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Created a minimal entry for Haskell Curry.
I discovered that there was no content in the entry path space, so I gave it some.
Am working on the entry higher Cartan geometry. Started writing a Motivation section.
This is just the first go, need to quit now, will polish tomorrow.
I’ve just edited topological concrete category to correct the claim that topological functors create limits, which is not quite true: for instance, the forgetful fails to reflect limits because choosing a finer topology on the limit vertex yields a non-limiting cone with the same image in . This is correctly reported on wikipedia and in Joy of Cats, p. 227.
It is true that topological functors allow you to calculate limits using the image of the diagram under the functor, which is quite powerful. In Joy of Cats, a topological functor is said to “uniquely lift limits” (definition p. 227, proven p. 363). There doesn’t seem to be an nlab page for this property – I suppose it’s not much used by most category theorists.
added to partition of unity a paragraph on how to build Cech coboundaries using partitions of unity (but have been lazy about getting the relative signs right).
It would be good (for me) if we could add some more about smooth partitions of unity, too, eventually.
I added a brief description of how the exotic 7-spheres are constructed at exotic smooth structure.
The term “flow of a vector field” used to redirect to exponential map, which however is really concerned with a somewhat different concept. So I have created now a separate entry flow of a vector field.
at monoidal adjunction the second item says
while the left adjoint is necessarily strong
but should it not say
while the left adjoint is necessarily oplax
?
I created article cotopology including a redirect from cocompact space.
I separated p-derivation from Fermat quotient.
created arithmetic jet space, so far only highlighting the statement that at prime these are (regarded so in Borger’s absolute geometry by applying the Witt ring construction to it).
This is what I had hoped that the definition/characterization would be, so I am relieved. Because this is of course just the definition of synthetic differential geometry with regarded as the th abstract formal disk.
Well, or at least this is what Buium defines. Borger instead calls itself already the arithmetic jet space functor. I am not sure yet if I follow that.
I am hoping to realize the following: in ordinary differential geometry then synthetic differential infinity-groupoids is cohesive over “formal moduli problems” and here the flat modality is exactly the analog of the above “jet space” construction, in that it evaluates everything on formal disks. Moreover, canonically sits in a fracture suare together with the “cohesive rationalization” operation and hence plays exactly the role of the arithmetic fracture square, but in smooth geometry. I am hoping that Borger’s absolute geometry may be massaged into a cohesive structure over the base that makes the cohesive fracture square reproduce the arithmetic one.
If Borger’s absolute direct image were base change to followed by the Witt vector construction, then this would come really close to being true. Not sure what to make of it being just that Witt vector construction. Presently I have no real idea of what good that actually is (apart from giving any base topos for , fine, but why this one? Need to further think about it.)
while writing out the proof of the fundamental product theorem in K-theory I had occasion to record that
I put a disambiguation block at the top of inverse image to point to preimage.
created induced metric, just for completeness
Created parametric right adjoint.
created diagram chasing lemmas - contents (with some information on what implies what) and included it as a floating TOC into the relevant entries.
Created twosets20170617. Contains an svg illustration of a full subcategory of consisting of a terminal object and a two-element set. Uses the convention that an identity arrow is labelled by its object. Intended for use in some graph-theoretical considerations from an nPOV. Sufficiently general to be possibly of use in some other nLab articles too.
Comment/question on terminology in the (?,1)-case at regular epimorphism
Created isotropy group of a topos, now that Simon’s preprint finally explains what it actually is!
I am looking for a decent account of the homotopy ring spectrum structure on with that would be self-contained for a reader with good point-set topology background, but not involving or model category theory.
What I find in the literature is all sketchy, but maybe I am looking in the wrong places.
First, a discussion of the H-space structure on in the first place I find on p. 205 (213 of 251) in A Concise Course in Algebraic Topology. But for the crucial step it there only says:
we merely affirm that, by fairly elaborate arguments, one can pass to colimits to obtain a product
Is there a reference that would spell this out?
Next, for the proof of the homotopy ring spectrum structure on , the idea is indicated on the first page of
James McLure, -ring spectra via space-level homotopy theory (pdf)
Is there a place where this would be spelled out in some detail?
I have started at topological K-theory a section “For non-compact spaces” (here).
Created assignment and operation.
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