# nLab Sierpinski topos

### Context

#### Topology

topology

algebraic topology

# Contents

## Definition

###### Definition

The Sierpinski topos is the arrow category of Set.

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

$\mathrm{Sh}\left(\mathrm{Sierp}\right)\simeq \mathrm{PSh}\left(\Delta \left[1\right]\right)\simeq {\mathrm{Set}}^{\Delta \left[1\right]}\phantom{\rule{thinmathspace}{0ex}}.$Sh(Sierp) \simeq PSh(\Delta[1]) \simeq Set^{\Delta[1]} \,.
###### Definition

Similarly, the Sierpinski (∞,1)-topos is the arrow category $\infty {\mathrm{Grpd}}^{\Delta \left[1\right]}$ of ∞Grpd.

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

${\mathrm{Sh}}_{\left(\infty ,1\right)}\left(\mathrm{Sierp}\right)\simeq {\mathrm{PSh}}_{\left(\infty ,1\right)}\left(\Delta \left[1\right]\right)\simeq \infty {\mathrm{Grpd}}^{\Delta \left[1\right]}\phantom{\rule{thinmathspace}{0ex}}.$Sh_{(\infty,1)}(Sierp) \simeq PSh_{(\infty,1)}(\Delta[1]) \simeq \infty Grpd^{\Delta[1]} \,.

## Properties

### Presentation and Homotopy type theory

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

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

### Connectedness, locality, cohesion

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

${H}^{I}\to H\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{H}^I \to \mathbf{H} \,.
###### Proposition

The Sierpinski topos is a cohesive topos.

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

$\left(\Pi ⊣Disc⊣\Gamma ⊣\mathrm{coDisc}\right):{H}^{I}\stackrel{\stackrel{\Pi }{\to }}{\stackrel{\stackrel{\mathrm{Disc}}{←}}{\stackrel{\stackrel{\Gamma }{\to }}{\underset{\mathrm{coDisc}}{←}}}}H\phantom{\rule{thinmathspace}{0ex}}.$(\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} \,.
###### Proof

For the first statement, see the detailed discussion at cohesive topos here.

For the second statement, see the discussion at cohesive (∞,1)-topos here.

###### Remark

The fact that the Sierpienski $\left(\infty ,1\right)$-topos is, therefore, in particular

all follow directly from the fact that it is the image, under localic reflection, of the Sierpinski space (hence that it is 0-localic, its (-1)-truncation being the frame of opens of the Sierpinski space).

That space $\mathrm{Sierp}$, in turn,

which implies the corresponding three properties of the Sierpinski $\infty$-topos above.

###### Remark

By the discussion at cohesive (∞,1)-topos every such may be thought of as a fat point, the abstract cohesive blob. In this case, this fat point is the Sierpinski space. This space can be thought of as being the abstract “point with open neighbourhood”.

###### Remark

Accordingly, the objects of the Sierpinski $\left(\infty ,1\right)$-topos may be thought of as ∞-groupoids (relative to $H$) equipped with the notion of cohesion modeled on this: they are bundles $\left[P\to X\right]$ of ∞-groupoids whose fibers are regarded as being geometrically contractible, in that

$\Pi \left(\left[P\to X\right]\right)\simeq X$\Pi([P \to X]) \simeq X

and so in particular

$\Pi \left(\left[Q\to *\right]\right)\simeq *\phantom{\rule{thinmathspace}{0ex}}.$\Pi([Q \to *]) \simeq * \,.

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 ${P}_{x}:=P{×}_{X}\left\{x\right\}$.

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

$\Gamma \left(\left[P\to X\right]\right)\simeq P$\Gamma([P \to X]) \simeq P

of the discrete base space and the inner structure of the fibers.

The discrete object in the Sierpinski $\left(\infty ,1\right)$-topos on an object $X\in H$ is the bundle

$\mathrm{Disc}\left(X\right)\simeq \left[X\stackrel{\mathrm{id}}{\to }X\right]$Disc(X) \simeq [X \stackrel{id}{\to} X]

which is $X$ with “no cohesive blobs attached”.

Finally the codiscrete object in the Sierpinski $\left(\infty ,1\right)$-topos on an object $X\in H$ is

$\mathrm{coDisc}\left(X\right)\simeq \left[X\to *\right]\phantom{\rule{thinmathspace}{0ex}},$coDisc(X) \simeq [X \to *] \,,

the structure where all of $X$ is regarded as one single contractible cohesive ball.

