nLab
Sierpinski topos

Definition

Definition

The Sierpinski topos is the arrow category of Set.

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

$$\mathrm{Sh}(\mathrm{Sierp})\simeq \mathrm{PSh}(\Delta [1])\simeq {\mathrm{Set}}^{\Delta [1]}.$$Sh(Sierp) \simeq PSh(\Delta[1]) \simeq Set^{\Delta[1]} \,.

Definition

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

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

$${\mathrm{Sh}}_{(\mathrm{\infty },1)}(\mathrm{Sierp})\simeq {\mathrm{PSh}}_{(\mathrm{\infty },1)}(\Delta [1])\simeq \mathrm{\infty }{\mathrm{Grpd}}^{\Delta [1]}.$$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 $[\Delta [1],\mathrm{sSet}]$.

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

$$H^I\to H.$$\mathbf{H}^I \to \mathbf{H} \,.

Proposition

The Sierpinski topos is a cohesive topos.

The Sierpinski $(\mathrm{\infty },1)$-topos is a cohesive (∞,1)-topos.

$$(\Pi ⊣Disc⊣\Gamma ⊣\mathrm{coDisc}):H^I\stackrel{\stackrel{\Pi }{\to }}{\stackrel{\stackrel{\mathrm{Disc}}{\leftarrow }}{\stackrel{\stackrel{\Gamma }{\to }}{\underset{\mathrm{coDisc}}{\leftarrow }}}}H.$$(\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 $(\mathrm{\infty },1)$-topos is, therefore, in particular

1. a locally ∞-connected (∞,1)-topos;

2. an ∞-connected (∞,1)-topos;

3. a local (∞,1)-topos

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,

1. is a contractible topological space;

2. a locally contractible topological space.

3. has a focal point

which implies the corresponding three properties of the Sierpinski $\mathrm{\infty }$-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 $(\mathrm{\infty },1)$-topos may be thought of as ∞-groupoids (relative to $H$) equipped with the notion of cohesion modeled on this: they are bundles $[P\to X]$ of ∞-groupoids whose fibers are regarded as being geometrically contractible, in that

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

and so in particular

$$\Pi ([Q\to *])\simeq *.$$\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 $\mathrm{\infty }$-groupoid $P_x:=P{\times }_X\{x\}$.

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

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

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

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

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

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

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

$$\mathrm{coDisc}(X)\simeq [X\to *],$$coDisc(X) \simeq [X \to *] \,,

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

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

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

on $[P\to X]$ is

$$\begin{array}{ccc}[P& \to & X]\\ \downarrow & & \downarrow \\ [X& \to & X]\end{array}.$$\array{ \mathllap{[}P &\to& X\mathrlap{]} \\ \downarrow && \downarrow \\ \mathllap{[}X &\to& X\mathrlap{]} } \,.

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

$$\begin{array}{ccc}[P& \to & P]\\ \downarrow & & \downarrow \\ [P& \to & X]\end{array}.$$\array{ \mathllap{[}P &\to& P\mathrlap{]} \\ \downarrow && \downarrow \\ \mathllap{[}P &\to& X\mathrlap{]} } \,.

Hence the canonical natural transformation

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

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

$$(\Gamma \to \Pi )([P\to X])=(P\to X).$$(\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 $(\mathrm{\infty },1)$-category of effective epimorphisms in the ambient $(\mathrm{\infty },1)$-topos, hence the $(\mathrm{\infty },1)$-category of groupoid objects in the ambient $(\mathrm{\infty },1)$-topos;

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

Cohesive structures

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

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

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

$$\begin{array}{ccc}\hat{G}& \to & B\hat{G}\\ \downarrow & & \downarrow \\ \hat{G}& \to & G\end{array}.$$\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

$${♭}_{\mathrm{dR}}B[\hat{G}\to G]=[*\to BA],$$\mathbf{\flat}_{dR} \mathbf{B}[\hat G \to G] = [* \to \mathbf{B}A] \,,

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

The corresponding Maurer-Cartan form

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

is

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

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

Infinitesimal cohesion

The above discussion generalizes immediately as follows.

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

$$H^{\Delta [1]}\stackrel{\stackrel{\mathrm{const}}{\hookleftarrow }}{\underset{\mathrm{dom}}{\to }}H.$$\mathbf{H}^{\Delta[1]} \stackrel{\overset{const}{\hookleftarrow}}{\underset{dom}{\to}} \mathbf{H} \,.

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

$$(i_{!}⊣i^{*}⊣i_{*}⊣i^{*}):H\stackrel{\stackrel{{\top }_{!}}{\hookrightarrow }}{\stackrel{\stackrel{{\top }^{*}}{\leftarrow }}{\stackrel{\stackrel{\mathrm{const}}{\hookrightarrow }}{\underset{{\perp }^{*}}{\leftarrow }}}}{H}^{\Delta [1]}$$(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 $H^{\Delta [1]}$ as an infinitesimal cohesive neighbourhood of $H$ (here $(\perp ,\top ):\Delta [0]\coprod \Delta [0]\to \Delta [1]$ denotes the enpoint inclusions, following the notation here).

(See also the corresponding examples at Q-category.)

Observation

We have for all $X\in H$ that

$$i_{!}(X)\simeq [\varnothing \to X].$$i_!(X) \simeq [\emptyset \to X] \,.

Proof

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

$$H(X,i^{*}[A\to B])\simeq H(X,\mathrm{cod}(A\to B))\simeq H(X,B),$$\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

$$H^{\Delta [1]}([\varnothing \to X],[A\to B]).$$\mathbf{H}^{\Delta[1]}([\emptyset \to X], [A \to B]) \,.

Therefore an object of $H^{\Delta [1]}$ given by a morphism $[P\to X]$ in $H$ is regarded by the infinitesimal cohesion $i:H\hookrightarrow H^{\Delta [1]}$ 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 $H$” (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 $i:H\hookrightarrow H^{\Delta [1]}$, 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

$$\begin{array}{ccc}{i}_{!}X& \stackrel{i_{!}f}{\to }& i_{!}Y\\ \downarrow & & \downarrow \\ i_{*}X& \stackrel{i_{*}}{\to }& i_{*}Y\end{array}$$\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 $H^{\Delta [1]}$ is

$$\begin{array}{ccc}[\varnothing \to X]& \to & [\varnothing \to Y]\\ \downarrow & & \downarrow \\ [X\stackrel{\mathrm{id}}{\to }X]& \to & [Y\stackrel{\mathrm{id}}{\to }Y]\end{array}.$$\array{ [\emptyset \to X] &\to& [\emptyset \to Y] \\ \downarrow && \downarrow \\ [X \stackrel{id}{\to} X] &\to& [Y \stackrel{id}{\to} Y] } \,.

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

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

and

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

are $(\mathrm{\infty },1)$-pullbacks in $H$. This is clearly always the case.

References

The Sierpinski topos is mentioned around remark B3.2.11 in

• Peter Johnstone, Sketches of an Elephant

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

• Mike Shulman, The univalence axiom for inverse diagrams (arXiv:1203.3253)

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

• Urs Schreiber, differential cohomology in a cohesive topos