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    • I have added some links to preprint on the entry Lascar group. I do not understand the model theory, but its link with Galois theory may be of use to someone looking at model theory and type theory elsewhere on the Lab, so I hope it is useful.

    • creatd connecting homomorphism with (just) the pedestrian description.

      (Relation to snake lemma and more generally to fiber sequences not there yet…)

    • As I said in another thread, I would like to see the nnLab entries related to universes be somehow better, more organized, more comprehensive.

      In order to get a handle on it I decided, as so often, to tabulate what we have and what we should have, so I am creating:

      universe - contents

      and will include it as a “floating table of contents” into the relevant entries

    • at inductive reasoning it says

      Induction here is not to be confused with mathematical induction.

      We should point out that, however, there is a close relation:

      one can see this still in the German tem for, “induction over the natural numbers” which is not Induktion, but vollständige Induktion: meaning ” complete induction” !

      I guess the reasoning is clear, mathematical induction (at least that over the natural numbers) is a special case of inductive reasoning, namely that where we can be sure that we are inducing from a complete set of instances of the general rule.

      Does anyone feel like touching the entry accordingly to clarify this?

    • turns out plenty of entries were asking for quotient group. I created something. But am running a bit out of steam for tonight.

    • I have touched cokernel, briefly adding some basics. More needs to be done here.

    • Using the LaTeX macro package TikZ, I’ve redrawn most of the SVGs on the knots and links pages. I hope that I haven’t trodden on any toes in so doing! I may have missed a few diagrams as well.

      I’ve shifted the actual SVGs to pages of their own. This makes it easier to edit the pages with them on - TikZ’s SVG export isn’t as compact as the inbuilt SVG editor - and easier to include on other pages. For example, I can imagine that the trefoil knot is going to appear again and again!

      (Incidentally, are the two trefoils distinct? If so, which have I drawn at trefoil knot - SVG).

      I’ve named the pages with - SVG in their name, though for the moment I’ve also put in redirects to the name without the SVG. When actually including the diagram, one should always use the canonical name (ie with the - SVG) since it may be that we actually write a page about the trefoil knot one day. But I thought that for the moment, a nice aspect of hyperlinks is that if we mention the trefoil knot in a page then we can put in a link to an actual picture.

      Diagrams done so far:

      Pages with includes include: link, Reidemeister move, colorability, bridge number.

      What would be fantastic here is if the “source” link took one to the actual LaTeX/TikZ source! I do intend to put that up on the nLab, but I need to clean it up a little as it depends on some customised style files that have a lot of crud in them.

    • I have been attacking some of the grey links in knot theory and the related pages. If someone has the time (and the patience) adding a few more links would be a good thing. I have added Crowell, Fox, Dehn, Alexander, Louis Kauffman, plus some non-people pages such as Alexander polynomial. That needs some diagrams if it is to do what it ’should’ and my svg skills are too slight to attempt that today. :-)

    • I finally gave the poor entry physics a bit of text.

    • Made some changes at logic and started inductive inference and George Polya.

      There are still things to change at logic

      As a discipline, logic is the study of methods of reasoning. While in the past (and often today in philosophical circles), this discipline was prescriptive (describing how one should reason), it is increasingly (and usually in mathematical circles) descriptive (describing how one does reason).

      Could whoever wrote it explain what they meant? Seems odd to me.

      Also I don’t think that category-theoretic logic should be there. Should it not appear in mathematical logic, or be a new page?

    • I’ve been editing second order arithmetic (I usually write “second-order arithmetic”, with a hyphen). I would appreciate someone taking a look and making corrections if necessary. There are probably some hyperlinks which could be added.

    • I ended up polishing type theory - contents (which is included as a floating table of contents in the relevant entries):

      1. expanded and re-arranged the list under “syntax”, created stubs for the missing items definition and program

      2. expanded the (logic/type theory)-table to a (logic/category theory/type theory)-table and subsumed some of the items into it that were floating around elsewhere.

    • at axiom of choice into the section In dependent type theory I have moved the explicit statement taken from the entry of dependent type theory (see there for what I am talking about in the following).

      One technical question: do we need the

        : true

      at the very end of the formal statement of the theorem?

      One conceptual question: I feel inclined to add the following Remark to that, on how to think about the fact that the axiom of choice is always true in this sense in type theory. But please let me know what you think:

      Heuristically, the reason that the axiom of choice is always true when formulated internally this way in dependent type theory is due to the fact that its assumption thereby is stated in constructive mathematics:

      Stated in informal but internal logic, the axiom of choice says:

      If B→AB \to A is a map all whose fibers are inhabited, then there is a section.

      But now if we interpret the assumption clause

      a map all whose fibers are inhabited

      constructively, we have to provide a constructive proof that indeed the fibers are inhabited. But such a constructive proof is a choice of section.

      So constructively and internally the axiom is reduced to “If there is a section then there is a section.” And so indeed this is always true.

      Would you agree that this captures the state of affairs?

    • I added a sentence to fundamental group which contains a link to an example for a fundamental group of an affine scheme.

    • split off total complex from double complex. Let the Definition-section stubby, as it was, but added a brief remark on exactness and on relation to total simplicial sets, under Dold-Kan and Eilenberg-Zilber. More to be done.

    • Dedekind completions of quasiorders (not just linear orders) may now be found at Dedekind completion. Example: the lower Dedekind completion of the quasiorder of continuous functions is the quasiorder of lower semicontinuous functions.

    • I put a bunch of stuff there that might be of interest to the logicians and foundationalists among us, although it’s still pretty trivial.

    • Am I correct in supposing that the first definition of Dedekind cuts at real numbers object is missing an openness condition (as given in the later, power object-using definition on the same page)?

    • I have started a floating table of contents group theory - contents, and started adding it to some relevant entries

      (the toc is neither meant to already be complete nor to be optimally organized, please expand and polish as you see the need)

    • Another new article: sequence space. I await the inevitable report that this term is also used for other things.

    • New page: Banach coalgebra.

      Hopefully you all know that l 1l^1 is a Banach algebra under convolution, but did you know that l ∞l^\infty is a Banach coalgebra under nvolution? (Actually, they are both Banach bialgebras!)

    • created a little table: chains and cochains - table and included it into the relevant entries (some of which still deserve to be edited quite a bit).

    • I have created a table relations - contents and added it as a floatic TOC to the relevant entries.

    • I added a few observations under a new section “Results” at bornological set. Bornological sets form a quasitopos. I don’t have a good reference for the theorem of Schanuel.

      Related is an observation which hadn’t occurred to me before: the category of sets equipped with a reflexive symmetric relation is a quasitopos. I’d like to return to this sometime in the context of thinking about morphisms of (simple) graphs.

    • I have started an entry (∞,n)-category with adjoints, prompted by wanting to record these slides:

      • Nick Rozenblyum, Manifolds, Higher Categories and Topological Field Theories, talk Northwestern University (2012) (pdf slides)

      If anyone can say more about the result indicated there, I’d be most grateful for a comment.

      Also, I seem to hear that at Luminy 2012 there was some extra talk, not appearing on the schedule (maybe by Nick Rozenblyum, but I am not sure) on something related to geometric quantization. If anyone has anything on that, I’d also be most grateful.

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