The $\left(\Pi ⊣\mathrm{Disc}\right)$-adjunction unit

$i:\mathrm{id}\to \mathrm{Disc}\Pi$i : id \to Disc \Pi

on $\left[P\to X\right]$ is

$\begin{array}{ccc}\left[P& \to & X\right]\\ ↓& & ↓\\ \left[X& \to & X\right]\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \mathllap{[}P &\to& X\mathrlap{]} \\ \downarrow && \downarrow \\ \mathllap{[}X &\to& X\mathrlap{]} } \,.

The $\left(\mathrm{Disc}⊣\Gamma \right)$-counit $\mathrm{Disc}\Gamma \to \mathrm{id}$ on $\left[P\to X\right]$ is

$\begin{array}{ccc}\left[P& \to & P\right]\\ ↓& & ↓\\ \left[P& \to & X\right]\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \mathllap{[}P &\to& P\mathrlap{]} \\ \downarrow && \downarrow \\ \mathllap{[}P &\to& X\mathrlap{]} } \,.

Hence the canonical natural transformation

$\begin{array}{ccccc}\Gamma & & \to & & \Pi \\ & {}_{\Gamma \left(i\right)}↘& & {↗}_{\simeq }\\ & & \Gamma \mathrm{Disc}\Pi \end{array}$\array{ \Gamma && \to && \Pi \\ & {}_{\mathllap{\Gamma(i)}}\searrow & & \nearrow_{\mathrlap{\simeq}} \\ && \Gamma Disc \Pi }

from “points to pieces” is on $\left[P\to X\right]$ simply the morphism $P\to X$ itself

$\left(\Gamma \to \Pi \right)\left(\left[P\to X\right]\right)=\left(P\to X\right)\phantom{\rule{thinmathspace}{0ex}}.$(\Gamma \to \Pi)([P \to X]) = (P \to X) \,.

Therefore

1. the full sub-(∞,1)-category on those objects in ${H}^{I}$ for which ”pieces have points”, hence those for which $\Gamma \to \Pi$ is an effective epimorphism, is the $\left(\infty ,1\right)$-category of effective epimorphisms in the ambient $\left(\infty ,1\right)$-topos, hence the $\left(\infty ,1\right)$-category of groupoid objects in the ambient $\left(\infty ,1\right)$-topos;

2. the full sub-$\left(\infty ,1\right)$-category on the objects with ”one point per piece” is the ambient $\left(\infty ,1\right)$-topos itself.

### Cohesive structures

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

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

The corresponding flat coefficient object $♭B\left[\stackrel{^}{G}\to G\right]\to B\left[\stackrel{^}{G}\to G\right]$ is

$\begin{array}{ccc}\stackrel{^}{G}& \to & B\stackrel{^}{G}\\ ↓& & ↓\\ \stackrel{^}{G}& \to & G\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \mathbf{\hat G} &\to& \mathbf{B} \hat G \\ \downarrow && \downarrow \\ \mathbf{\hat G} &\to& \mathbf{G} } \,.

Hence the corresponding de Rham coefficient object is

${♭}_{\mathrm{dR}}B\left[\stackrel{^}{G}\to G\right]=\left[*\to BA\right]\phantom{\rule{thinmathspace}{0ex}},$\mathbf{\flat}_{dR} \mathbf{B}[\hat G \to G] = [* \to \mathbf{B}A] \,,

where $A\to \stackrel{^}{G}\to G$ exhibits $\stackrel{^}{G}$ has an $\infty$-group extension of $G$ by $A$ in $H$.

The corresponding Maurer-Cartan form

$\left[\stackrel{^}{G}\to G\right]\to {♭}_{\mathrm{dR}}B\left[\stackrel{^}{G}\to G\right]$[\hat G \to G] \to \mathbf{\flat}_{dR}\mathbf{B}[\hat G \to G]

is

$\begin{array}{ccc}\stackrel{^}{G}& \to & G\\ ↓& & ↓\\ *& \to & BA\end{array}$\array{ \hat G &\to& G \\ \downarrow && \downarrow \\ * &\to& \mathbf{B}A }

exhibiting the $A$-cocycle that classifies the extension $\stackrel{^}{G}\to G$.

### Infinitesimal cohesion

The above discussion generalizes immediately as follows.

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

${H}^{\Delta \left[1\right]}\stackrel{\stackrel{\mathrm{const}}{↩}}{\underset{\mathrm{dom}}{\to }}H\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{H}^{\Delta[1]} \stackrel{\overset{const}{\hookleftarrow}}{\underset{dom}{\to}} \mathbf{H} \,.

