added a proof to *Urysohn’s lemma*

I am splitting off *Zariski topology* from *Zariski site*, in order to have a page for just the concept in topological spaces.

So far I have spelled out the details of the old definition of the Zariski topology on $\mathbb{A}^n_k$ (here).

]]>I noticed that presently *topological basis* redirects to *basis in functional analysis* instead of to the entry *topological base*. This seems dangerous. I’d like to change that redirect.

partition of unity, locally finite cover

Will put up some stuff about Dold’s trick of taking a not-necessarily point finite partition of unity and making a partition of unity. There is a case when I know it works and a case I’m really not sure about - I need to find where the argument falls down because I get too strong a result. I’ll discuss this in the thread soon, and port it over when it is stable.

]]>Wrote out a proof for *paracompact Hausdorff spaces are normal*.

(By the way, I also looked at TopoSpaces here to check what they offer, and am a bit dubious about their step 5. But maybe I am misreading it. In any case, I feel there is a simpler way to state the proof.)

]]>at *separation axiom* I have expanded the Idea section here, trying to make it more introductory and expository.

I have spelled out the proof at *paracompact Hausdorff spaces equivalently admit subordinate partitions of unity*.

This uses Urysohn’s lemma and the skrinking lemma, whose proofs are not yet spelled out on the $n$Lab.

]]>I have edited *support* to say that in topology the support of a function is usually to be the topological closure of the naive support.

I have added to *Galois connection* some more remarks to the Idea section, and expanded the Examples-section with the material that Todd wrote here.

I recently put on the arXiv this preprint:

https://arxiv.org/abs/1704.00303

that stemmed from a question I posed on MathOverflow a few months ago (the title is the same, googling gives both the arxiv preprint and the MO-thread), and that received some attention and positive comments (I hope).

We authors are in the phase of polishing some details, and improving the clarity of the discussion. Once this process is finished, I’d like to have it published: what is, in your opinion, a good journal where to send the preprint?

Thanks!

]]>There is an obvious similarity between the four adjoints describing cohesion for the Sierpinski $(\infty, 1)$-topos in Example 6.1.2 of dcct and four of the seven adjoints (third to sixth) for arrow categories of pointed categories having pullbacks and pushouts MO answer+comment. So

$\Pi([P \to X]) = X$ ; $Disc(X) = [X \to X]$; $\Gamma([P \to X]) = P$ ; $coDisc(Q) = [Q \to \ast]$.

$[f: A \to B] \mapsto coker(f)$; $A \mapsto [0 \to A]$; $[A \to B] \mapsto B$; $A \mapsto [A \to A]$; $[A \to B] \mapsto A$; $A \mapsto [A \to 0]$; $[f: A \to B] \mapsto ker(f)$.

It seems there’s a linearization happening, and we might want to consider, as we did a while ago, whether there is a Freyd-like result for AT-$(\infty, 1)$-categories between stable ones and toposes.

Forming the (co)monads from the adjunction strings, the Sierpinski case gives us the expected three for cohesion, ʃ $\dashv\; \flat \;\dashv\; \sharp$. The other case gives us six. Now, are there traces of these extra three in the nonlinear case, trying to show themselves?

Consulting the cohesion - table, ʃ does have a left adjoint, that is, if one lifts it up to *infinitesimal* shape $\Im$. It’s the reduction modality, $\Re$. And by the same move, its lift to $\rightsquigarrow$ has a further left adjoint $\rightrightarrows$.

So is there a connection between $\Re$ and the modality $[A \to B] \to [0 \to B]$? Maybe we’re not so far from the tangent (infinity,1)-category construction as a halfway house, with an element above a space $X$, a parameterized spectrum, as an infinitesimal there.

And what of the other two modalities, $[A \to B] \mapsto [0 \to coker f]$ and $[A \to B] \mapsto [ker f \to 0]$?

]]>Late last night I was reading in *Science of Logic* vol 1, “The objective logic”.

I see that the idea of cohesion is pretty explicit there, not in the first section of the first book (*Determinateness*, which has the discussion of “being and becoming” that Lawvere is alluding to in the Como preface) but in the second section of the first book, “The magnitude”.

There the discussion is all about how the continuous is made up from discrete points with “repulsion” to prevent them from collapsing to a single and with “attraction” that keeps them together nevertheless.

This “attraction” is clearly just the same idea as “cohesion”. One can play this a bit further and match Hegel’s *Raunen* to formal expressions involving the flat modality and the shape modality pretty well. I made some quick notes in the above entry.

On the other hand, that section 1 about being and becoming seems to be more about the underlying type system itself. Notably about the empty type and the unit type, I think

]]>The usual notion of Peano curve involves continuous images of the unit interval, not the whole real line (which could be considered as well, of course).

