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Looking at this entry only now (promted by activity in another thread here), I must say that I find the entry, in its current form, hard to read. Which seems odd, as the topic should be completely elementary.
One reason is that the lifting conditions advertized in title and abstract are never mentioned again in the entry!
At the same time, besides plenty of non-standard notation, crucial notation remains undefined, notably the intended meaning of the ““-symbol is never defined!
So I went now and checked the article arXiv:1408.6710. That’s clearer. And there I see that the above two problems are related: It’s the symbol “” that is meant to indicate that a lifting exists.
Also, I see that the lifting diagrams that the reader expects to see finally are displayed – as a handwritten addendum to the article appended on the last pages.
Apparently the OP didn’t know how to typeset the lifting diagrams – so let me show how it’s done, in the next comment. Maybe we can jointly beautify this entry to the enjoyable mathematical entertainment that it should it be.
To get a lifting diagram in Instiki of this form
one can type this code:
$$
X \; \text{is}\; T_1
\;\;\;\;\;\;
\Leftrightarrow
\;\;\;\;\;\;
\array{
CoDisc(\{0,1\})
&\xrightarrow{\;\forall\;}&
X
\\
\big\downarrow
& {}^{\mathllap{\exists}} \nearrow &
\big\downarrow
\\
\ast
&\xrightarrow{\;\;\;\;}&
\ast
}
$$
This kind of typesetting would already go a long way in the entry.
But we can beautify this further using TikZ, and I suggest we do that. But TikZ doesn’t work here in the nForum, so I will prepare an example for that now in the Sandbox entry. Just a minute…
Okay, I have now prepared TikZ-typesetting of the first few handwritten examples in arXiv:1408.6710, p. 7.
The typeset diagrams are, for the time being,
I suggest that we use this kind of typesetting in the entry here – at least in the Idea-section, where the general idea is (or should be) explained in expository form.
The source code of the Sandbox is visible here.
One could proceed by copy-and-pasting (part of) this. (It should be clear from the code how to produce any number of further variants.)
Now I have touched the section “Separation axioms and lifting properties” (here). I have:
Added definition of the notation “” for “lifting property”
added a link to Joyal-Tierney calculus
added a fair bit of whitespace around each of the following occurrences of , to make it clear to the eye that it does not bind the objects next to it, but the functions (this had previously made the formulas almost unreadable)
adjusted indention in a couple of places in the following list by adding missing whitespace
also cleaned up the code a little, to make it easier to see beginning, structure, and end of bullet items and of the displayed formulas they contain
(notice that Instiki expects the user to be aware of the role of whitespace in typesetting. It’s a whitespace driven language, really, and that is one of the rare positive aspects about it.)
finally I have added missing arXiv-links (in the first case) and missing publication data (in the second) to the two reference items:
Misha Gavrilovich, Point set topology as diagram chasing computations, The De Morgan Gazette. 2014. Vol. 5. No. 4. P. 23-32. (arXiv:1408.6710, pdf)
Misha Gavrilovich, The unreasonable power of the lifting property in elementary mathematics, 2017 (arXiv:1707.06615, pdf)
<blockquote>
<p>Maybe we can jointly beautify this entry to the enjoyable mathematical entertainment that it should it be.</p>
</blockquote>
<p>Indeed let us try!</p>
<p>Finite topological spaces are (i.e. can be viewed as) preorders, and for readability it is important to picture them in this way. Also, for redability it is important to combine the picture of the preorder with Venn diagram of open subsets (or closed subsets, or both, whatever turns out more readable).</p>
<p>For example, we need a way to represent readably on the page
the following preorder {a<b>c<d>e} as a picture
of the shape /\/\ with an indication which subsets are open.</p>
<p>I have also added a later reference</p>
<ul>
<li>Misha Gavrilovich, Konstantin Pimenov. <em>A naive diagram chasing approach to formalisation of tame topology.</em>, 2018 <a href="http://mishap.sdf.org/mintsGE.pdf">pdf</a>)</li>
</ul>
<p>Anonymous</p>
<p><a href="https://ncatlab.org/nlab/revision/diff/separation+axioms+in+terms+of+lifting+properties/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/separation+axioms+in+terms+of+lifting+properties/23">v23</a>, <a href="https://ncatlab.org/nlab/show/separation+axioms+in+terms+of+lifting+properties">current</a></p>
Okay!