Conversely, we may think of ${H}^{\Delta \left[1\right]}$ as being an “infinitesimal thickening” of $H$, as formalized at infinitesimal cohesion, where we regard

$\left({i}_{!}⊣{i}^{*}⊣{i}_{*}⊣{i}^{*}\right):H\stackrel{\stackrel{{\top }_{!}}{↪}}{\stackrel{\stackrel{{\top }^{*}}{←}}{\stackrel{\stackrel{\mathrm{const}}{↪}}{\underset{{\perp }^{*}}{←}}}}{H}^{\Delta \left[1\right]}$(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]}

as exhibiting ${H}^{\Delta \left[1\right]}$ as an infinitesimal cohesive neighbourhood of $H$ (here $\left(\perp ,\top \right):\Delta \left[0\right]\coprod \Delta \left[0\right]\to \Delta \left[1\right]$ denotes the enpoint inclusions, following the notation here).

###### Observation

We have for all $X\in H$ that

${i}_{!}\left(X\right)\simeq \left[\varnothing \to X\right]\phantom{\rule{thinmathspace}{0ex}}.$i_!(X) \simeq [\emptyset \to X] \,.
###### Proof

For all $\left[A\to B\right]$ in ${H}^{\Delta \left[1\right]}$ we have

$H\left(X,{i}^{*}\left[A\to B\right]\right)\simeq H\left(X,\mathrm{cod}\left(A\to B\right)\right)\simeq H\left(X,B\right)\phantom{\rule{thinmathspace}{0ex}},$\mathbf{H}(X, i^*[A \to B]) \simeq \mathbf{H}(X, cod(A \to B)) \simeq \mathbf{H}(X, B) \,,

which is indeed naturally equivalent to

${H}^{\Delta \left[1\right]}\left(\left[\varnothing \to X\right],\left[A\to B\right]\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{H}^{\Delta[1]}([\emptyset \to X], [A \to B]) \,.

Therefore an object of ${H}^{\Delta \left[1\right]}$ given by a morphism $\left[P\to X\right]$ in $H$ is regarded by the infinitesimal cohesion $i:H↪{H}^{\Delta \left[1\right]}$ 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 $H$” (hence not necessarily discrete in $H$) 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

###### Proposition

With respect to the above infinitesimal cohesion $i:H↪{H}^{\Delta \left[1\right]}$, every morphism in $H$ is a formally étale morphism.

###### Proof

By definition, given a morphism $f:X\to Y$, it is formally étale precisely if

$\begin{array}{ccc}{i}_{!}X& \stackrel{{i}_{!}f}{\to }& {i}_{!}Y\\ ↓& & ↓\\ {i}_{*}X& \stackrel{{i}_{*}}{\to }& {i}_{*}Y\end{array}$\array{ i_! X &\stackrel{i_! f}{\to}& i_! Y \\ \downarrow && \downarrow \\ i_* X &\stackrel{i_*}{\to}& i_* Y }

is an (∞,1)-pullback.

By prop. 2 the above square diagram in ${H}^{\Delta \left[1\right]}$ is

$\begin{array}{ccc}\left[\varnothing \to X\right]& \to & \left[\varnothing \to Y\right]\\ ↓& & ↓\\ \left[X\stackrel{\mathrm{id}}{\to }X\right]& \to & \left[Y\stackrel{\mathrm{id}}{\to }Y\right]\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ [\emptyset \to X] &\to& [\emptyset \to Y] \\ \downarrow && \downarrow \\ [X \stackrel{id}{\to} X] &\to& [Y \stackrel{id}{\to} Y] } \,.

Since $\left(\infty ,1\right)$-pullbacks of $\left(\infty ,1\right)$-presheaves are computed objectwise, this is an $\left(\infty ,1\right)$-pullback in ${H}^{\Delta \left[1\right]}$ precisely if the “back and front sides”

$\begin{array}{ccc}\varnothing & \to & \varnothing \\ ↓& & ↓\\ X& \stackrel{f}{\to }& Y\end{array}$\array{ \emptyset &\to& \emptyset \\ \downarrow && \downarrow \\ X &\stackrel{f}{\to}& Y }

and

$\begin{array}{ccc}X& \stackrel{f}{\to }& Y\\ {↓}^{\mathrm{id}}& & {↓}^{\mathrm{id}}\\ X& \stackrel{f}{\to }& Y\end{array}$\array{ X &\stackrel{f}{\to}& Y \\ \downarrow^{\mathrm{id}} && \downarrow^{\mathrm{id}} \\ X &\stackrel{f}{\to}& Y }

are $\left(\infty ,1\right)$-pullbacks in $H$. This is clearly always the case.

## References

The Sierpinski topos is mentioned around remark B3.2.11 in

The homotopy type theory of the Sierpinski $\left(\infty ,1\right)$-topos is discussed in

Cohesion of the Sierpinski $\infty$-topos is discussed in section 2.2.4 of

Revised on April 17, 2012 16:41:13 by Urs Schreiber (89.204.154.100)