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

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

Yet another description is that it is the Freyd cover of Set.

###### Definition

Similarly, the Sierpinski (∞,1)-topos is the arrow 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 $Sierp$:

$Sh_{(\infty,1)}(Sierp) \simeq PSh_{(\infty,1)}(\Delta[1]) \simeq \infty Grpd^{\Delta[1]} \,.$

## 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 $[\Delta[1], sSet]$.

Specifically the Reedy model structure of simplicial presheaves on the interval category $[\Delta[1], sSet]_{Reedy}$ provides a univalent model for homotopy type theory in the Sierpinski $(\infty,1)$-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 $\mathbf{H}$, hence regard the global section geometric morphism

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

The Sierpinski topos is a cohesive topos.

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

$(\Pi \dashv \Disc \dashv \Gamma \dashv coDisc) : \mathbf{H}^I \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\leftarrow}}}} \mathbf{H} \,.$
###### 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 $(\infty,1)$-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 $Sierp$, 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 $(\infty,1)$-topos may be thought of as ∞-groupoids (relative to $\mathbf{H}$) 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

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

and so in particular

$\Pi([Q \to *]) \simeq * \,.$

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 $P_x := P \times_X \{x\}$.

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

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

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

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

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

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

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

The $(\Pi \dashv Disc)$-adjunction unit

$i : id \to Disc \Pi$

on $[P \to X]$ is

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

The $(Disc \dashv \Gamma)$-counit $Disc \Gamma \to id$ on $[P \to X]$ is

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

Hence the canonical natural transformation

$\array{ \Gamma && \to && \Pi \\ & {}_{\mathllap{\Gamma(i)}}\searrow & & \nearrow_{\mathrlap{\simeq}} \\ && \Gamma Disc \Pi }$

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

$(\Gamma \to \Pi)([P \to X]) = (P \to X) \,.$

Therefore

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

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

### Cohesive structures

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

$\array{ \mathbf{\hat G} &\to& \mathbf{B} \hat G \\ \downarrow && \downarrow \\ \mathbf{\hat G} &\to& \mathbf{G} } \,.$

Hence the corresponding de Rham coefficient object is

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

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

The corresponding Maurer-Cartan form

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

is

$\array{ \hat G &\to& G \\ \downarrow && \downarrow \\ * &\to& \mathbf{B}A }$

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

### Infinitesimal cohesion

The above discussion generalizes immediately as follows.

For $\mathbf{H}$ 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

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

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

$(i_! \dashv i^* \dashv i_* \dashv i^*) : \mathbf{H} \stackrel{\overset{\top_!}{\hookrightarrow}}{\stackrel{\overset{\top^*}{\leftarrow}}{\stackrel{\overset{const}{\hookrightarrow}}{\underset{\bot^*}{\leftarrow}}}} \mathbf{H}^{\Delta[1]}$

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

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

###### Observation

We have for all $X \in \mathbf{H}$ that

$i_!(X) \simeq [\emptyset \to X] \,.$
###### Proof

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

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

which is indeed naturally equivalent to

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

Therefore an object of $\mathbf{H}^{\Delta[1]}$ given by a morphism $[P \to X]$ in $\mathbf{H}$ is regarded by the infinitesimal 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

###### Proposition

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

###### Proof

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

$\array{ i_! X &\stackrel{i_! f}{\to}& i_! Y \\ \downarrow && \downarrow \\ i_* X &\stackrel{i_*}{\to}& i_* Y }$

is an (∞,1)-pullback.

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

$\array{ [\emptyset \to X] &\to& [\emptyset \to Y] \\ \downarrow && \downarrow \\ [X \stackrel{id}{\to} X] &\to& [Y \stackrel{id}{\to} Y] } \,.$

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”

$\array{ \emptyset &\to& \emptyset \\ \downarrow && \downarrow \\ X &\stackrel{f}{\to}& Y }$

and

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

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

## References

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

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

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

Revised on April 3, 2015 09:20:57 by Thomas Holder (89.204.155.21)