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2-category 2-category-theory abelian-categories accessible adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categories category category-theory chern-weil-theory cobordism-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology deformation-theory derived-geometry descent differential differential-cohomology differential-geometry duality education elliptic-cohomology enriched factorization-system fibration foundations functional-analysis functor galois-theory gauge-theory gebra general topology geometric-quantization geometry goodwillie-calculus graph gravity group-theory higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory hypercovers index-theory infinity integration-theory k-theory kan lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory model-theory monoidal monoidal-category-theory morphism motives motivic-cohomology newpage nforum nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory subobject superalgebra supergeometry symplectic-geometry synthetic-differential-geometry terminology theory topological topology topos topos-theory tqft type type-theory universal web

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- Discussion Type
- discussion topicUrysohn's lemma
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Todd_Trimble
- Last Active 5 hours ago

added a proof to

*Urysohn’s lemma*

- Discussion Type
- discussion topicZariski topology
- Category Latest Changes
- Started by Urs
- Comments 35
- Last comment by Todd_Trimble
- Last Active 5 hours ago

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).

- Discussion Type
- discussion topictopological basis
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by DavidRoberts
- Last Active 6 hours ago

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.

- Discussion Type
- discussion topicpartitions of unity and locally finite cover
- Category Latest Changes
- Started by DavidRoberts
- Comments 12
- Last comment by DavidRoberts
- Last Active 7 hours ago

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.

- Discussion Type
- discussion topicparacompact Hausdorff spaces are normal
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

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.)

- Discussion Type
- discussion topicseparation axiom
- Category Latest Changes
- Started by Urs
- Comments 22
- Last comment by Urs
- Last Active 1 day ago

at

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

- Discussion Type
- discussion topicparacompact Hausdorff spaces equivalently admit subordinate partitions of unity
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

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.

- Discussion Type
- discussion topicsupport
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

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.

- Discussion Type
- discussion topicGalois connection
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

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.

- Discussion Type
- discussion topic"Science of Logic"
- Category Latest Changes
- Started by Urs
- Comments 138
- Last comment by Urs
- Last Active 2 days ago

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

- Discussion Type
- discussion topicPeano curve
- Category Latest Changes
- Started by Todd_Trimble
- Comments 1
- Last comment by Todd_Trimble
- Last Active 2 days ago

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.

- Discussion Type
- discussion topictopological invariance of dimension
- Category Latest Changes
- Started by Urs
- Comments 28
- Last comment by Urs
- Last Active 3 days ago

started a stub for

*topological invariance of dimension*

- Discussion Type
- discussion topicgeometric realization of categories and Quillen's theorems A and B
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by Urs
- Last Active 3 days ago

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)

- Discussion Type
- discussion topicschemes are sober
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 3 days ago

at

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

- Discussion Type
- discussion topicmaximal ideal
- Category Latest Changes
- Started by Todd_Trimble
- Comments 19
- Last comment by Todd_Trimble
- Last Active 4 days ago

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.

- Discussion Type
- discussion topicevil
- Category Latest Changes
- Started by DavidRoberts
- Comments 37
- Last comment by David_Corfield
- Last Active 4 days ago

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.

- Discussion Type
- discussion topicalternative algebra
- Category Latest Changes
- Started by Urs
- Comments 21
- Last comment by TobyBartels
- Last Active 4 days ago

I have added to

*alternative algebra*the characterization in terms of skew-symmetry of the associator.

- Discussion Type
- discussion topicanalysis
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by TobyBartels
- Last Active 4 days ago

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*

- Discussion Type
- discussion topicTychonoff theorem
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by m
- Last Active 4 days ago

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.

- Discussion Type
- discussion topicstuff, structure, property
- Category Latest Changes
- Started by Urs
- Comments 20
- Last comment by Urs
- Last Active 4 days ago

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]

- Discussion Type
- discussion topiccompact space
- Category Latest Changes
- Started by Todd_Trimble
- Comments 12
- Last comment by Mike Shulman
- Last Active 6 days ago

As an outcome of recent discussion at Math Overflow here, Mike Shulman suggested some nLab pages where comparisons of different definitions of compactness are rigorously established. I have created one such page: compactness and stable closure. (The importance and significance of the stable closure condition should be brought out better.)

- Discussion Type
- discussion topicfinite intersection property
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 6 days ago

gave

*finite intersection property*an entry, added statement and proof of the corresponding characterization of compact topological spaces. Pointed to this from*Tychonoff theorem*, where the statement is used.

- Discussion Type
- discussion topicReflective and coreflective subcategores
- Category Latest Changes
- Started by Bartek
- Comments 4
- Last comment by Todd_Trimble
- Last Active 7 days ago

I added recollement as an example to the pages on reflective and coreflective subcategories.

- Discussion Type
- discussion topicempty set
- Category Latest Changes
- Started by Todd_Trimble
- Comments 4
- Last comment by Urs
- Last Active Apr 21st 2017

I did a little editing over at empty set; the query-box discussion of $0^0$ looked like it could be summarized with dispatch and relegated to a remark. Revert back or re-edit if you don’t like it.

- Discussion Type
- discussion topicCW-complex
- Category Latest Changes
- Started by Urs
- Comments 25
- Last comment by Urs
- Last Active Apr 21st 2017

added in CW-complex in the Examples section something about noncompact smooth manifolds.

Eventually it would be good to state here precisely Milnor’s theorem etc. Googling around I seem to see a lot of misleading imprecision in the usual statements along these lines (on Wikipedia and MO) concerning the distinctions between countably generated and general CW-complexes and concerning homotopy equivalence vs weak homotopy equivalence.

- Discussion Type
- discussion topicgeneral topology
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 21st 2017

I have edited a bit at

*general topology*, trying to stream-line for readability.

- Discussion Type
- discussion topicHausdorff topological space
- Category Latest Changes
- Started by Urs
- Comments 40
- Last comment by TobyBartels
- Last Active Apr 20th 2017

added to

*Hausdorff topological space*a brief paragraph*Beyond topological spaces*

- Discussion Type
- discussion topicdiagram
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 20th 2017

I have substantially expanded the exposition at

*diagram*. In the section*Diagrams shaped like categories*.First I expanded on the quick functorial definition of diagrams and their ((co-)limiting) (co-cones) and then I added a detailed component-based definition meant for readers not fluent in unwinding the former definition.

Haven’t proof-read yet, as my battery is dying now.

- Discussion Type
- discussion topicfuzzy dark matter
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 20th 2017

Added to the entry

*fuzzy dark matter*pointer to Lee 17 which appeared today on the preprint server. This is just a concise 2.5 page survey of all the available literature, but as such is very useful. For instance it points out this Nature-article:- Hsi-Yu Schive, Tzihong Chiueh, Tom Broadhurst,
*Cosmic structure as the quantum interference of a coherent dark wave*, Nature Physics 10, 496–499 (2014) (doi:10.1038/nphys2996)

which presents numerical simulation of the fuzzy dark matter model compared to experimental data.

- Hsi-Yu Schive, Tzihong Chiueh, Tom Broadhurst,

- Discussion Type
- discussion topicclub
- Category Latest Changes
- Started by Todd_Trimble
- Comments 5
- Last comment by Mike Shulman
- Last Active Apr 20th 2017

I began writing the article club, with more to come.