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brief note: Sierpinski topos
It’s not quite right that $0 \to 1$ is the category of opens of Sierpinski space, since Sierpinski space has three open sets.
However, it is true that $Set^{0 \to 1}$ is equivalent to sheaves on Sierpinski space.
It is the category of non-empty opens, though, and the trivial topology on $\{ 0 \to 1 \}$ happens to be the same as the subcategory topology.
Does it classify the open subobjects of a topos, like the Sierpiński space classifies open sets?
Right, thanks. I have fixed the wording now.
I would imagine so, as by Diaconescu’s theorem $Topos(E, Set^{\mathbf 2}) \simeq Flat(\mathbf{2}, E)$, and $1$ is terminal and $0 \to 1$ monic in $\mathbf{2}$, properties preserved (I would guess) by flat functors. Right?
@Finn: Yes, that looks right. So the Sierpinski topos is the classifying topos of subterminal objects.
For completeness, I have added to cohesive (infinity,1)-topos a remark (here) that for $\mathbf{H}$ any cohesive $(\infty,1)$-topos and $D$ a diagram $(\infty,1)$-category with terminal and initial object, then also $\mathbf{H}^D$ is cohesive. So in particular the Sierpinski $(\infty,1)$-topos is cohesive, and I added a remark on that also to Sierpinski topos.
How should I think of the thickened point corresponding to the Sierpinski $(\infty,1)$-topos? As one of the points of the Sierpinski space?
Good question.
In this case it is really easy to think of this point, because the topos is localic! So that fat point is the Sierpinski space. Which, indeed, is contractible.
I’ll add some more comments on this to the entry in a short moment…
I don’t really think about “thickened points”, but the “cohesion” of an object of the Sierpinski topos can be thought of as specifying, in addition to a set of points, a set of “components” such that each point falls into exactly one component. Discrete cohesion puts every point in its own component; codiscrete cohesion puts everything in a unique component; and $\Pi$ is just the set of components.
And for the Sierpinski $(\infty,1)$-topos one has that an object is given by an $\infty$-bundle and the cohesion is such that it regards all the fibers of this bundle as being contractible blobs, sitting over a discrete base $\infty$-groupoid.
I have started adding some more remarks to Sierpinski topos. But need to interrupt now.
So you can have a topological, smooth and super Sierpinski $(\infty, 1)$-toposes too presumably.
Yes, for $\mathbf{H}$ any cohesive $\infty$-topos, also $\mathbf{H}^{\Delta[1]}$ is cohesive.
Incidentally, I just added further discussion related to this, before even seeing your comment here. This was promted by private email discussion with Stephan Spahn, who pointed out that we should relate the Sierpinski $(\infty,1)$-topos also to infinitesimal cohesion.
I have now added some first simple remarks on this in a new section here.
Does anything interesting come from other choices of diagram, or does any choice remain describable as
a very crude notion of infinitesimal extension?
Does anything interesting come from other choices of diagram, or does any choice remain describable as
a very crude notion of infinitesimal extension?
For all the diagrams as discussed there (having initial and terminal object) the conclusion (in fact the whole discussion) is the same (since everything only depends on and only uses the existence of initial and terminal objects in the diagram).
For diagrams not having both initial and terminal objects the situation may be different, but of course there are fairly strong conditions on the diagram shape $D$ such that $\mathbf{H}^D$ is still cohesive, if $\mathbf{H}$ is.
But, I suppose the Sierpinski $\infty$-topos is still important in the theory of the formally etale morphisms induced by the infinitesimal cohesion, as a classifying object: as Stephan Spahn points out in private email, this is an observation due to Dubuc and Kock, see section 3 of Dubuc’s article now listed at etale map.
Here is a simple remark on the simple topic of the Sierpinski (infinity,1)-topos, which nevertheless may be worth saying out loud: let $\mathbf{H}$ be an $\infty$-topos, and $\mathbf{H}^I$ its Sierpinski $\infty$-topos, regarded as a cohesive (∞,1)-topos over $\mathbf{H}$, as discussed here. Then:
the full sub-$\infty$-category on those objects for which pieces have points is that of groupoid objects in $\mathbf{H}$;
the full sub-$\infty$-category on those objects with one piece per point is $\mathbf{H}$ itself.
