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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 Σ∞(−)+:Ho(Top)⟶Ho(Spectra).
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 ℂ-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 C are interpretations of theories in the internal logic of C 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 ω-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 T and expand it to a new theory T′ by naming one of its models M with constant symbols for each element of M while additionally stipulating those constant symbols have to behave like they came from M.
If those additional stipulations were only quantifier-free, the models of T′ are those models of T containing M as an induced substructure.
If those additional stipulations were all the first-order sentences satisfied by the elements of M, then the models of T′ are those models of T containing M 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 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.
at separation axiom I have expanded the Idea section here, trying to make it more introductory and expository.
I saw tube lemma, and decided to bulk it up.
Many books (such as the famous topology text by Munkres) give proofs which involves multiple subscripts and multiple choices; I’ve written arguments to mostly eliminate that.
we have a page semi-locally simply connected space but no page locally simply connected space. I am not sure I know the difference! And there is no mention of it in the entry that we have. How is this resolved?
I have written an article closed-projection characterization of compactness, so as to record a nice way of proving the Tychonoff theorem that is due to Clementino and Tholen. It’s rather direct and elementary, which doesn’t involve ultrafilters or nets or any such machinery. This might make it a possibility for a strong undergraduate classroom. (Munkres also has a proof which I haven’t cross-checked; the one I wrote up involves a smidge of categorical terminology, notably inverse limits.)
I didn’t want to stick in the proof at Tychonoff theorem as that article might be getting a bit bloated, but I did link there to the new article.
I created Cantor space to record its definition as a locale, but goodness knows there is no end to what might be written about it.
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.
At limits and colimits by example an entry for Kuratowski pair was requested, so I created it. The place where this should really be linked to is the entry ordered pair, so I added a link there. Also, I made that entry mention Cartesian product, which it didn’t.
I have given product topological space an semi-informal Idea-section that means to quickly and transparently tell the reader what they need to know.
This is in reaction to what we presently have at Tychonoff product which presently seems needlesly intransparent for the purpose of a reader who just wants to know what the open subsets actually are.
I was thinking about merging product topological space with Tychonoff product. I haven”t yet, but mostly just due to lack of energy.
The entry for infinitesimal extension said that an infinitesimal extension of rings was an epimorphism of rings with nilpotent kernel. I’ve changed this to say a quotient map of rings with nilpotent kernel. I hope this is correct: for example, localization maps are ring epimorphisms, and often have zero kernel (so in particular, nilpotent kernel) but geometrically these correspond to dense open inclusions, which are in no sense infinitesimal extensions.
Finally created stereographic projection. Lots more could be said, including metric/conformal aspects.
Somebody anonymous has created codiagonal comodule structure with no content. But let’s leave the it there and simply fill in some.
Presently I am concerned with the following: I want to teach some basics of limits and colimits of topological spaces to undergraduates. I had tried to gently introduce some category theoretic terminology as I introduced topological spaces as such, but a little testing reveals that part of the audience would rather not see these side remarks turn out to become required reading.
But now since all the shapes of diagrams that I’d be about to consider anyway are free, I figured I circumvent the need to speak of diagrams as functors by restricting to free domains and simply giving everything explicitly in components, with the underlying category theory again relegated to side-remarks that may be ignored at will.
I am trying my hands at an exposition of this kind in a new entry free diagram.
Eventually this kind of material might also be worthwhile as introductory exposition at limits and colimits by example.
I have a request to the logic experts:
The entry classical logic is a bit thin. I would like to be able to point to it so that readers who don’t already know about it all, get away with a decent idea of what is meant. Might somebody have the energy to add a few lines?
Of course I could add a few lines myself, but I imagine it would be more efficient if some expert here did it from scratch, instead of having to improve on what I would come up with.
Especially it would be nice if the following keywords were at least mentioned and maybe briefly commented on in the entry:
Also for instance the keyword constructive anything is presently missing from the entry.
Thanks!
had occasion to give attaching space its own little entry. Cross-linked with Top and CW-complex.
I added a sentence on ’eigenfunction’ to eigenvector.