For Venn diagrams we can use TikZ in its full versiontikzpicture. If you know how to use TikZ, you can experiment in the Sandbox by enclosing ordinary Tikzcode inside
\begin{tikzpicture}
...
\end{tikzpicture}
I can look into it later tonight. Busy now…
Thank you! But before coding in tikz, we would need to think a little about a readable notation for maps of preorders… The current notation is based on the view that : a preorder is a category, and a monotone map is a functor “adding” morphisms including, somewhat weirdly, identity morphisms. This does lead to a concise syntax, but the problem is that you don’t see at a glance what “new morphisms”/relations are added, and in a picture you should be able to… Maybe in a picture, one can use a new colour for the arrows and objects added …
Another issue is that it would help to explpcitly introduce the idea of lifting wrt simplest “counterexamples” (i.e. a notion being defined by the lifting property wrt a simplest counterexample) and calling it the lifting property Quillen negation conveys this intuition and makes for expressive language. I shall add this to the page carefully later.
Anonymous
Okay, here is a way to typeset maps of finite posets in a more readable way, using inline Instiki code:
Output of this form:
is obtained by typing the following code, using the array-environment
$$
\left\{
\;\;
\array{
& & U && && V
\\
& \swarrow && \searrow && \swarrow && \searrow
\\
a && && x && && b
}
\;\;
\right\}
\;\;\;\;\;\;\;\;
\xrightarrow{\phantom{------}}
\;\;\;\;\;\;\;\;
\left\{
\;\;
\array{
&& U &=& x &=& V
\\
& \swarrow & && && & \searrow
\\
a & & && && && b
}
\;\;
\right\}
$$
I have now implemented this particular example rendering in the entry here.
Oh, and to get color, use \color. For instance this output
comes from adding two \color{red}-commands to the previous example:
$$
\left\{
\;\;
\array{
&& U &\color{red}=& x &\color{red}=& V
\\
& \swarrow & && && & \searrow
\\
a & & && && && b
}
\;\;
\right\}
$$
Thanks, this looks readable, especially with colours. I suppose downward closed sets (=closed subsets) are visually apparent enough so there is no need to mark them ? And the names of points suggest whether these points are open or closed so that the convention (whether downward closed sets are either open or closed) is clear.
But this code cannot be used within the lifting diagrams ?
Anonymous
But this code cannot be used within the lifting diagrams ?
Apparently, some time ago, the TikZ developers made some unfortunate design choice, and ever since it is impossible to include into a TikzCD diagram an environment that itself uses ampersand symbols.
But here I think we are fine without: If we just declare a preorder in one line by showing its generating graph, and then use the pre-order’s symbol in a lifting diagram in the next line, all will be nicely readable, and understandble, I expect.
I have now worked a little on beautifying the section Background and notation.
Throughout, I have adjusted the text a little for flow and clarity (I hope).
But mostly I have touched the typesetting of the examples. For instance, it now shows this part of the dictionary FinSpace/SpecOrder:
| finite topological space | open subsets | specialization order |
|---|---|---|
| discrete space | ||
| Sierpinski space | ||
| codiscrete space | ||
| point space |
and the canonical maps/functors between these examples I have now made to render like this:
<p>Thank you, this reads much better now. I added a couple of words about the lifting property with respect to counterexamples. Even though it breaks the flow of text, I find it clarifying.</p>
<p>Should we try to use the preorder notation for the two-point spaces introductory bit ? We can just denote them as a preorder with letter names indicating which points are open so it would be clear to the reader even before reading the preorder section, and not much a distruction.
