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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 24th 2010
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   Author: Todd_Trimble
   Format: MarkdownItexStubby start to [[topological topos]]. Will be adding material by and by.

   Stubby start to topological topos. Will be adding material by and by.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMay 24th 2010
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   Author: Mike Shulman
   Format: MarkdownItexCould we call this something else? Johnstone's article was just called "on A topological topos"; I think there could be many different toposes that deserve to be thought of as "topological" for various reasons. Johnstone remarked slightly tongue-in-cheek that the objects of this topos might be called "consequential spaces," so if we wanted to go with that we could write about them on a page called [[consequential space]]. Or we could think of another name.

   Could we call this something else? Johnstone’s article was just called “on A topological topos”; I think there could be many different toposes that deserve to be thought of as “topological” for various reasons. Johnstone remarked slightly tongue-in-cheek that the objects of this topos might be called “consequential spaces,” so if we wanted to go with that we could write about them on a page called consequential space. Or we could think of another name.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 24th 2010
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   Author: Todd_Trimble
   Format: MarkdownItexThere could be, but there isn't. :-) Seriously: if someone wanted to look this up or google this, they'd type "topological topos", wouldn't they? At least that's what I tried. I doubt they would try consequential space. I could go along with something like "Johnstone's topological topos".

   There could be, but there isn’t. :-) Seriously: if someone wanted to look this up or google this, they’d type “topological topos”, wouldn’t they? At least that’s what I tried. I doubt they would try consequential space.

   I could go along with something like “Johnstone’s topological topos”.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeMay 25th 2010
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   Author: Mike Shulman
   Format: MarkdownItexWhat about the topos of sheaves on some small subcategory of Top, or the topos of sheaves on a category of (ultra)filters? In the article Johnstone lists a few other contenders like that. I've never heard anyone refer to this as "the topological topos." And that name isn't really correct anyway; it's not the topos that's topological, its the objects of the topos.

   What about the topos of sheaves on some small subcategory of Top, or the topos of sheaves on a category of (ultra)filters? In the article Johnstone lists a few other contenders like that. I’ve never heard anyone refer to this as “the topological topos.” And that name isn’t really correct anyway; it’s not the topos that’s topological, its the objects of the topos.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 25th 2010
   • (edited May 25th 2010)
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   Author: Todd_Trimble
   Format: MarkdownItexOkay, let's focus less on my first sentence of #3 (which seems to be leading to an argument I didn't intend), and more on what might be the most useful title. You'll notice I didn't say "the topological topos". On the other hand, despite your last sentence, it was Johnstone's phrase, not mine. So, what say you to "Johnstone's topological topos"? I'm thinking of the person trying to find out what this thing is that's being referred to in, for example, the page on [[subsequential space]]. Edit: and it might not be just from reading that page on the nLab -- it could be some poor slob like me who's heard of the darned thing, googles it, and comes up with little except bibliographic references and maybe a link to the paper for a price. I wanted to give it a title which is more likely to attract hits.

   Okay, let’s focus less on my first sentence of #3 (which seems to be leading to an argument I didn’t intend), and more on what might be the most useful title.

   You’ll notice I didn’t say “the topological topos”. On the other hand, despite your last sentence, it was Johnstone’s phrase, not mine. So, what say you to “Johnstone’s topological topos”? I’m thinking of the person trying to find out what this thing is that’s being referred to in, for example, the page on subsequential space.

   Edit: and it might not be just from reading that page on the nLab – it could be some poor slob like me who’s heard of the darned thing, googles it, and comes up with little except bibliographic references and maybe a link to the paper for a price. I wanted to give it a title which is more likely to attract hits.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeMay 25th 2010
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   Author: Mike Shulman
   Format: MarkdownItexOkay, fair enough. I can certainly go along with "Johnstone's topological topos."

   Okay, fair enough. I can certainly go along with “Johnstone’s topological topos.”