Hmm, that’s an interesting way of thinking about it.
I’m still in the exegesis of Urs’ statement: With the notation here pieces have points means that
$\bottom^* x\to \bottom^* p^* \top^* x\simeq \top^* x$(or if we look at it in $\infty$Grpd
$\Gamma \bottom^* x\to \Gamma \bottom^* p^* Disc \Pi \top^* x\simeq \Pi \top^* x$)
is an effective epimorphism; and the statement is that this is the case precisely if $\bottom^* x=dom \, x$ is a groupoid object in $\mathbf{H}$ for $x:X\to X^\prime$.
…
Hi Stephan,
right, I should have been more explicit. In fact, I was, though not here but in the $n$Lab entry here (scroll down to the end of that section).
So an object in the Sierpinski $\infty$-topos $\mathbf{H}^I$ is a morphism $[P \to X]$ in $\mathbf{H}$. For the cohesive structure
$(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : \mathbf{H}^I \to \mathbf{H}$we have
$\Pi([P \to X]) = X$ $\Gamma([P \to X]) = P$and one checks that the value of the canonical natural transformation $\Gamma \to \Pi$ on the object $[P \to X]$ is the morphism in $\mathbf{H}$
$(\Gamma \to \Pi) ([P \to X]) = P \to X \,,$hence just the morphism $P \to X$ itself. So $(\Gamma \to \Pi)([P \to X])$ being an effective epi means that $P \to X$ is an effective epi.
So the $\infty$-category of objects in the Sierpinski $\infty$-topos $\mathbf{H}^I$ for which “pieces have points” is that of effective epimorphisms in $\mathbf{H}$.
Finally, just in order to amplify the meaning of this a bit more, I used that by the $\infty$-Giraud axioms the $\infty$-category of of effective epis in $\mathbf{H}$ is equivalent to that of groupoid objects in $\mathbf{H}$.
For this recall that a groupoid object in an (infinity,1)-category is a simplicial object $\mathcal{G} : \Delta^{op} \to \mathbf{H}$ that satisfies all the groupoidal Segal conditions. One way to get such groupoid objects is as the “Čech nerves” of morphisms $P \to X$ in $\mathbf{H}$ by setting
$\mathcal{G} : [n] \mapsto P \times_X \cdots \times_X P \;\; (n \; factors) \,.$The $\infty$-Giraud axiom (or Giraud-Rezk-Lurie-axiom, or whatever) then says that this construction of groupoid objects from morphisms actually establishes an euivalence from effective epimorphisms to groupoid objects in $\mathbf{H}$.
So we may regard any effective epimorphism as a placeholder for the groupoid object that it induces, as above. This is what I was alluding to in my telegraphic message above.
Re #6: I have added to Sierpinski topos and to subterminal object the remark that the Sierpienski topos is the classifying topos for subterminal objects, with pointer to p. 117 of Johnstone’s “Topos theory”, for the record.
Disappointingly, I must be missing something fundamental.
$CoDisc(P) = [P \to \ast]$, so that $\sharp([P \to X]) = [P \to \ast]$, and the empty modality is $\emptyset [P \to X] = [\emptyset \to \emptyset]$, so this isn’t a case of $\emptyset \lt \sharp$, as required by a resolution. And that tallies with there being some objects for which pieces don’t have points, e.g., $[\empty \to \ast]$.
But doesn’t cohesion require this resolution?
Actually it is an example of resolution: by the formulas which you just recalled you have $\sharp([\emptyset \to \emptyset]) = [\emptyset \to \emptyset]$. That’s what $\emptyset \lt \sharp$ means: if an object is sent to itself by $\emptyset$, then it is also sent to itself by $\sharp$.
Regarding whether resolution of the initial opposition is part of the definition of cohesion: not in the original axioms. But of course you may decide that it is an axiom that one wants to add.
According to those formulas, doesn’t $\sharp([\emptyset \to \emptyset]) = [\emptyset \to \ast]$?
We also know that only for effective epimorphisms do “pieces have points”, e.g., $\flat[\emptyset \to \ast] = [\emptyset \to \emptyset]$ while $\esh [\emptyset \to \ast] = [\ast \to \ast]$.
