Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I had need to record the traditional point-set construction of cofiber sequences in more detail. Now at topological cofiber sequence.
made closed cofibration a redirect to Hurewicz cofibration
Then I added the statement of the theorem that a morphisms of homotopy pullback diagrams along closed cofibrations induces a closed cofibration on the pullbacks.
I am starting to add statement and discussion of basic properties of to Top. Spelled out the basics about limits and colimits, added some basic examples, some first remarks on the characterization over , etc.
created little entries
Pi modality flat modality sharp modality
for completeness. The first one redirects to what used to be called Pi-closed morphism, but which should be generalized a bit.
[edit: later we renamed the “Pi modality” to shape modality ]
On the page countable choice there seemed to be an unsubstantiated claim that weak countable choice proves that the Cauchy and the Dedekind reals coincide. I have cleaned it up a bit. It’s not perfect yet.
I improved magma. Entry quasigroup is reworked with some new ideas incorporated and part of the entry delegated to new entry, historical notes on quasigroups which also feature (terminological, historical and opinionated issues on) other nonassociative binary algebraic structures. This delegated also part of what Tom Leinster called in another occasion mathematical bitching (in his example used about categories, here about quasigroups), i.e. opinionated attack on some field of mathematics. Some parts of theory of quasigroups and loops are now very hot in connection to new classes of examples and applications. In particular, analytic loops (like Lie groups) appear to have rich tangent structures, Sabinin algebras (sorry, the entry still under construction) and (augmented) Lie racks (=left distributive left quasigroups) appeared as a solution to local Lie integration problem for Leibniz algebras (nonassociative algebras which satisfy the Leibniz identity, just like Lie algebras, but without skew-symmetry, with lots of applications and relation to the Leibniz homology of Lie algebras and to the conjectural noncommutative K-theory envisioned by Jean-Louis Loday).
I encountered that Borceux-Bourn call magma what wikipedia and Lab would call unital magma. I discussed origin of word groupoid at historical notes on quasigroups (which are now a proposed subject of discussion) and created a related name entry Øystein Ore.
I wish we had a decent account at sheaf toposes are the accessible left exact reflective subcategories of presheaf toposes.
Presently the account that the Lab gives of the important fact is not very good. It’s stated at category of sheaves with proof by pointer to reflective (infinity,1)-category, which in turn points to Lurie’s HTT. The crucial point about the accessibility condition is presently discussed, very briefly, without cross-links at reflective subcategory here.
So all this should be collected coherently in one place, which we can then link to.
I seem to remember that the statement was a bit scattered in the Elephant, and I thought we had pointers to the relevant propositions in the Elephant on the Lab, but now I don’t find them anymore.
Created isomorphic functors, with some references.
I’ve created an entry on Lindström’s theorem and readjusted a bit the entries on predicate logic, Löwenheim-Skolem theorem. I guess the most valuable thing in the entry is the link to free version of the Barwise-Feferman handbook. Hopefully somebody with a bird’s eye view of the nlab knows a better context than ’foundation’ for the entry. Any other improvement or expansion would be appreciated as well.
Started hyperimaginary element to record the observation that hyperimaginaries aren’t usually preserved by ultraproducts. I also touched up definable set a bit while adding a referenceable definition for type-definable set.
I have added discussion of how the “superfields” in the physics literature are generalized elements of internal homs in the topos over supermanifolds: here
At linking number, I added a diagrammatic/combinatorial proof that the linking number is an integer, and hence that there must be an even number of crossings between a pair of components of a link. It is surprisingly hard to find a diagrammatic proof: the typical argument is geometric, using something like the Jordan curve theorem. I stumbled across the argument I have added yesterday and thought it was rather nice, so decided to add it here before I forget it!
I’ve rather greatly expanded differentiable map, defining variations.
