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started an entry cubic curve,
For the moment I wanted to record (see the entry) a pointer to Akhil Mathew’s identification of that eight-fold cover of (hence of ) which is analogous to the 2-fold cover of the “moduli stack of formal tori” that ends up being the reason for the -action on .
So here is the question that I am after: that cover is classified by a map , hence we get a double cover of the moduli space of elliptic curves .
Accordingly there is a spectrum equipped with a -action whose homotopy fixed points is , I suppose: . (Hm, maybe I need to worry about the compactification…).
I’d like to say that is to as is to . This is either subject to some confusion (wich one?) or else is an old hat. In the second case: what would be a reference?
started some minimum at KSC-theory
started a table-for-inclusion-in-relevant-entries: string theory and cohomology theory – table
Right now it reads like this:
cohomology theories of string theory fields on orientifolds
have created geometric infinity-stack
gave Toën’s definition in detail (quotient of a groupoid object in an (infinity,1)-category in ) and indicated the possibility of another definition, along the lines that we are discussing on the Café
I made a new page called twisted form. Unfortunately, this stole the redirect from a sub-heading on differential form. The page is still pretty much a stub. I hope to enlarge it soon.
inverse semigroups behave very well, in some aspects almost like groups, and have close relation to etale groupoids and quantales. I added few references to its stubby entry.
briefly started Brauer infinity-group with a quick remark on the relation to and cross-link to Picard infinity-group and infinity-group of units
I happened upon our entry ETCS again (which is mostly a pointer to further entries and further resources) and found that it could do with a little bit more of an Idea-section, before it leaves the reader alone with the decision whether to follow any one of the many further links offered.
I have expanded a bit, and now it reads as follows. Please feel invited to criticize and change. (And a question: didn’t we have an entry on ETCC, too? Where?)
The Elementary Theory of the Category of Sets (Lawvere 65), or ETCS for short, is a formulation of set-theoretic foundations in a category-theoretic spirit. As such, it is the prototypical structural set theory.
More in detail, ETCS is a first-order theory axiomatizing elementary toposes and specifically those which are well-pointed, have a natural numbers object and satisfy the axiom of choice. The idea is, first of all, that traditional mathematics naturally takes place “inside” such a topos, and second that by varying the axioms much of mathematics may be done inside more general toposes: for instance omitting the well-pointedness and the axiom of choice but adding the Kock-Lawvere axiom gives a smooth topos inside which synthetic differential geometry takes place.
Modern mathematics with emphasis on concepts of homotopy theory would more directly be founded in this spirit by an axiomatization not just of elementary toposes but by elementary (∞,1)-toposes. This is roughly what univalent homotopy type theory accomplishes, for more on this see at relation between type theory and category theory – Univalent HoTT and Elementary infinity-toposes.
Instead of increasing the higher categorical dimension (n,r) in the first argument, one may also, in this context of elementary foundations, consider raising the second argument. The case is the elementary theory of the 2-category of categories (ETCC).
New references at symmetric function and new stub noncommutative symmetric function. An (unfinished?) discussion query from symmetric function moved here:
David Corfield: Why does Hazewinkel in his description of the construction of on p. 129 of this use a graded projective limit construction in terms of projections of polynomial rings?
John Baez: Hmm, it sounds like you’re telling me that there are ’projections’
given by setting the st variable to zero, and that Hazewinkel defines to be the limit (= projective limit)
rather than the colimit
Right now I don’t understand the difference between these two constructions well enough to tell which one is ’right’. Can someone explain the difference? Presumably there’s more stuff in the limit than the colimit.
Mike Shulman: I think the difference is that the limit contains “polynomials” with infinitely many terms, and the colimit doesn’t. That’s often the way of these things.
Actually, on second glance, I don’t understand the description of the maps in the colimit system; are you sure they actually exist? What exactly does it mean to “add in new terms with the new variable to make the result symmetric”?
David Corfield: The two constructions are explained very well in section 2.1 of the Wikipedia article.
