}
Hence the corresponding de Rham coefficient object is

\begin{displaymath}
\mathbf{\flat}_{dR} \mathbf{B}[\hat G \to G]
  = 
  [* \to \mathbf{B}A]
  \,,
\end{displaymath}
where $A \to \hat G \to G$ exhibits $\hat G$ has an $\infty$-group extension of $G$ by $A$ in $\mathbf{H}$.

The corresponding Maurer-Cartan form

\begin{displaymath}
[\hat G \to G] \to \mathbf{\flat}_{dR}\mathbf{B}[\hat G \to G]
\end{displaymath}
is

\begin{displaymath}
\itexarray{
    \hat G &\to& G
    \\
    \downarrow && \downarrow
    \\
    * &\to& \mathbf{B}A
  }
\end{displaymath}
exhibiting the $A$-cocycle that classifies the extension $\hat G \to G$.

\hypertarget{DifferentialCohesion}{}\subsubsection*{{Differential cohesion}}\label{DifferentialCohesion}

The leftmost adjoint, $\Pi$, of the string of four adjoints exhibiting cohesion \hyperlink{Cohesion}{above} itself possesses a left adjoint, $X \mapsto [0 \to X]$. Taking this functor along with the leftmost three adjoints of this previous string yields a second quadruple adjunction which exhibits [[differential cohesion]].

For $\mathbf{H}$ any [[cohesive (∞,1)-topos]], we have the ``Sierpinski $(\infty,1)$-topos relative to $\mathbf{H}$'' given by the [[arrow category]] $\mathbf{H}^{\Delta[1]}$, whose [[geometric morphism]] to the [[base topos]] is the [[domain cofibration]]

\begin{displaymath}
\mathbf{H}^{\Delta[1]}
   \stackrel{\overset{const}{\hookleftarrow}}{\underset{dom}{\to}}
   \mathbf{H}
  \,.
\end{displaymath}
Conversely, we may think of $\mathbf{H}^{\Delta[1]}$ as being an ``infinitesimal thickening'' of $\mathbf{H}$, as formalized at \emph{[[differential cohesion]]}, where we regard

\begin{displaymath}
(i_! \dashv i^* \dashv i_* \dashv i^!)
  : 
  \mathbf{H}
    \stackrel{\overset{\top_!}{\hookrightarrow}}{\stackrel{\overset{\top^*}{\leftarrow}}{\stackrel{\overset{const}{\hookrightarrow}}{\underset{\bot^*}{\leftarrow}}}}
  \mathbf{H}^{\Delta[1]}
\end{displaymath}
as exhibiting $\mathbf{H}^{\Delta[1]}$ as an infinitesimal cohesive neighbourhood of $\mathbf{H}$ (here $(\bot, \top) : \Delta[0] \coprod \Delta[0] \to \Delta[1]$ denotes the endpoint inclusions, following the notation \href{http://ncatlab.org/nlab/show/cohesive+%28infinity,1%29-topos#CohesiveDiagramToposes}{here}).

(See also the corresponding examples at \emph{[[Q-category]]}.)

\begin{prop}
\label{iInclusion}\hypertarget{iInclusion}{}
We have for all $X \in \mathbf{H}$ that

\begin{displaymath}
i_!(X) \simeq [\emptyset \to X]
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
For all $[A \to B]$ in $\mathbf{H}^{\Delta[1]}$ we have

\begin{displaymath}
\mathbf{H}(X, i^*[A \to B])
  \simeq
  \mathbf{H}(X, cod(A \to B))
  \simeq
  \mathbf{H}(X, B)
  \,,
\end{displaymath}
which is indeed naturally equivalent to

\begin{displaymath}
\mathbf{H}^{\Delta[1]}([\emptyset \to X], [A \to B])
  \,.
\end{displaymath}
\end{proof}
Therefore an object of $\mathbf{H}^{\Delta[1]}$ given by a morphism $[P \to X]$ in $\mathbf{H}$ is regarded by the differential cohesion $i : \mathbf{H} \hookrightarrow \mathbf{H}^{\Delta[1]}$ as being an infinitesimal thickening of $X$ by the fibers of $P$: where before we just had that the fibers of $P$ are ``contractible cohesive thickenings'' of the discrete object $X$, now $X$ is ``discrete relative to $\mathbf{H}$'' (hence not necessarily discrete in $\mathbf{H}$) and the fibers are in addition regarded as being infinitesimal.

This is of course a very crude notion of infinitesimal extension. Notice for instance the following

\begin{prop}
\label{}\hypertarget{}{}
With respect to the above differential cohesion $i : \mathbf{H} \hookrightarrow \mathbf{H}^{\Delta[1]}$, every morphism in $\mathbf{H}$ is a [[formally étale morphism]].

\end{prop}
\begin{proof}
By definition, given a morphism $f : X \to Y$, it is formally \'e{}tale precisely if

\begin{displaymath}
\itexarray{
    i_! X &\stackrel{i_! f}{\to}& i_! Y
    \\
    \downarrow && \downarrow
    \\
    i_* X &\stackrel{i_*}{\to}& i_* Y
  }
\end{displaymath}
is an [[(∞,1)-pullback]].