I.e. something like this: denote the Sierpinski space by {o->c} and say o is an open point and c is closed, and the discrete space by {u,v} where both points are open, and codiscrete space by {a <-> b} where neither is open or closed (the latter is not very clear…) I’m not sure how clear would that be.</p>
<p>Anonymous</p>
<p><a href="https://ncatlab.org/nlab/revision/diff/separation+axioms+in+terms+of+lifting+properties/28">diff</a>, <a href="https://ncatlab.org/nlab/revision/separation+axioms+in+terms+of+lifting+properties/28">v28</a>, <a href="https://ncatlab.org/nlab/show/separation+axioms+in+terms+of+lifting+properties">current</a></p>
Before I reply, let’s sort out the formatting issue with your comments:
I suspect that the reason that your comments #16 and #8 confuse the parser is that they contain the symbols < and/or > un-escaped.
I believe that to display these symbols here with our Instiki parser you need to instead type \lt or \gt, respectively, inside a maths environment.
You should be able to still edit comments #16 and #8: See if you find a button “edit” on the top right of these entries.
(But if you do edit, you might just as well replace -> by \rightarrow and <- by \leftarrow.)
Sorry for the trouble. It shouldn’t happen that our software throws unhandled errors in this way. But that’s what we have.
That said, I like the comments that the required lifting property is always against the “simplest counterexample”. That’s insightful! Maybe we could call it the “archetypical counterexample”.
This meshes well with the general perspective that lifting properties exhibit the left class of morphisms as being “orthogonal” to the right class, as reflected in the terminology “orthogonal factorization system”.
In fact, most examples of interest for the separation axioms seem to have unique lifting if there is a lifting at all, which would further strengthen this terminological point.
it is helpul to call the weak orthogonal Quillen negation
Sure, sounds good!
I see, that’s a good hack. I have now (here):
adjusted the spacing a little,
moved the “” to the left vertical map,
removed the label “injective” and instead gave the arrow a hook.
But please check if you agree.
re #22:
I don’t know of a variant of \boxed, sorry. But if something like \circled existed, it would probably take up too much space. We’d need something like \boxedwithroundedcorners instead… But I think the rectangular boxes look fine enough.
Is section 3 useful in your opinion ?
Shall we leave section 4 more or less as it is ? Should we rewrite the \rightthreetimes expressions as diagrams ?
I do suspect that section 3 is useful, but it is terse and hard to read and so I haven’t really tried to absorb its content.
I gather what you are doing there is to establish a dictionary between separation conditions and lifting properties, but I wonder how many reader will even see that this is what the section is offering: the section never says that this is what it does! :-)
I think to make this section properly reflect your insights to readers it should be brought into a form closer to what we now have in sections 1 and 2: There should be clear statements of “this property is equivalent to this lifting” and then the lifting condition should be displayed in a form that is more easily discernible by the eye.
Similar comments apply to section 4, though I think this is closer to readable now, with the extra whitespace around the .
I was initially planning to progress further through the entry and re-rendering it item-by-item. But I am afraid that now I am too absorbed with other tasks, for the time being.
I have added a reformulation of extremally disconnected, for the \boxed notation is enough to picture this.
I am happy to work the examples below one by one but it would help me if you would do one in each section so that I could follow an examlpe (your way of writing is very clear and I am unable to write as well).
Anonymous
I have added a reformulation of extremally disconnected, for the \boxed notation is enough to picture this.
I am happy to work the examples below one by one but it would help me if you would do one in each section so that I could follow an examlpe (your way of writing is very clear and I am unable to write as well).
Anonymous
I rewrote T3-T5 using more graphic notation of preorders, and also add some explanatory remarks relating the pictures/preorders and the usual wording. Does it seem helpful to the reader ? Unfortunately, \boxed cant really be hacked to picture open subsets with complicated intersections, so in T5 I do not use it, for example. Also, the picture for extremallydisconnected was incorrect and I fixed it.
Should I replace by the liftning diagram ?
I am not sure I understand your question.
Anonymous
I am not sure I understand your question.
I was wondering what exactly you were asking me (in #28) to do.
