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    • 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 Top to Top. Spelled out the basics about limits and colimits, added some basic examples, some first remarks on the characterization over Set, 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 nLab 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 nLab 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 nLab, but now I don’t find them anymore.

    • 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.

    • 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 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.

    • 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 a page on the set-theoretical meaning of the “class function”.

    • I have cross-linked the entries forcing and classifying topos just a tad more by

      1. adding a half-sentence at the end of the paragraph in the Idea-section at “forcing” which mentions the word “classifying topos”

      2. 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 -algebra).

    • 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.

    • 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.

    • 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?

    • 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.

    • 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.

    • 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.

    • 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?

    • 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.