(\esh isn’t available, I guess.)
Sorry, I am clearly not paying attention. Am distracted elsewhere.
Am distracted elsewhere.
Of course, you still have to teach!
Regarding whether resolution of the initial opposition is part of the definition of cohesion: not in the original axioms. But of course you may decide that it is an axiom that one wants to add.
So that is perhaps news to me, and maybe, if and when there’s time, it could be made clearer in a few places that different conventions are being used for ’cohesive’. E.g., here the resolution is taken as part of cohesiveness. I guess it would help to have different terms to distinguish the uses.
Right, so Aufhebung of the initial opposition is a consequence of the axiom “pieces have points” (here). Lawvere included this axiom since “Axiomatic cohesion”, but not before, I think.
To get an esh ʃ type ʃ
.
@#22: The Aufhebung of $\emptyset\dashv \ast$ in the Sierpinski topos is given by $\nabla\dashv$ʃ where $\nabla = L\circ\Pi$ obtains from the further left adjoint $L\dashv\Pi$ with $L(Z)=[\emptyset\to Z]$ whence $\nabla([X\to Y])=[\emptyset\to Y]$. This corresponds to the open (in particular essential) subtopos with objects $[X\overset{id}{\to} X]$ which is the copy of $Set$ on the open point of the Sierpinski space. This obviously contains the initial object $[\emptyset\to\emptyset]$ of $Set^2$ whence is $Sh_{\neg\neg}(Set^2)$. The topos of $\sharp$-sheaves is the complementary closed subtopos with objects $[X\to 1]$.
More generally, in a $\bot$-scattered topos $\mathcal{E}$ the Aufhebung of $\emptyset\dashv \ast$ is $\mathcal{E}_{\neg\neg}$ since by definition this is open whence essential in particular.
Ah. So that comparison I have here continues to the left.
It’s more linear than the general situation where one needs infinitesimals to form a reduction modality, $\Re$.
I find it surprising too, that $\Pi$ has a left adjoint in $Set^2$ but since it is a petit topos and hence ’banned’ as a gros topos from a Lawverian perspective perhaps strange things are bound to happen here e.g. $\Pi$ is logical due to the openness of the subtopos.
Perhaps we should say on this page that there’s a string of five adjoints. At the moment it treats the lower four as providing cohesion in the section ’Connectedness, locality, cohesion’, and then the upper four adjoints as providing ’infinitesimal cohesion’, without noting that they are part of the same string.
Also that Aufhebung of $\emptyset \vdash \ast$ coms in the form of the leftmost (co)monads.
So I’ve added something along those lines.
I pointed out that cohomology in the Sierpinski $(\infty, 1)$-topos, $\mathbf{H}^I$, is relative cohomology in $\mathbf{H}$, but then I remembered we have at twisted cohomology:
Given an $\infty$-topos $\mathbf{H}$, then also its arrow $\infty$-category $\mathbf{H}^I$ is an $\infty$-topos, over $\infty Grpd^I$ and it also sits over $\mathbf{H}$ by the codomain fibration, constituting an “extension” of $\mathbf{H}$ by itself:
$\array{ \mathbf{H} \\ \downarrow^{\mathrlap{incl}} \\ \mathbf{H}^I \\ \downarrow^{\mathrlap{cod}} \\ \mathbf{H} } \,.$The intrinsic cohomology of $\mathbf{H}^I$ under this fibration is nonabelian twisted cohomology as discussed in some detail in Principal ∞-bundles – theory, presentations and applications (schreiber).
I guess that should be added too. But how to express the relation between twisted and relative cohomology?
We also have arrow (∞,1)-topos making the point about relative cohomology.
Hmm, why ’def’ and ’bulk’ there?
This comes from interpreting the relative cohomology in terms of fields in a QFT with defects.
