Thanks, I see.

So I have fixed it here

$y \,\leftarrow\, x \;\;\;\text{iff}\;\;\; y \,\leq\, x\;$and added a mnemonic:

“view the arrow head of “$\leftarrow$” as a “$\lt$“-symbol”

]]>Alternatively: $y\rightarrow x$ iff $x\leq y$ (with a right arrow this time).

]]>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 $x\leftarrow y$ iff $y\leq x$ (notice it’s a left arrow and not a right arrow), it should instead be $x\leftarrow y$ iff $x\leq y$. In other words, in your notation you want the arrows to go down and not up in the specialization order poset, right?

]]>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

$cl\big(\{0\}\big) \,=\, \{0,1\} ,\;\; cl\big(\{1\}\big) \,=\, \{1\}$and so $1 \in cl\big(\{0\}\big)$ hence $1 \leq 0$ hence $0 \to 1$.

But below that example box I did edit the subsequent sentence as suggested (here).

]]>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!

]]>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

]]>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!

]]>Reading a little further into the article, the next paragraph specifies how to interpret a preordered set as a category. Specifically it seems that you intend the morphisms to "go down in the specialization preorder" so that further down in the text the closed sets C are the downward closed sets in the proset and there are no morphisms going out of C in the corresponding category. But then something should be fixed when specifying the morphisms: instead of "x <-- y iff y <= x", it should be "x <-- y iff x <= y" (which also makes it easier for the reader to follow as it matches "x <= y" in the definition of specialization order further up).

Also a little further there is a first table of examples. Right after it says: "in {0→1} the point 0 is open (as there do emanate arrows form it) ..." But in general having an arrow emanating from a point does not mean it's an open point. It just means it's not a closed point. So maybe remove that parenthetical clause. ]]>

Right, thanks for saying.

I have fixed it (here)

]]>It is helpful to the reader trying to parse these pictures,

to point out that if the horizontal arrows are taken

as indicated by pictures (so $\bullet_x$ goes to $\bullet_x$, etc),

then the diagrams reads as the actual definition.

And if you take other horizontal morphisms,

you get a weaker condition. But I cant formulate this readably...

Also, it might help to explicitly point out that the picture of a finite topological space

can be kind of read as a picture of a decomposition of X into a union of open subsets...

Are the pictures readable ? There is an annoying issue with box hights in T4 ...

Also, probably it should be two or three diagrams on a line. ]]>

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. :-)

]]>>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.

Perhaps one should make a table with lifting property reformulations with pictures for T0-T4 or T0-T5, the original Tietze axioms,

analogous to

[norrmal space]

> !include main separation axioms -- table

Will that be readable enough to a general reader ? The boxed diagrams seem rather readable for someone who does not know the notation.

Though for T4 and T5 I'm so not sure (and I have not yet typeset that)

What do you think ? ]]>

Sure, feel free to copy the Idea section over!

(Myself, I am on my phone now, can’t do any nontrivial edits right now.)

]]>re #59: yes, this is smooth, though the reference breaks the flow. Except for that, it may be in its final form...

re:57: the idea section in the Sandbook looks good to me. Could it also be moved to [lift] page?

Perhaps only the first paragraph ? Not sure what is appropriate.

I'll try polishing out \lrr things in next couple of days, thanks for pointing this out. ]]>

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”).

(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.)

]]>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 $(-)^{l l r}$ 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…

]]>Okay, will do. But later tonight, off for dinner now…

]]>to the lifting property page ? I am afraid I cannot do it myself --- my edit is refused by spam filter. ]]>

Yes, sounds good!

]]>>Intuitively, the lifting property is a negation in category theory:

Taking the class of morphisms having the lifting property with respect to each morphism in a class $P$

is a simple, and unexpectedly common, way to define a class of morphisms excluding non-isomorphisms from $P$,

in a way which is useful in a diagram chasing computation.

A number of elementary notions may be expressed by iteratively

using the lifting property starting from a list of simple or archetypal (counter)examples.

>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.

Perhaps most or almost all of this entry should go to the entry lifting property ? With wording as above,

one could remove the terminology "Quillen negation" from the idea section, and introduce it after the definition

when talking about orthogonals.

>The entry on “Quillen negation” could the focus more on bringing out the logical aspect.

Indeed, the entry on Quillen negation could focus on the logical intuition and some very elementary examples. ]]>

Thanks for pointing to your draft in the Sandbox.

I suggest that an Idea-section should start out with a really brief $\leq 2$-sentence-idea. Here it could be these two sentence (before anything else is said):

Given a property $P$ 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 $P$, 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 negationof $P$.

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.