Mike Shulman: Thanks! Here’s what I get from the Wikipedia article: the projections are easy to define. They are surjective and turn out to have sections (as ring homomorphisms). The ring of symmetric functions can be defined either as the colimit of the sections, or as the the limit of the projections in the category of graded rings. The limit in the category of all rings would contain too much stuff.
recorded some references on equivariant complex oriented cohomology theory at equivariant cohomology – References – In complex oriented cohomology.
hm, it seems I never announced it: there is an old table-for-inclusion-in-relevant-entries called
genera and partition functions - table
which I have been editing a bit more lately.
started Eisenstein series with some formulas.
I have added to K-orientation pointers to the articles by Atiyah-Bott-Shapiro and to Joachim (2004), together with a brief paragraph.
I added some categorical POV on structure in model theory (which is being touched upon in another thread).
Given the series of entries lately, I naturally came to the point that I started to want a “floating context” table of contents. So I started one and included it into relevant entries:
But this needs more work still, clearly.
Created a minimum at Jacobi form.
Missing from braid group was the precise geometric definition, so I put that in.
am starting something at logarithmic cohomology operation, but so far there are just some general statements and some references
Finally created thick subcategory theorem with a quick statement of the theorem and a quick pointer to how this determines the prime spectrum of a monoidal stable (∞,1)-category of the (∞,1)-category of spectra.
Cross-linked vigorously with related entries.
Created some minimum at Bousfield-Kuhn functor, for the moment just so as to record some references.
Created BICEP2, currently with the following text:
BICEP2 is the name of an astrophysical experiment which released its data in March 2014. The experiment claims to have detected a pattern called the “B-mode” in the polarization of the cosmic microwave background (CMB).
This data, if confirmed, is widely thought to be due to a gravitational wave mode created during the period of cosmic inflation by a quantum fluctuation in the field of gravity which then at the era of decoupling left the characteristic B-mode imprint on the CMB. This fact alone is regarded as further strong evidence for the already excellent experimental evidence for cosmic inflation as such (competing models did not predict such gravitational waves to be strong enough to be detectable in this way).
What singles out the BICEP2 result over previous confirmations of cosmic inflation is that the data also gives a quantitative value for the energy scale at which cosmic inflation happened (the mass of the hypothetical inflaton), namely at around GeV. This is ntoeworthy as being only two order of magnituded below the Planck scale, and hence 12 or so orders of magnitude above energies available in current accelerator experiments (the LHC). Also, it is at least a curious coincidence that this is precisely the hypothetical GUT scale.
It is thought that this value rules out a large number of variant models of cosmic inflation and favors the model known as chaotic inflation.
am creating a table modalities, closure and reflection - contents and adding it as a floating table of contents to relevant entries
just for completeness so that I don’t have gray links elsewhere, I have created some minimum (nothing exciting) at quantum fluctuation.
Started something at chaotic inflation.
Created a webpage for Florian Ivorra, to eliminate a grey link.
I did some editing at exponential function, to restore what I had believed to be a clear argument, which had been edited out by Colin Tan in favor of his own argument. His argument has been moved to a footnote.
added a category:reference entry for
Loop Groups, Characters and Elliptic Curves
talk at ASPECTS of Topology, 17-19 December 2012, Oxford,
on (geometric) representation theory reflected in and incarnated as low-dimensional extended gauge field theory and specifically on formalizations of geometric Langlands duality in terms of 2-dimensional QFT motivated from compactification of N=4 D=4 super Yang-Mills theory.
added to chiral differential operator a paragraph briefly summarizing how the Witten genus of a complex manifolds is constructed by Gorbounov, Malikov and Schechtman. Copied the same paragraph also into the Properties-section at Witten genus
created a minimum at Hirzebruch-Riemann-Roch theorem (I really only wanted to un-gray links at Todd genus)
As we discussed in another thread but maybe it should be announced separately: I have started an entry Dirac operator on smooth loop space with some bare minimum.
Wrote up more stuff at pi.