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeOct 8th 2016
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   Author: Mike Shulman
   Format: MarkdownItexI added to [[Johnstone's topological topos]] a proof that the *Cauchy* real numbers object therein also has the usual topology. (Johnstone's paper only discusses the Dedekind real numbers.) It would be nice to have a more abstract proof --- e.g. does this topos possess some general property that is sufficient for the Cauchy and Dedekind to coincide? --- but I can't think of one.

   I added to Johnstone’s topological topos a proof that the Cauchy real numbers object therein also has the usual topology. (Johnstone’s paper only discusses the Dedekind real numbers.) It would be nice to have a more abstract proof — e.g. does this topos possess some general property that is sufficient for the Cauchy and Dedekind to coincide? — but I can’t think of one.

8. • CommentRowNumber8.
   • CommentAuthorspitters
   • CommentTimeOct 8th 2016
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   Author: spitters
   Format: MarkdownItexNice. Could you include a reference that it does not support countable choice?

   Nice. Could you include a reference that it does not support countable choice?

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeOct 9th 2016
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   Author: Mike Shulman
   Format: MarkdownItexHmm... no, I don't think I could. Actually I'm not completely sure, but my instinct is that it doesn't. (Though it does satisfy "crisp countable choice", i.e. $\mathbb{N}$ is externally projective.)

   Hmm… no, I don’t think I could. Actually I’m not completely sure, but my instinct is that it doesn’t. (Though it does satisfy “crisp countable choice”, i.e. $\mathbb{N}$ is externally projective.)

10. • CommentRowNumber10.
    • CommentAuthorspitters
    • CommentTimeOct 10th 2016
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    Author: spitters
    Format: MarkdownItex[Fourman - continuous truth II](http://homepages.inf.ed.ac.uk/mfourman/research/publications/pdf/fourman2013-continuous-truth-II.pdf) shows that the gross topos of sheaves over separable locales has local choice. Both this gross topos and Johnstone's topological topos are ways to cut down the Giraud gross topos. It would be nice to understand what breaks countable choice in Johnstone's topos.

    Fourman - continuous truth II shows that the gross topos of sheaves over separable locales has local choice. Both this gross topos and Johnstone’s topological topos are ways to cut down the Giraud gross topos. It would be nice to understand what breaks countable choice in Johnstone’s topos.

11. • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeOct 10th 2016
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    Author: Mike Shulman
    Format: MarkdownItexWhat is "local choice"?

    What is “local choice”?

12. • CommentRowNumber12.
    • CommentAuthorspitters
    • CommentTimeOct 10th 2016
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    Author: spitters
    Format: MarkdownItexIt's defined in [Continuous truth I](http://homepages.inf.ed.ac.uk/mfourman/research/publications/pdf/fourman84-continuous-truth-I.pdf) below Prop 4.3. Locally, we have a choice function, i.e. there is an open cover U_i on which we can define a choice function on each U_i. This does not seem enough to derive countable choice. However, it is suggested that CAC holds in this model. I am surprised it is not clearly stated. Perhaps, I am overlooking something.

    It’s defined in Continuous truth I below Prop 4.3. Locally, we have a choice function, i.e. there is an open cover U_i on which we can define a choice function on each U_i.

    This does not seem enough to derive countable choice. However, it is suggested that CAC holds in this model. I am surprised it is not clearly stated. Perhaps, I am overlooking something.

13. • CommentRowNumber13.
    • CommentAuthorspitters
    • CommentTimeOct 11th 2016
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    Author: spitters
    Format: MarkdownItexA similar model is described by [van der Hoeven, Moerdijk - sheaf models for choice sequences](http://www.sciencedirect.com/science/article/pii/0168007284900356). Countable choice holds in this topos (2.2.6) and the real number object is computed on p81. The topological monoid of endomorphism of Baire space seems to be a [[topological site]] as in continuous truth II, but I haven't checked it formally yet. Similarly, for Johnstone's monoid of endomorphisms on $N_\infty$. I expect this to be folklore.