By prop. \ref{iInclusion} the above square diagram in $\mathbf{H}^{\Delta[1]}$ is

\begin{displaymath}
\itexarray{
    [\emptyset \to X] &\to& [\emptyset \to Y]
    \\
    \downarrow && \downarrow
    \\
    [X \stackrel{id}{\to} X]
    &\to&
    [Y \stackrel{id}{\to} Y]
  }
  \,.
\end{displaymath}
Since $(\infty,1)$-pullbacks of $(\infty,1)$-presheaves are computed objectwise, this is an $(\infty,1)$-pullback in $\mathbf{H}^{\Delta[1]}$ precisely if the ``back and front sides''

\begin{displaymath}
\itexarray{
    \emptyset &\to& \emptyset
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{f}{\to}& Y
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
    X &\stackrel{f}{\to}& Y
    \\
    \downarrow^{\mathrm{id}} && \downarrow^{\mathrm{id}}
    \\
    X &\stackrel{f}{\to}& Y
  }
\end{displaymath}
are $(\infty,1)$-pullbacks in $\mathbf{H}$. This is clearly always the case.

The adjunction $i_{!} \circ i^{\ast} \vdash i_{\ast} \circ i^{\ast}$ forms the [[Aufhebung]] of $\emptyset \vdash \ast$ for the Sierpinski topos.

\end{proof}
\hypertarget{AsAClassifyingTopos}{}\subsubsection*{{As a classifying topos}}\label{AsAClassifyingTopos}

The Sierpinski topos is the [[classifying topos]] for [[subterminal objects]] in [[toposes]] (see e.g. \hyperlink{Johnstone77}{Johnstone 77, p. 117}).

The generic subterminal inclusion in the Sierpinski topos is the unique inclusion of $[\emptyset \to \ast]$ into $[\ast  \to \ast]$.

The (non-full) inclusion

\begin{displaymath}
\Delta^1 = \{\ast \to S^0\} \longrightarrow FinSet^{\ast/} \hookrightarrow \infty Grpd_{fin}^{\ast/}
\end{displaymath}
induces via restriction and right [[Kan extension]] an [[essential geometric morphism|essential]] [[geometric morphism]]

\begin{displaymath}
\mathbf{H}^{\Delta^1} \stackrel{\longleftarrow}{\hookrightarrow}
  \mathbf{H}[X_\ast]
\end{displaymath}
of the Sierpinski topos into the [[classifying topos]] for [[pointed objects]], $\mathbf{H}[X_\ast]$. (This becomes a [[geometric embedding]] if we refined the Sierpinski topos of bundles to the topos of bundles with sections).

The pointed object in $\mathbf{H}^{\Delta^1}$ which is classified by this [[geometric morphism]] is $[\ast \to S^0]$ (with $S^0 = \ast \coprod \ast$ the [[0-sphere]]) with its canonical map from $[\ast \to \ast]$.

In summary, the generic subterminal object and the generic pointed object fit into a sequence of the form

\begin{displaymath}
\itexarray{
     \emptyset &\hookrightarrow& \ast &\longrightarrow& \ast
     \\
     \downarrow && \downarrow && \downarrow
     \\
     \ast &\longrightarrow& \ast &\longrightarrow& S^0
  }
\end{displaymath}
\hypertarget{cohomology}{}\subsubsection*{{Cohomology}}\label{cohomology}

According to the general idea of [[cohomology]], for $\mathbf{H}$ an [[(∞,1)-topos]], and $X, A \in \mathbf{H}$ two [[objects]], cohomology classes of $X$ with coefficients in $A$ are the connected components

\begin{displaymath}
H(X,A) := \pi_0 \mathbf{H}(X,A)
  \,.
\end{displaymath}
Cohomology in the Sierpinski $(\infty, 1)$-topos, $\mathbf{H}^{I}$, corresponds to [[relative cohomology]] in $\mathbf{H}$.

Indeed, let $i : Y \to X$ and $f : B \to A$ be two [[morphisms]] in $\mathbf{H}$. Then the \textbf{relative cohomology} of $X$ with coefficients in $A$ relative to these morphisms is the connected components of the $\infty$-groupoid of relative cocycles

\begin{displaymath}
H_{Y}^B(X,A) := \pi_0 \mathbf{H}^I(Y \stackrel{i}{\to} X\;,\; B \stackrel{f}{\to} A)
  \,.
\end{displaymath}
In terms of the [[codomain fibration]], $\mathbf{H}^I \to \mathbf{H}$, intrinsic cohomology can be considered as nonabelian [[twisted cohomology]] (see there).

\hypertarget{references}{}\subsection*{{References}}\label{references}

The Sierpinski topos is mentioned around remarks A2.1.12, B3.2.11 (p.83, p.387f) in

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]] vol. I}, Oxford UP 2002.

\end{itemize}
See also

\begin{itemize}%
\item [[Peter Johnstone]], \emph{Topos theory}, London Math. Soc. Monographs \textbf{10}, Acad. Press 1977, xxiii+367 pp.

\end{itemize}
The [[homotopy type theory]] of the Sierpinski $(\infty,1)$-topos is discussed in

\begin{itemize}%
\item [[Mike Shulman]], \emph{The univalence axiom for inverse diagrams} (\href{http://arxiv.org/abs/1203.3253}{arXiv:1203.3253})

\end{itemize}
Cohesion of the Sierpinski $\infty$-topos is discussed in section 2.2.4 of

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]}

\end{itemize}
[[!redirects Sierpinski (∞,1)-topos]] [[!redirects Sierpinski (infinity,1)-topos]]



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