Incidentally, there are some statements at irrational number that look a little peculiar to me. For example:
In the early modern era, Latin mathematicians began work with imaginary numbers, which are necessarily irrational. They subsequently proved the irrationality of pi, (…)
I suppose Legendre could qualify as a “Latin mathematician of the early modern era” if we take a sufficiently broad view (e.g., he spoke a language in the Latin “clade”), but somehow I feel this is not what the author really had in mind; there were those Renaissance-era Italians who began work with imaginaries IIRC. :-) Probably it would be good to rephrase slightly.
Also this:
There is an easy nonconstructive proof that there exist irrational numbers and such that is rational; let be and let be either or , depending on whether the latter is rational or irrational. A constructive proof is much harder
Not that hard actually: take and , where if I did my arithmetic correctly. Pretty sure that can be made constructive. (Again, I think it’s probably just a case of several thoughts being smooshed together.)
started some minimum at KO-dimension
added a new Examples-section Integral versus real cohomology to fiber sequence
(and renamed the original fibration sequence and made it a redirect to that – but the cache bug is in the way,as usual).
created stub for spectral action
Back in the days I had made several web postings on the “FRS formalism” and how it may be understood as rigorously implementing “holography” in the form of CS/WZW-correspondence. Ever since the nLab came into being there was a stub entry FFRS-formalism which collected some (not all) of these links.
Now I got a question on how it works. (As a student one cannot imagine yet that communication in academia/maths often has latency periods of several years….) While I have absolutely no time for this now, this afternoon I went and expanded that stub entry a bit more (and maybe it’s at least good for my own sanity in these days). Also renamed it to something more suggestive, now it is titled
There is still plenty and plenty of room to expand further (urgent would be to mention the tensor produc of the MTC with its dual, which currently the entry is glossing over), but I am out of time now.
I have added to SimpSet a list of a few properties of the internal logic of the 1-topos of simplicial sets.
Thomas Nikolaus recently gave an impressive talk in which he announced a number of impressive results in topological T-duality. I have already been referring to this from the page T-duality 2-group and I just felt I want to refer to it from elsewhere, too. So therefore I now gave it a category:reference entry:
started some minimum at geometric engineering of quantum field theory
Started some minimum at Stolz conjecture.
In the category:people entry Vladimir Voevodsky I’d like to have a brief statement on the motivic work. I have now put in the following, but experts are please asked to fine tune this where necessary:
Владимир Воеводский (who publishes in English as Vladimir Voevodsky) (web site) is a famous mathematician.
he received a Fields medal in 2002 for a proof of the Milnor conjecture. The proof crucially uses A1-homotopy theory and motivic cohomology developed by Voevodsky for this purpose. In further development of this in 2009 Voevodsky announced a proof of the Bloch-Kato conjecture.
After this work in algebraic geometry, cohomology, homotopy theory Voevodsky turned to the foundations of mathematics and is now working on homotopy type theory which he is advertizing as a new “”univalent foundations” for modern mathematics.
I'm trying to keep some results where I can get at them at Taylor's theorem.
created computable function (analysis) with the definition of “continuously realizable functions”.
I ended up giving Baire space (computability) its (minimal) stand-alone entry after all
I had created a stub for effective topological space and cross-linked with equilogical space. Added some pointers to the literature, but otherwise no real content yet.
An entry which defines both the local category and the local Grothendieck category, two notions which generalize the notion of a category of modules over a commutative local ring.
stub for clutching construction
New entry for Samuel compactification of uniform spaces, and some references at uniform space.
started something at Church-Turing thesis, please see the comments that go with this in the thread on ’computable physics’.
This is clearly just a first step, to be expanded. For the moment my main goal was to record the results about physical processes which are not type-I computable but are type-II computable.
Old discussion at star-autonomous category, which I think was addressed in the entry, and which I’m now moving here:
+–{: .query} Mike: Can someone fill in some examples of -autonomous categories that are not compact closed?
Finn: Blute and Scott in ’Category theory for linear logicians’ (from here) give an example: reflexive topological vector spaces where the topologies are ’linear’, i.e. Hausdorff and with 0 having a neighbourhood basis of open linear subspaces; ’reflexive’ meaning that the map as above is an isomorphism. It seems this category is -autonomous but not compact. I don’t know enough topology to make much sense of it, though.
