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    • 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.

    • Missing from braid group was the precise geometric definition, so I put that in.

    • 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 1016GeV. 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.

    • just for completeness so that I don’t have gray links elsewhere, I have created some minimum (nothing exciting) at quantum fluctuation.

    • 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 statement of the Schmidt decomposition to the page "tensor product of Banach spaces". It seemed the right place for it, since this is the page where the tensor product of Hilbert spaces is discussed.
    • 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

    • 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 a and b such that ab is rational; let b be 2 and let a be either 2 or 22, depending on whether the latter is rational or irrational. A constructive proof is much harder

      Not that hard actually: take a=2 and b=2log3log2, where ab=3 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.)

    • Added the proof that every positive operator is self-adjoint.
    • For my first contribution to nLab, I've typed up my notes on effect algebras, with the definitions of a generalized effect algebra and morphism of effect algebras. The proofs here are more basic than most that I've seen on the wiki, but I've decided to include them in the spirit of this being a public lab book.
    • 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.

    • Hello all,
      I liked very much the nLab-entry "well-founded relation": concise and informative.
      Do you think "lexicographic order" may be included in the section Examples as another, practically relevant example of well-founded relation?
      If yes, I would be very grateful if somebody could do that (I am not an expert).

      Best regards from Germany
    • Hello,
      I have just created a page on C*-correspondences (http://ncatlab.org/nlab/show/C-star-correspondence). I will add a few stuff about the weak 2-category of C*-algebras built upon those later.
    • 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.

    • 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 dV 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. =–

    • Am starting an entry computable physics. For the moment this is essentialy a glorified lead-in for

      • Thomas Streicher, Computability Theory for Quantum Theory, talk at Logic Seminar Univ. Utrecht in July 2012 (pdf)

      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 RT(𝒦2).

    • 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.

    • just in case you are watching the logs and are wondering:

      I think we should have another “floating table of contents” for collecting the topic cluster

      • Constructivism, Realizability, Computability

      so I am starting one at constructivism - contents and am including it into relevant entries.

      But right now there is nothing much there yet. This is going to be expanded.

    • Created a brief entry transfer context in order to record an observation by Haugseng.

      He defines a transfer context to be a linear homotopy-type theory aka Wirthmüller context in which not only f! but also f* satisfies its projection formula. Then he observes that a natural Umkehr map that may be built with this projection formula is (the abstract generalization of) the Becker-Gottlieb transfer.

      (Have briefly cross-linked with these related entries.)

      Thanks to Thomas Nikolaus for being reminded of Haugseng’s work when Joost Nuiten and me talked about something closely related as ESI yesterday.

    • I added some basic definitions to stability in model theory. No attempt yet to motivate them.

      Some of the logic entries seem to be in a slight state of neglect, e.g., theory. I might want to get in there sometime soon, but anyone should please feel free to precede me.

    • for some reason it seems we never had an entry compactly generated (infinity,1)-category (and out of all sections listed at HTT just 5.5.7 had been missing for some reason which is a mystery to me now).

      I gave it a minimum of content. But this alerts one that there is a distinction being made here which we don’t have in the corresponding 1-categorical entries.

    • I’ve been adding material to Polish space, and plan on adding more (mostly in view of model-theoretic considerations).