So I made some adjustments and stated some relevant facts at Peano curve, with a few pointers to proofs and to literature.

]]>started a stub for *topological invariance of dimension*

found it necessary to split off geometric realization of categories as a separate entry, recorded Quillen’s theorems A and B there

all very briefly. I notice that David Roberts has more on his personal web (have included it as a reference)

]]>at *schemes are sober* I have added pointer to a comprehensive proof, here

I added a little bit to maximal ideal (first, a first-order definition good for commutative rings, and second a remark on the notion of scheme, adding to what Urs wrote about closed points).

The second bit is almost a question to myself: I don’t feel I really grok the notion of scheme (why it’s this and not something slightly different that’s the natural definition, the Tao if you like). In particular, it’s where *fields* – simple objects in the category of commutative rings – make their entrance in the notion of covering by affine opens that I don’t feel I really understand.

I added a reference in the section on terminology to Makkai’s ’Towards a categorical foundation of mathematics’, where he defines what he calls the ’Principle of Isomorphism’. This is essentially what ’evil’ captures, I think, and it is handy to have a published version with a sensible name to which to refer people.

Here’s a wild thought: what about renaming the page principle of isomorphism and having evil redirect there. It would necessitate a rewrite of the page, but still contain material about the jokey names (evil, kosher etc). I recall that someone here told how some of these in-jokes are off-putting to outsiders or newcomers (Zoran, maybe?). Just an idea.

]]>I see there a preprint just out

- Norihiro Yamada,
*Dependent Cartesian Closed Categories*, (arXiv:1704.04747)

]]>We present a generalization of cartesian closed categories (CCCs) for dependent types, called dependent cartesian closed categories (DCCCs), which also provides a reformulation of categories with families (CwFs), an abstract semantics for Martin-Löf type theory (MLTT) which is very close to the syntax. Thus, DCCCs accomplish mathematical elegance as well as a direct interpretation of the syntax. Moreover, they capture the categorical counterpart of the generalization of the simply-typed lambda-calculus (STLC) to MLTT in syntax, and give a systematic perspective on the relation between categorical semantics for these type theories. Furthermore, we construct a term model from the syntax, establishing the completeness of our interpretation of MLTT in DCCCs.

I have added to *alternative algebra* the characterization in terms of skew-symmetry of the associator.

If x is an object of a category C, one usually says that x is if finite presentation (or compact) if for any direct filtered system (y_i) in C, the canonical map

f : colim_i Hom(x,y_i) -> Hom(x, colim y_i)

is bijective. One usually says x is of finite type if this holds only for direct filtered systems of monomophisms.

However, I am interested in objects x for which f is injective (without additional condition on the direct filtered system). It seems to me that, in the category of (right) modules over a ring, such objects are exactly finitely generated modules. My question is, has this been considered, and is there a name for this property of x?

Thanks in advance

Alain Bruguières ]]>

I noticed that the entry *analysis* is in a sad state. I now gave it an Idea-section (here), which certainly still leaves room for expansion; and I tried to clean up the very little that is listed at *References – General*

I have edited at *Tychonoff theorem*:

tidied up the Idea-section. (Previously there was a long paragraph on the spelling of the theorem before the content of the theorem was even mentioned)

moved the proofs into a subsection “Proofs”, and added a pointer to an elementary proof of the finitary version, here

Notice that there is an ancient query box in the entry, with discussion between Todd and Toby. It would be good to remove this box and turn whatever conclusion was reached into a proper part of the entry.

At then end of the entry there is a line:

More details to appear at Tychonoff theorem for locales

which however has not “appeared” yet.

But since the page is not called “Tychonoff theorem for topological spaces”, and since it already talks about locales a fair bit in the Idea section, I suggest to remove that line and to simply add all discussion of localic Tychonoff to this same entry.

]]>I have removed the following discussion box from *stuff, structure, property* – because the entry text above it no longer contained the word that the discussion is about :-)

[begin forwarded discussion]

+–{: .query} Mike: Maybe you all had this out somewhere that I haven’t read, but in the English I am accustomed to speak, “property” is not a mass noun. So you can “forget a property” or “forget properties” but you can’t “forget property.”

*Toby*: Well, ’property’ *can* be a mass noun in English, but not in this sense. Also, if we were to invent an entirely new word for the concept, it would surely be a mass noun. Together, these may explain why it's easy to slip into talking this way, but I agree that it's probably better to use the plural count noun here.
=–

[end forwarded discussion]

]]>I have touched *net*, just adding some more hyperlinks and cross-references within the page. Also *filter*, where I made *eventuality filter* come out as a link.