Todd: Finn, I expect that example is in Barr’s book, which would then probably have a lot of details. But I must admit I have not studied that book carefully. Also, the Chu construction was first given as an appendix to that book.
John: I get the impression that a lot of really important examples of -autonomous categories — important for logicians, anyway — are of a more ’syntactical’ nature, i.e., defined by generators and relations. =–
I am working on entries related to the (oo,1)-Grothendieck construction
started adding a bit of structure to (oo,1)-Grothendieck construction itself, but not much so far
added various technical details to model structure on marked simplicial over-sets
created stub for model structure for left fibrations to go in parallel with that
added to Kleene’s second algebra under “Properties” the sentence:
Kleeen’s second algebra is an abstraction of function realizability introduced for the purpose of extracting computational content from proofs in intuitionistic analysis. (e.g. Streicher 07, p. 17)
Am starting an entry computable physics. For the moment this is essentialy a glorified lead-in for
I had had the feeling that most previous literature on computability in physics is suffering from being not well informed of the relevant mathematical concepts, but then I found
which seems to be sober, well-informed and sensible. The main drawback seems to be, to me, that the author looks only at type-I computability and not really seriously at quantum physics. Both of this is what Streicher’s note above aims to do!
If anyone has more pointers to decent literature on this topic, please drop me a note.
Here is what it currently has in the entry text computable physics:
The following idea or observation or sentiment has been expressed independently by many authors. We quote from Szudzik 10, section 2:
The central problem is that physical models use real numbers to represent the values of observable quantities, Careful consideration of this problem, however, reveals that the real numbers are not actually necessary in physical models. Non-negative integers suffice for the representation of observable quantities because numbers measured in laboratory experiments necessarily have only finitely many digits of precision.
Diverse conclusions have been drawn from this. One which seems useful and well-informed by the theory of computability in mathematics is the following (further quoting from Szudzik 10, section 2)
So, we suffer no loss of generality by restricting the values of all observable quantities to be expressed as non-negative integers — the restriction only forces us to make the methods of error analysis, which were tacitly assumed when dealing with real numbers, an explicit part of each model.
In type-I computability the computable functions are partial recursive functions and in view of this some authors conclude (and we still quote Szudzik 10, section 2) for this:
To show that a model of physics is computable, the model must somehow be expressed using recursive functions.
However, in computability theory there is also the concept type-II computable functions used in the field of “constructive analysis”, “computable analysis”. This is based on the idea that for instance for specifying computable real numbers as used in physics, an algorithm may work not just on single natural numbers, but indefinitely on sequences of them, producing output that is in each step a finite, but in each next step a more accurate approximation.
!include computable mathematics – table
This concept of type-II computability is arguably closer to actual practice in physics.
Of course there is a wide-spread (but of course controversial) vague speculation (often justified by alluding to expected implications of quantum gravity on the true microscopic nature of spacetime and sometimes formalized in terms of cellular automata, e.g. Zuse 67) that in some sense the observable universe is fundamentally “finite”, so that in the end computability is a non-issue in physics as one is really operating on a large but finite set of states.
However, since fundamental physics is quantum physics and since quantum mechanics with its wave functions, Hilbert spaces and probability amplitudes invokes (functional) analysis and hence non-finite mathematics even when describing the minimum of a physical system with only two possible configurations (a “qbit”) a strict finitism perspective on fundamental physics runs into severe problems and concepts of computable analysis would seem to be necessary for discussing computability in physics.
This issue of computable quantum physics has only more recently been considered in (Streicher 12), where it is shown that at least a fair bit of the Hilbert space technology of quantum mechanics/quantum logic sits inside the function realizability topos .
I have started something at computability.
Mainly I was after putting some terms in organized context. That has now become
which I have included under “Related concepts” in the relevant entries.
created a minimum at computable real number, for the moment just so as to record the references with section numbers as given there.
happened to need Type Two Theory of Effectivity