Would you like me to typeset the example “topologically disjoint”? I was just thinking that, meanwhile, you seem to have picked up all the techniques for how to best do this, so I wasn’t sure what else I could contribute. Your last diagrams look really good.
Regarding the symbol for lifting property: how about “” (\perp) ?
re #31: is already used on other pages to denote orthogonality (the unique lifting property); unfortunately iTeX doesn’t allow \boxslash (I tried in the Sandbox), i.e. a (nicely visual) square with a slash in it (which is used by e.g. Riehl’s Algebraic model structures)
— a legally distinct Anonymous
Yes, I think would be confusing for a non-unique lifting property. Richard, any chance of getting boxslash?
I experimented with this this evening (actually by modifying itex2MML itself for the first time, not through a post-processing hack), but there is an issue with regard to what unicode symbol to render to. Ideally, it would belong to the following list, because these symbols have a little space around them and have the correct semantics.
https://www.compart.com/en/unicode/block/U+2200
There is the following symbol, but it does not seem to be available on most systems.
https://www.compart.com/en/unicode/U+29C4
The closest I found otherwise is the following
https://www.compart.com/en/unicode/U+2341
but since it does not belong to the ’mathematical operator’ block that I linked to above, there is no space around it, and it does not look too good. I can probably add some space in itex2MML if desired, but maybe somebody has a better idea?
I see now that a circle slash exists:
https://www.compart.com/en/unicode/U+2298
Could that be used instead of boxslash if I implement it?
Instiki does have (\oslash) already.
It can also typeset this: (\underline{\overline{\vert\!/\!\vert}}) :-)
Well spotted regarding \oslash.
Using ⍁, namely U+2341, is a little better than the hack in your second sentence I think.
Let me know if I should implement \boxslash as U+2341 with a little space around it.
Sure, if you could give us ⍁, that would be useful in any case!
re #32:
I see. Okay, I have now worked on the beginning of section 3 (here).
Apart from expanding the text a little for clarity (I hope) and from adding TikZ-diagrams, I have taken the liberty of slightly adjusting the notation for the elements of the finite spaces that we are lifting against. This is just a suggestion and is not so important, please feel free to undo if desired.
On a different note, I see that now the table in section 2 shows two different conventions for the arrow direction. That seems unnecessarily burdensome, since the arrow direction is an arbitrary convention in any case, no? If you feel it was me who chose a conflicting convention (but I thought I did follow yours) then please feel free to change it and harmonize.
Thank you for you help! Section 3 is now reworked, with diagrams.
The notation is left inconsistent, yet.
Here are questions/remarks about notation.
I would prefer to denote the Sierpinski space by
{0->1} where 0 is open and 1 is closed.
This is the way it is done in references so another convention would be confusing.
vs is a matter of readability and not a source of confusion. For me seems more natural for an isomorphism whereas suggest a pair which does not necessarily commutate.
It is clear that we consider and to be the same ? I find a mix of \boxed and/or \overset{}{} much more readable (even though it is a hack…). Is there a less hacking way to do \overset{}{} notation, e.g. ?
vs other symbols: it is not a source of confusion (whenever the symbol is asymmetric) so whatever convention you find more readable. Though I feel and (rarely used) are more distinctive than a symbol in a box.
Anonymous
Thanks, looks.
About Sierpinski space: Sure. I see that you harmonized my notation now. Looks good.
About the double arrows: I chose because it is understood that our categories here are thin, which should make this notation unambiguous, while is not established notation in category theory. But if you want I can change all back to , it’s an easy matter of search-and-replace.
Regarding typesetting of diagonal arrows and Venn diagrams: We could use \begin{tikzpicture}...\end{tikzpicture} instead of \begin{tikzcd}...\end{tikzcd}. Inside tikzpicture there is no limit to the possibilities of drawing! (their manual is here: pdf). But it requires some dedication and time.