I started a new article hereditary property. In so doing I inadvertently created a number of gray links (some of which I found surprising). Comments are welcome.
at Postnikov tower in the section Construction for simplicial sets I made explicit three different models. Two of them were discussed there before, the third I have now added. Briefly. Deserves to be expanded.
Created SomePage > naturalitysquare20170605, following some instruction in the HowTo pages. Diagram is planned to be used in the next few days in analytical part of the unfinished article unnatural isomorphism, the writing of which had to be broken off.
Added to our entry on corepresentable functors the notion of corepresentability used in the study of moduli problems, for instance in Definition 2.2.1 of Huybrecht and Lehn’s notes.
Stub for Peirce. Very quick write-up without any pretense of being super-precise or super-accurate. Needs more links and redirects.
At CW-complexes are paracompact Hausdorff spaces
I wrote out proof of the lemma that the result of attaching a cell to a paracompact Hausdorff space is still paracompact Hausdorff (here).
Not very nice yet. Needs polishing and maybe some more general lemmas.
Created essentially pointwise isomorphic, with a reference.
Created a page on the set-theoretical meaning of the “class function”.
I had begun writing classifying topos of a localic groupoid and sheaves on a simplicial topological space, but am (naturally) being distracted from nLab work now, so this is left in somewhat unfinished form…
created a minimum at Kadison-Singer problem
I have cross-linked the entries forcing and classifying topos just a tad more by
adding a half-sentence at the end of the paragraph in the Idea-section at “forcing” which mentions the word “classifying topos”
adding to “classifying topos” the references (grabbed from “forcing”) on the relation between the two: here.
I imagine any categorical logician who would write a pedagogical exposition at forcing on how this concept appears from the point of view of topos theory could have some effect on the community. The issue keeps coming up in discussions I see, and so if we had a point to send people to really learn about the relation (instead of just being bluntly old that there is one) that might have an effect.
I have given infinity-group infinity-ring its own entry (it used to be redirecting to infinity-group of units)
Then I added a section “H-group ring spectra” with some details on the simpler version .
the entry neighbourhood base was in a funny state. I have edited a little. No mention of filters yet.
I added a few words to address an oversight noted by Sridar Ramesh at topological ring, and corrected also a second oversight in the formulation of topological algebra (a standard mistake which would imply that the quaternions are a -algebra).
at general linear group we only had some sentences on its incarnation as an algebraic group. I have started a subsection Definition – As a topological group with some basics.
The entry cofibration is need of some attention. It wasn’t even linked to from codiscrete cofibration, so I’ve remedied that. There’s also Hurewicz cofibration to bring into the fold.
I wrote out some elementary details at basic complex line bundle on the 2-sphere.
I’ve started filling out elimination of quantifiers, adding some initial remarks and sketching the usual proof that algebraically closed fields eliminate quantifiers.
I have spelled out a chunk of elementary details at Grothendieck group – For commutative monoids:
wrote out a second version of the definiton, made explicit the proof that it is all well defined and satisfies the universal property of the group completion, added remark on how the definition simplifies in the cancellative case, and wrote out the most basic examples in some detail.
In the course of this I created an entry cancellative monoid with a bare minimum of content.
I also slightly re-structured the remaining bit of the entry. The small section on -group completion I simply removed, because that belongs to group completion where in fact the content of the paragraphed that I removed is kept in more polished form.
created inner product of vector bundles with the construction over paracompact Hausdorff spaces
Created stubby orthogonal factorization system in a derivator just to record a reference.
I have given topological vector bundle an expository Defintion-section with details on the (re-)construction of topological vector bundle from transition function data, and have now used pointer to that in order to give actual definitions at
and eventally I should also touch dual vector bundle accordingly.