I suppose it’s hardly surprising that there are 5 adjoints between $\mathbf{H}^I$ and $\mathbf{H}$, given
$(\bottom \dashv p \dashv \top) : I \stackrel{\overset{\bottom}{\hookleftarrow}}{\stackrel{\overset{p}{\to}}{\underset{\top}{\hookleftarrow}}} * \,.$Curiously, we never seem to see all five together. Sometimes it’s the upper 4
$\mathbf{H} \stackrel{\overset{\top_!}{\hookrightarrow}}{\stackrel{\overset{\top^*}{\leftarrow}}{\stackrel{\overset{const}{\hookrightarrow}}{\underset{\bot^*}{\leftarrow}}}} \mathbf{H}^{\Delta[1]} \,,$and sometimes the lower 4
$\mathbf{H}^D \stackrel{\overset{\top^*}{\to}}{\stackrel{\overset{p^*}{\hookleftarrow}}{\stackrel{\overset{\bottom^*}{\to}}{\underset{\bottom_*}{\hookleftarrow}}}} \mathbf{H}.$But don’t we have a mismatch at Sierpinski topos. Elsewhere, ’infinitesimal cohesion’ is used for when $\flat$ = ʃ, e.g., the point is being made here that $\mathbf{H}^{sec}$ has infinitesimal cohesion because the diagram ’sec’ has a zero object, unlike the interval.
Surely the point about the Sierpinski situation is that with the two overlapping quadruples in the string of 5 adjoints, there are the three cohesive monads and the three differentially cohesive monads.
don’t we have a mismatch
Yes, at some point I switched terminolog. What had been called “infinitesimal cohesion” originally is now called differential cohesion.
That happened way back, but apparently this entry wasn’t updated yet. Now it is.
@#34: The above argument shows that $\nabla\dashv$ ʃ resolves $\emptyset\dashv \ast$ but the argument that it is the Aufhebung uses the fact that $Set$ is Boolean. In the more general case, if $H_{\neg\neg}$ is merely essential in $H$ with $n_! \dashv n^{\ast} \dashv n_{\ast} : H_{\neg\neg} \to H$ then it is essential in $H^{\Delta(1)}$ by $L\circ n_!\dashv n^\ast \circ \Pi \dashv \Delta \circ n_\ast: H_{\neg\neg} \to H^{\Delta (1)}$ and, accordingly, the Aufhebung of $\emptyset\dashv \ast$ is $(L\circ n_!)\circ (n^\ast \circ \Pi) \dashv (\Delta\circ n_\ast)\circ (n^\ast\circ\Pi)$.
Note that if $H_{\neg\neg}$ is essential in $H$, it is the Aufhebung of $\emptyset \dashv \ast$ in $H$, and, for the completely general case, the subtopos of $H^{\Delta (1)}$ corresponding to the Aufhebung of $\emptyset \dashv \ast$ in $H$ in a similar way then seems to be the natural candidate for the Aufhebung of $\emptyset \dashv \ast$ in $H^{\Delta (1)}$.
I don’t feel confident to say anything concerning the higher topos case, and therefore I probably shouldn’t have used $H^{\Delta (1)}$ in #39 since I think of the ’general’ Sierpinski topos there entirely in 1-categorical terms of Artin gluing along $id_H$.
I intend to eventually add a section to Aufhebung spelling the details of the Sierpinski 1-topos out, since I think this gives a nice&concrete illustration of the concepts at work. One way to avoid the question how much of this generalizes to $(\infty , 1)$-toposes would then be to simply link to this from Sierpinski topos until somebody attends to the higher categorical case.
In Remarks on Punctual Local Connectedness Proposition 1.4, Johnstone highlights that a a condition for a shaef topos to be precohesive over sets is that the site of definition $(\mathcal{C},J)$ satisfies that every object in $\mathcal{C}$ has a point. In the proof, this condition is needed in the Nullstellensatz. Therefore, the statement "the Sierpinski topos is cohesive" is a little sloppy. It is true that satisfies the continuity axiom as it is, but it is not precohesive over sets, so it is not cohesive. To obtain a cohesive category (it will be extensive and cartesian closed, but not a topos) over sets generated from the Sierpinski topos it is needed to consider the full subcategory generated from the objects in the Sierpinski topos that satisfy the Nullstellensatz
Just to highlight that throughout the nLab, notably in the linked pages for cohesive topos, we regard the pieces-have-points-axiom as an extra axiom, on top of cohesion.
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