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 $\Delta[1] := \mathbf{2} = \{0 \to 1\}$, 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 (∞,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 $Sierp$:

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

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

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 $Sierp$, 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 $\infty$-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 $\hat G \to G$.

Differential cohesion

The leftmost adjoint, $\Pi$, of the string of four adjoints exhibiting 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

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

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

Proposition

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.

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. 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.

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

As a classifying topos

The Sierpinski topos is the classifying topos for subterminal objects in toposes (see e.g. 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

$$\Delta^1 = \{\ast \to S^0\} \longrightarrow FinSet^{\ast/} \hookrightarrow \infty Grpd_{fin}^{\ast/}$$

induces via restriction and right Kan extension an essential geometric morphism

$$\mathbf{H}^{\Delta^1} \stackrel{\longleftarrow}{\hookrightarrow} \mathbf{H}[X_\ast]$$

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

$$\array{ \emptyset &\hookrightarrow& \ast &\longrightarrow& \ast \\ \downarrow && \downarrow && \downarrow \\ \ast &\longrightarrow& \ast &\longrightarrow& S^0 }$$

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

$$H(X,A) := \pi_0 \mathbf{H}(X,A) \,.$$

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 relative cohomology of $X$ with coefficients in $A$ relative to these morphisms is the connected components of the $\infty$-groupoid of relative cocycles

$$H_{Y}^B(X,A) := \pi_0 \mathbf{H}^I(Y \stackrel{i}{\to} X\;,\; B \stackrel{f}{\to} A) \,.$$

In terms of the codomain fibration, $\mathbf{H}^I \to \mathbf{H}$, intrinsic cohomology can be considered as nonabelian twisted cohomology (see there).

References

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

• Peter Johnstone, Sketches of an Elephant vol. I, Oxford UP 2002.

See also

• Peter Johnstone, Topos theory, London Math. Soc. Monographs 10, Acad. Press 1977, xxiii+367 pp.

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

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

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

• Urs Schreiber, differential cohomology in a cohesive topos