    A similar model is described by van der Hoeven, Moerdijk - sheaf models for choice sequences. Countable choice holds in this topos (2.2.6) and the real number object is computed on p81. The topological monoid of endomorphism of Baire space seems to be a topological site as in continuous truth II, but I haven’t checked it formally yet. Similarly, for Johnstone’s monoid of endomorphisms on $N_\infty$. I expect this to be folklore.

14. • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeOct 14th 2016
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    Author: Mike Shulman
    Format: MarkdownItexAfter some conversation with Martin and trying to build a counterexample, I'm no longer sure that the topological topos fails countable choice. In fact, here is an argument claiming that it *satisfies* countable choice. Please poke a hole in it. We have to show that $\mathbb{N}$ is internally projective, i.e. that dependent product along $A \times \mathbb{N} \to A$ preserves epis. Since $\mathbb{N}$ is a countable coproduct of copies of 1, $A \times \mathbb{N}$ is a countable coproduct of copies of $A$, which means that this dependent product can be identified with the countably infinite product functor on the slice category $E/A$. Now, Johnstone's explicit description of the Grothendieck topology defining this topos implies that for a map $f:B \to A$ to be epi means that (1) it is surjective on points and (2) for any convergent sequence $(a_n) \rightsquigarrow a_\infty$ in $A$, and any infinite subsequence $n_k$, there is a further subsequence $m_l$ and a convergent sequence $(b_l) \rightsquigarrow b_\infty$ in $B$ that maps via $f$ to $(a_{m_l}) \rightsquigarrow a_\infty$. Thus, suppose we have epis $f_i : B_i \to A$ for natural numbers $i$; we want to show that $g : P \to A$ is epi, where $P$ is the pullback of all the $B_i$'s over $A$. It's certainly surjective on points. Let $(a_n) \rightsquigarrow a_\infty$ be a convergent sequence in $A$, and $n_k$ an infinite subsequence. Since $f_0$ is epi, there is a subsequence $m^0_k$ of $n_k$ and a convergent sequence $(b^0_k) \rightsquigarrow b^0_\infty$ mapping via $f_0$ to $(a_{m^0_k}) \rightsquigarrow a_\infty$. Now since $f_1$ is epi, there is a subsequence $m^1_k$ of $m^0_k$ and a convergent sequence $(b^1_k) \rightsquigarrow b^1_\infty$ mapping via $f_1$ to $(a_{m^1_k}) \rightsquigarrow a_\infty$. And so on. (Of course, in choosing particular such subsequences $m^i_k$ we are using countable/dependent choice in Set.) Now define $p_k = m^k_k$, and a sequence $c_k$ in the pullback $P$ by $(c_k)_i = b^i_{p_k}$. Since convergence in $P$ is defined in each factor, $(c_k) \rightsquigarrow c_\infty$ where $(c_\infty)_i = b^i_\infty$, and it maps via $g : P \to A$ to $(a_{p_k}) \rightsquigarrow a_\infty$, with $p_k$ a subsequence of the given $n_k$. Thus, $g$ is also epi.

    After some conversation with Martin and trying to build a counterexample, I’m no longer sure that the topological topos fails countable choice. In fact, here is an argument claiming that it satisfies countable choice. Please poke a hole in it.

    We have to show that $\mathbb{N}$ is internally projective, i.e. that dependent product along $A \times \mathbb{N} \to A$ preserves epis. Since $\mathbb{N}$ is a countable coproduct of copies of 1, $A \times \mathbb{N}$ is a countable coproduct of copies of $A$, which means that this dependent product can be identified with the countably infinite product functor on the slice category $E/A$.

    Now, Johnstone’s explicit description of the Grothendieck topology defining this topos implies that for a map $f:B \to A$ to be epi means that (1) it is surjective on points and (2) for any convergent sequence $(a_n) \rightsquigarrow a_\infty$ in $A$, and any infinite subsequence $n_k$, there is a further subsequence $m_l$ and a convergent sequence $(b_l) \rightsquigarrow b_\infty$ in $B$ that maps via $f$ to $(a_{m_l}) \rightsquigarrow a_\infty$.

