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Stream: deprecated: topos theory

Topic: pullback stability with enough points

Joshua Wrigley (Jun 06 2022 at 09:59):

In the category $\bf Top$ of Grothendieck toposes and geometric morphisms there are two well-known factorisation systems:

• the (hyperconnected,localic) factorisation scheme,
• the (surjection,embedding) factorisation scheme.

The former is well-behaved with pullbacks. For example, localic geometric morphisms are stable under pullback - i.e. given a pullback diagram

$$\begin{CD} \mathcal{E} \times_{\mathcal{G}} \mathcal{F} @>{f}>> \mathcal{E}\\ @VVV @VVV\\ \mathcal{F} @>{g}>>\mathcal{G} \end{CD}$$

in which $g$ is localic, then so too is $f$. Inclusions are also preserved under pullback (see Example A4.5.14(e) of the Elephant). The same is not true for surjections. However, we do have that open surjections are stable under pullback.

Let $\bf TopWP$ denote the full subcategory of $\bf Top$ of Grothendieck toposes with enough points (a topos $\mathcal{E}$ has enough points if there exists a set $X$ and a geometric surjection ${\bf Sets}^X \to \mathcal{E}$). The category $\bf TopWP$ clearly inherits both factorisation schemes from $\bf Top$. Let $\mathcal{E} \to \mathcal{F}$ be a geometric morphism between two toposes with enough points, and let

$$\begin{CD} \mathcal{E} @>{hyp}>> \mathcal{E}' @>{loc}>> \mathcal{F} \end{CD}$$

be the (hyperconnected,localic) factorisation in $\bf Top$. As hyperconnected morphisms are surjections, there exists a surjection ${\bf Sets}^X \to \mathcal{E} \xrightarrow{hyp} \mathcal{E}'$. The (surjection,embedding) case is identical.

So the natural next question is whether we obtain similar pullback stability properties in $\bf TopWP$. We know, by Corollary 7.18 in Johnstone's Topos theory, that for each topos $\mathcal{E}$ there exists a unique largest subtopos ${\rm Pt}(\mathcal{E})$ that has enough points. Thus, for each geometric morphism $\mathcal{F} \xrightarrow{f} \mathcal{E}$ where $\mathcal{F}$ has enough points, $f$ factors through the subtopos ${\rm Pt}(\mathcal{E})$. Therefore the pullback in $\bf TopWP$ of a pair of arrows

$$\mathcal{E} \longrightarrow \mathcal{G} \longleftarrow \mathcal{F}$$

is given by ${\rm Pt}(\mathcal{E} \times_{\mathcal{G}} \mathcal{F})$, where $\mathcal{E} \times_{\mathcal{G}} \mathcal{F}$ is the pullback in $\bf Top$. As ${\rm Pt}(\mathcal{E} \times_{\mathcal{G}} \mathcal{F})$ is a subtopos of $\mathcal{E} \times_{\mathcal{G}} \mathcal{F}$, and inclusions are localic, it immediately follows that localic geometric morphisms and geometric embeddings are stable under pullback in $\bf TopWP$.

My question to the forum is this: is there an easy way to see whether or not

• hyperconnected morphisms and
• open surjections

are also stable under pullback in $\bf TopWP$? In the lack of an easy way, is there a hard way?

Morgan Rogers (he/him) (Jun 06 2022 at 12:41):

@Ivan Di Liberti and I are working on understanding which toposes have enough points with enough clarity to answer that kind of question, although I have been distracted enough by other things that the progress has been very slow lately.

I expect there to be counterexamples to at least the second bullet point. A potential strategy would be to exploit the following two observations:

1. while the codomain of a surjection from a topos with enough points necessarily has enough points, not all of its points need factor through the surjection.
2. A point of the pullback is a pair of points which complete the square defining the pullback.

I reckon that with a clever gluing of two toposes you could construct a pullback of a surjection whose subtopos with enough points is the degenerate subtopos.

Morgan Rogers (he/him) (Jun 06 2022 at 12:43):

iirc a hyperconnected morphism which is not surjective on points appears in one of @Jens Hemelaer 's papers, maybe he can remind me which one.

Jens Hemelaer (Jun 06 2022 at 19:55):

Morgan Rogers (he/him) said:

iirc a hyperconnected morphism which is not surjective on points appears in one of Jens Hemelaer 's papers, maybe he can remind me which one.

Yes, there's an example at the end of Subsection 3.1 in "A topological groupoid representing the topos of presheaves on a monoid". As pointed out by the reviewer, it is not the first example. There's also an example in the Elephant, Example D3.4.14, of an atomic topos that is hyperconnected (over $\mathbf{Sets}$) and does not have any points.

Jens Hemelaer (Jun 06 2022 at 20:00):

The example from my paper goes more or less like this:
Take a cardinal $\alpha$ and let $M_1$ and $M_2$ be the free monoid on $\alpha$ generators and the free commutative monoid on $\alpha$ generators, respectively. The projection map $M_1 \to M_2$ induces a hyperconnected, essential geometric morphism at the level of presheaf toposes. As soon as $\alpha$ is big enough (e.g. $\alpha=|\mathbb{R}|$), then this geometric morphism is not surjective on points.

Jens Hemelaer (Jun 06 2022 at 20:07):

For any hyperconnected geometric morphism $f : \mathcal{F} \to \mathcal{E}$ that is not surjective on points, you can take a point $p : \mathbf{Sets} \to \mathcal{E}$ that is not in the pointwise image of $f$. The pullback of $f$ along $p$ in $\mathbf{TopWP}$ is then the inclusion of the empty topos in $\mathbf{Sets}$, and this is not surjective anymore. So hyperconnected geometric morphisms and open surjections both fail to be stable under pullback in $\mathbf{TopWP}$.

Jens Hemelaer (Jun 06 2022 at 20:11):

Thanks @Morgan Rogers (he/him) for writing out the strategy!

Morgan Rogers (he/him) (Jun 07 2022 at 06:19):

Aha so the clever gluing isn't even necessary! iirc the example in the Elephant is unbounded over sets (since every bounded connected atomic topos has a point by a result in C3.5), and in any case using a topos with no points wouldn't give a counterexample here, so your example is really necessary :wink:

Jens Hemelaer (Jun 07 2022 at 06:35):

The example in the Elephant is bounded over $\mathbf{Sets}$ (there are easier unbounded examples as well). Maybe the result from C3.5 you are thinking of is that every connected atomic topos with a point is bounded?

Joshua Wrigley (Jun 07 2022 at 07:11):

Thanks @Jens Hemelaer @Morgan Rogers (he/him) for the counter example!

Morgan Rogers (he/him) (Jun 07 2022 at 11:52):

Jens Hemelaer said:

The example in the Elephant is bounded over $\mathbf{Sets}$ (there are easier unbounded examples as well). Maybe the result from C3.5 you are thinking of is that every connected atomic topos with a point is bounded?

Ah yep, that's what I was thinking of. I did not recall correctly!