At vector bundle all the way back from rev 1 there is a request for an entry “sheaf semantics”, at the bottom, where it says
sheaf semantics (Kripke-Joyal semantics)
Should we just make “sheaf semantics” redirect to Kripke-Joyal semantics?
at connected space I have started a section Properties with statement and proof that connected components are always closed subsets.
at locally connected space in the Definition section I used this to write out the proof that the equivalent characterizations of local connectedness are indeed equivalent.
finally, for completeness, there is also locally compact and second-countable spaces are sigma-compact
As promised (and following what has been done for topology - contents), I’ve made the previously puny model theory - contents somewhat less puny. I’ll try to fill the grey links therein before I start travelling for the summer.
I’ve expanded Urs’s entry on interpretation: models of theories in are interpretations of theories in the internal logic of are (cartesian, regular, coherent, first-order, geometric) functors out of the syntactic categories of those theories. I also mention a concrete notion of interpretation (of models—of possibly different theories—in each other) that came into use among model theorists in the 80s and 90s, when people were trying to reconstruct -categorical theories from the automorphism groups of the countable model. I then mention some facts (I have proofs spelled out somewhere, but I’m sure they’re folklore) linking these concrete interpretations to the first notion of interpretation.
If I recall correctly, there isn’t a unified treatment in the various pages under the contexts of type theory and categorical logic treating the adjunction between taking syntactic categories and internal logics. I’ll try to get all that down in a single page in the future.
At well-ordering theorem there had been a request for “transfinite arithmetic”. I have now made that a redirect to ordinal arithmetic, but this entry is nothing but a stub at the moment.
In cross-linking, I realized that presently “arithmetic” redirects to number theory. Is that a good idea?
in the course of writing out proofs in elementary topology, i found it useful to be able to easily link to elementary statements in elementary set theory. So I created entries like
Nosing around a bit I came across the concept of absolute extensor at a neglected page quasi-finite CW-complex, so started the entry. Do people know much about this? There seems to be a related ’absolute neighborhood extensor’. Does that merit a separate page?
Some notes are to be found here.
created for completeness: cofinite subset, cofinite topology
For the sake of having a reference to link to later, I’ve written diagram of a first-order structure. This is just a construction where you take a theory and expand it to a new theory by naming one of its models with constant symbols for each element of while additionally stipulating those constant symbols have to behave like they came from .
If those additional stipulations were only quantifier-free, the models of are those models of containing as an induced substructure.
If those additional stipulations were all the first-order sentences satisfied by the elements of , then the models of are those models of containing as an elementary substructure.
To fill a grey link (and towards eventually writing down the characterization of the theory of the countable random graph as the expansion of the theory of an infinite set by a “generic” predicate), I created the page existentially closed model.
The corresponding notion for a theory (“all models are existentially closed models”) has been given a page at model complete theory.
I have added to Michael’s theorem the statement that was apparently the first to be called “Michael’s theorem”: here
I was going to write out the proof. But now I am out of steam.
I had started a stub entry fractal just so that I could link to Mandelbrot set. David C. has kindly added already references to Tom Leinsters work on fractals via terminal coalgebras.
In reaction I have now added also a corresponding minimum Examples-section at terminal coalgebra for an endofunctor for cross-linking purposes.
I have added the example of the rational numbers (here) at totally disconnected topological space
created a minimum at Mandelbrot set, added statement and proof of the compactness
Some remarks and a reference to finish later at extreme value theorem. But I need to learn more about semicontinuous maps on locales.
i have split off (copied over) the paragraph on the first uncountable ordinal from countable ordinal to first uncountable ordinal, just in order to make it possible to link to “first uncountable ordinal” more directly. Cross-linked with long line.
I have split off a little entry embedding of topological spaces from “subspace toplogy” to make it easier to point to either concept with more precision.
a really tiny lemma, but I gave it an entry such as to have an easier time pointing to it: closed injections are embeddings
stub for embedding of smooth manifolds
Created closed map to satisfy a few links. I notice that there’s quite a lot at open map. Does any of the more general picture carry over to closed maps?
wrote out statement and proof at
open subspaces of compact Hausdorff spaces are locally compact
and referenced this where the claim is stated at locally compact Hausdorff space.
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.