    Thus, suppose we have epis $f_i : B_i \to A$ for natural numbers $i$; we want to show that $g : P \to A$ is epi, where $P$ is the pullback of all the $B_i$’s over $A$. It’s certainly surjective on points. Let $(a_n) \rightsquigarrow a_\infty$ be a convergent sequence in $A$, and $n_k$ an infinite subsequence. Since $f_0$ is epi, there is a subsequence $m^0_k$ of $n_k$ and a convergent sequence $(b^0_k) \rightsquigarrow b^0_\infty$ mapping via $f_0$ to $(a_{m^0_k}) \rightsquigarrow a_\infty$. Now since $f_1$ is epi, there is a subsequence $m^1_k$ of $m^0_k$ and a convergent sequence $(b^1_k) \rightsquigarrow b^1_\infty$ mapping via $f_1$ to $(a_{m^1_k}) \rightsquigarrow a_\infty$. And so on. (Of course, in choosing particular such subsequences $m^i_k$ we are using countable/dependent choice in Set.) Now define $p_k = m^k_k$, and a sequence $c_k$ in the pullback $P$ by $(c_k)_i = b^i_{p_k}$. Since convergence in $P$ is defined in each factor, $(c_k) \rightsquigarrow c_\infty$ where $(c_\infty)_i = b^i_\infty$, and it maps via $g : P \to A$ to $(a_{p_k}) \rightsquigarrow a_\infty$, with $p_k$ a subsequence of the given $n_k$. Thus, $g$ is also epi.

15. • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeOct 14th 2016
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    Author: Mike Shulman
    Format: MarkdownItexAlso, a (un?)related question. In Johnstone's paper he cites an abstract by Isbell as showing that the pair of maps $x\mapsto \frac{1}{2} x$ and $x\mapsto \frac{1}{2}(x+1)$ are not universally effective-epimorphic in the monoid of continuous endomorphisms of the topological unit interval, and claims that this is related to the fact that the interval is locally connected. Can anyone reproduce (or know a citation for) such a proof? I'm guessing it has something to do with pulling back along some continuous map that behaves very badly near $\frac{1}{2}$, but I haven't been able to get any further, or get any intuition for what it has to do with local connectedness.

    Also, a (un?)related question. In Johnstone’s paper he cites an abstract by Isbell as showing that the pair of maps $x\mapsto \frac{1}{2} x$ and $x\mapsto \frac{1}{2}(x+1)$ are not universally effective-epimorphic in the monoid of continuous endomorphisms of the topological unit interval, and claims that this is related to the fact that the interval is locally connected. Can anyone reproduce (or know a citation for) such a proof? I’m guessing it has something to do with pulling back along some continuous map that behaves very badly near $\frac{1}{2}$, but I haven’t been able to get any further, or get any intuition for what it has to do with local connectedness.

16. • CommentRowNumber16.
    • CommentAuthorspitters
    • CommentTimeOct 14th 2016
    • PermaLink
    Author: spitters
    Format: MarkdownItex[Moerdijk-Reyes](http://www.sciencedirect.com/science/article/pii/0022404984900495) have some remarks about the failure of countable choice in the Euclidean topos. For them a topological topos is a topos of sheaves on a topological site. This should connect with Johnstone's topological topos, but I haven't seen the details spelled out.

    Moerdijk-Reyes have some remarks about the failure of countable choice in the Euclidean topos. For them a topological topos is a topos of sheaves on a topological site.

    This should connect with Johnstone’s topological topos, but I haven’t seen the details spelled out.

17. • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeOct 14th 2016
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    Author: Mike Shulman
    Format: MarkdownItexInteresting; they specifically say it is because of the "connectedness" of the euclidean site.

    Interesting; they specifically say it is because of the “connectedness” of the euclidean site.

18. • CommentRowNumber18.
    • CommentAuthorMike Shulman
    • CommentTimeOct 15th 2016
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    Author: Mike Shulman
    Format: MarkdownItexI added a remark to [[topological topos]] that while LPO fails, LLPO holds.

    I added a remark to topological topos that while LPO fails, LLPO holds.