Regarding the symbol for lifting: I’d suggest we should prefer symbols whose meaning is easy to guess for readers who happens upon them without having read all preceding explanations line-by-line. The problem I see (and had myself) with is that it looks mysterious and is unsuggestive of its intended meaning (that extra half-diagonal breaks the intended intuition). If Richard could implement ⍁ then that would be the perfect symbol to use here, I’d think.
(That Guest in #42 was me. Not sure what happened here.)
I have now made all the cross-links to equation numbers work (there were some broken ones, appearing as question marks, and some missing ones).
For example, in “Separated by neighbourhoods” (here) there is now this code:
\[
\label{PairOfSeparatedNeighbourhoods}
U \subseteq V
\;\text{and}\;
V \subseteq G
\;\;\;\text{such that}\;\;\;
U \cap V = \varnothing
\,.
\]
and, afterwards, this condition may be referred to by saying:
see (eq:PairOfSeparatedNeighbourhoods)
Regarding symbols: I see now that we can use ⧄ inside math mode inside entries. This comes out as “”. I have just tried that out at Joyal-Tierney calculus, see here.
Is there some larger story of what’s going on in this entry? Are there lessons beyond ?
Perhaps there’s a whiff of cohesion around?
Is there some larger story of what’s going on in this entry?
I don’t know. Probably there is the implicit idea that some proofs in general topology might be more transparent via lifting properties, but I haven’t seen them yet. It would be good to add commentary on that to the entry.
But it’s quite enjoyable all in itself. I am thinking that if I am ever to teach Introduction to Topology again, I will make this separation via lifting a running thread of examples/exercises. They are perfect for illustrating basic topology and basic category theory all at once.
Okay, I have replaced all “” by “”.
I have also replaced now all “” by “”.
Regarding section 4: I’ll try to look into it later (am already spending more time with this entry than is good for me… :-)
Regarding applications of the lifting-reformulation: If you have examples of such applications, it would be good to add them to this entry here, maybe in a new section “Applications”.
But in #49 you mentioned:
Urysohn lemma by iterating the lifting property for normality and passing to the limit in a careful way.
This sounds like an interesting application. If you could spell out in the entry some applications/reformulations of this kind, this might help to illustrate the usefulness of shifting perspective to lifting properties.
Thanks.
There is/was brief remarks in this direction at the end of section 4. I assume these remarks are superceded by your new addtion and should be removed?
Where you have the big diagram with crossing arrows I have:
replaced the first ampersand & by &[-60pt] to reduce the huge size of the left rectangle
gave the big ddll-arrow the argument crossing over to make the crossings easier to the eye
I also started editing section 4.
But none of this is visible for the moment, since submitting the edits gives me “timed out”-errors…
Okay, the rendering engine seems to have recovered and the entry saves now.
I have now also given section 4 Subsection-headers, in line with what we did to section 3, and then I touched the text and formatting of hte first item Kolmogorov spaces.
I guess it’s okay without more diagrams, since at this point of the entry the reader has gotten the idea. But where it helps to see what it is you are claiming, it doesn’t hurt to make it as reader-friendly as possible.
In this vein, one might want to add more explanation/proof to some of the claimed equivalences. Even if it’s straightforward…
I have cross-linked (here) with Gleason’s theorem and tried to adjust the wording a little, for clarity.
I think there is still room to make the wording run more transparently.
Let me say that your separation-by-lifting theory is really neat, and that I am glad that you are contributing it to the nLab. It well deserves to be further interconnected with other our entries on general topology, as you have started doing now. If you have energy to proceed, I’d appreciate it, this is a great contribution to the nLab.
Much as I would enjoy to continue helping out with typesetting this stuff, I do need to cut down on the time I am investing in this, as other issues need my attention. But i have now posted a little advertisement of your undertaking to here. Maybe somebody with inclination and energy sees this and joins in.
Thanks for pointing to your draft in the Sandbox.
I suggest that an Idea-section should start out with a really brief -sentence-idea. Here it could be these two sentence (before anything else is said):
Given a property characterizing a class of morphisms in some category, the classes of morphism with the respective left or right lifting property tend to be characterized by a property which is “opposite” to , in some sense. Since lifting properties were brought to the forefront of mathematician’s attention with Quillen’s formulation of model categories, Gavrilovish (cite…) has proposed to call these “opposite properties” the Quillen negation of .
This to orient the reader before anything is really being explained.
Then one should think about how much this entry needs to overlap with the entry lifting property. Many of the examples you offer would deserve to be mentioned as examples at “lifting property”! The entry on “Quillen negation” could the focus more on bringing out the logical aspect.
Yes, sounds good!
Okay, will do. But later tonight, off for dinner now…
All right, I have now done some tweaking of your material in the Sandbox:
reorganized the sub-section layout of the examples, for more systematics (I hope)
added more hyperlinks (such as to Serre fibrations in the classical model structure on topological spaces etc.)
It’s still all rather telegraphic most of the time, and could use lots more polishing for reader-friendliness. But it’s great to have a long list of examples!
Towards the end you keep omitting the lifting-symbol and just write etc. It’s clear what is meant, but if you have the energy, I’d urge to put the lifting symbol back in (or else remove it everywhere in the superscripts!).
In any case, I have now copied over the material – from here on – to the entry lift into a new section there: Examples of lifting properties.
Will announce this edit now also in that entry’s thread…
(By the way, our Spam filter gets triggered by, among other things, large edits. When you get stopped by the spam filter next time, and I am not around to help out, you can try to copy-and-paste material in smaller chunks, bit bby bit. That usually works.)
Seeing that/how the lead-in paragraph (here) received further edits since I last touched it, I have tweaked it once more. I think this still reflects what you had in mind here, but please check if you agree that this flows (more) smoothly:
In point-set topology, most of the separation axioms that are traditionally considered on topological spaces turn out (Gavrilovich 2014) to have an equivalent reformulation in terms of lifting properties, namely of the given space against, typically, a map of finite topological spaces which reflects the “opposite property” or the “archetypical counterexample” to the given separation condition, in a sense (“Quillen negation”).
Sure, feel free to copy the Idea section over!
(Myself, I am on my phone now, can’t do any nontrivial edits right now.)
A table would be great. I can’t quite say beforehand if it will be readable, but I trust that once you start something, we can make it readable.
(I am impressed to see that you managed to typeset overlapping boxes with Instiki. I don’t understand yet how that is even possible. :-)
Thanks. I might not have the leisure to do further adjustments, soon. If you have the energy and feel like you know what you are doing, please feel invited to edit!
All edits get recorded in
https://ncatlab.org/nlab/latest_revisions
as well as in the history for this page
https://ncatlab.org/nlab/history/separation+axioms+in+terms+of+lifting+properties
When you edit you could add a comment explaining what you did in your edit and it would lead to the comment being posted here on the nForum.
Charles Battenburg
Just hit “edit” at the bottom of the page and have a go at it!
All you’ll be asked to is sign with any name or pseudonym (now mandatory) and to add a brief comment on what edit you made (not technically mandatory but clearly desirable).
No need to worry about breaking anything. As Charles B. said, we can see your edits in the page history; and if anything goes really wrong we can always revert to previous versions.
Your input is appreciated!
I hope I did not kill the nice energy by asking you to make an edit.
I will try to email Misha Gavrilovich, who is the originator of the bulk of material in this entry, he will probably be happy to react to further comments.
But looking again at the entry now in view of the above comments (busy with other things, though, please bear with me):
The direction of the arrows seems correct to me, even if the convention might be confusing.
For instance, in the example of the Sierpienski space (here) we have
and so hence hence .
But below that example box I did edit the subsequent sentence as suggested (here).
Thanks for making the updates. What you are saying here about the Sierpinski space is correct, but it does not match the definition of the “arrow” notation a few paragraphs above. Where it says iff (notice it’s a left arrow and not a right arrow), it should instead be iff . In other words, in your notation you want the arrows to go down and not up in the specialization order poset, right?
Alternatively: iff (with a right arrow this time).
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