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    • I fixed up an obvious error in a claim at Dold fibration: the counterexample there showed the converse of what was claimed (and I think the error was due to me). Now I'd like to find a proper counterexample, so I'll ask on MathOverflow, in case anyone has one up their sleeve.

    • I began working on a page for the GNS construction but started dozing off at my keyboard (so it needs someone to help finish it). Also I added a note on QBism and category theory to the page on quantum states. In it I assert that category theory is, in essence, a theory about relations (bear with me on this one - read the entry).

    • Started monoidal bicategory with a bunch of references to the tortuous literature on braided and symmetric things. Please correct me if I left anything out. It would be nice to have the "accepted/correct" definitions here eventually, for reference.

    • Didn't have much free time, but managed to add a little bit to Stokes phenomenon and responded to Zoran's query about Birkhoff's theorem. I suggested we rename it "Birkhoff-von Neumann theorem" so as not to confuse it with other similarly titled theorems.

    • I have started an entry on group presentations. This will lead to a discussion of identities among relations as well as feeding into the rewriting page.

    • I broke out a very rough page on what I'm tentatively calling graphical quantum channels. If anyone has some time or interest, it's really rough and I could use some help further developing this. Essentially, my aim is to develop a fully robust theory of quantum channels in category theoretic terms, but with an eye for pedagogy and simplicity.

    • created multisymplectic geometry by effectively reproducing a useful survey website (see references given). But added a few wrapping sentences on the nLab perspective

    • I have given a list of chapters and section headings for the Menagerie notes (first 10 chapter).

    • there have been recent edits at partially ordered dagger category. i edited a bit in an attempt to polish.

      Tim Porter mentions parially ordered groupoids here. I am not sure why. These are not dagger categories, are they? This should go in another entry then, I suppose?

    • added to cartesian morphism

      • in the section for ordinary categories the definition in terms of pullbacks of over-categories

      • in the section on (oo,1)-categories more details on the definition and a very useful equivalent reformulation

    • I have to admit that I simply cannot parse many of the entries on type theory and related.

      Now, this is certainly my fault, as I am not spending any considerable time to follow this. But on the other hand my impression is that many statements here are not overly complicated, and that I ought to be following at least roughly what's going on. But I don't.

      One thing is that when I try to look up precise definitions such as at type theory I run into long pieces of text. I am not sure what to make of this.

      My understanding was at some point that all of type theory is really just another way of speaking about categories. Instead of "object" A we say "type" A. Instead of morphism p : U \to A we say p : A " p is of type A" and the like.

      Can we have some Rosetta-stone entry where all the type-theoretic language is translated into plain category theory this way?

      For instance I am suspecting that what is going on at identity type is somehow another way of saying equalizer. But I am not sure. Can anyone help me?

    • This comment is invalid XML; displaying source. <p>motivated by Domenico's <a href="http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=905&page=1#Item_40">latest comment</a> I copied the material on Whitehead towers in (oo,1)-toposes from the end of <a href="https://ncatlab.org/nlab/show/universal+covering+space">universal covering space</a> into a dedicated entry:</p> <ul> <li><a href="https://ncatlab.org/nlab/show/Whitehead+tower+in+an+%28infinity%2C1%29-topos">Whitehead tower in an (infinity,1)-topos</a></li> </ul>
    • Due to popular demand (well, maybe not) I have uploaded my presentation to the APS March Meeting from Friday. It can be found here. I linked it from the bottom of the quantum channel page.
    • Based on a discussion I had with someone after my talk today, I tossed an idea up on the entanglement page concerning how to use categories to model the process of entangling something which I think could be extremely useful to physicists. But it needs a bit of work and I have a plane to catch. I will note that the idea came to me during the conversation when I recalled p. 36 in Steve Awodey's book.

    • polished and expanded the Idea-section at AQFT

    • Zoran,

      concerning your paper with Durov and the sheaf category defined on p. 22, I am wondering:

      it would almost seem as if something essentially equivalent is obtained if we would very slightly change the definition of the site (Rings with a chosen nilpotent ideal) and think of it as the tangent category of the category of rings, i.e. of Mod, thought of as being the category of square-0-extensions of rings.

      So I am suggesting that we look at sheaves on (the opposite of) Mod

      Do you see what I mean?

    • Why the pluralized title in cochains on simplicial sets, unlike in the rest of nlab ? In addition the second plural "on simplicial sets" is misleading, as if it we were talking about cochains defined on a collection of simplicial sets, rather than cochains on a single simplicial set.

      Typoi discussoin, collectoin...

    • Started smooth structure of the path groupoid in response to Theo Something-Or-Other's question on MO. Initial input concerns the structure of the path groupoid in Euclidean space with a - perhaps surprising - conclusion.

    • added the original references that discuss how a spin strucvture on a space is the quantum anomaly cancellation condition for the superparticle sigma-model to spin structure

    • I have created an entry ind-scheme. This is a slightly wider topic than formal scheme, hence it deserves a separate entry, at least to record interesting references. Kapranov and Vasserot wrote a series of 4 articles in which they studied loop schemes, in a setup wider than those classifying loops in affine schemes (passage from affine to nonaffine situation is very nontrivial here, as the loops do not need to be localized so there is no descent property reducing it to loops in affine case), and an interesting result is the factorization monoid structure which is eventually responsible for factorization algebras in CFT. This should be compared to the approach via derived geometry a la Lurie and Ben-Zvi where topological loop spaces are used to obtain a similar structure.

    • in fibration sequence, changed the second diagram after "But the hom-functor has the crucial property..."

      please someone check with the previos version to see if my correction is correct.
    • I filled in a bit of stuff on open systems and reversibility under quantum channels and operations. There's some category-theoretic stuff I have to add to it including figuring out a category-theoretic proof for one of the lemmas. Don't have time to do it right now.

    • brief remark on my personal web on Whitehead systems in a locally contractible (oo,1)-topos.

      So the homotopy fibers of the morphism A \to \mathbf{\Pi}(A)\otimes R that induces the Chern character in an (oo,1)-topos are something like the "rationalized universal oo-covering space": all non-torsion homotopy groups are co-killed, or something like that.

      Is there any literature on such a concept?

    • Based on Urs' comments, I have tentatively merged "partial trace" with the article on "trace" and included a redirect. What do people think about that? If we agree we like the change, can we delete the old partial trace page and, if so, how?

      Also, the partial trace needs a diagram. I'm a little sketchy at this point on how to draw them in itex so if someone else is interested in taking a crack at it, it would be appreciated.
    • Based on where the discussion was headed, I renamed the quantum channels page quantum operations and channels (but included suitable redirects) and added a few To Do items (including describing quantum operations) since, in order to fully understand the reversibility stuff, open quantum systems should be discussed. I don't have time right now to fill in all the details, but will hopefully get a chance to sometime in the next few days (spring break is rapidly approaching its end which means my time will get eaten up again...).

      Incidentally, from the open systems stuff I will eventually link to a new page on closed time-like curves (CTCs) since they are (or can be) related and I think category theory might serve to help shed some light on how they function. This brings up the question: why isn't there a relativity section on nLab? I thought John Baez had done some work applying categories to quantum gravity? Maybe no one ever got to it?

    • edited homotopy coherent nerve a bit

      I tried to bring out the structure more by adding more subsections. Have a look at the new table of contents. Then I did a bunch of trivial edits like indenting some equations etc. Have a look at "See changes" if you want to see it precisely.

    • I put a summary of the Chapman complement theorem at shape theory. I remember a discussion about duality on the blog some time ago and this may be relevant.

    • I just added a page on unitary operators. I also have a query there about whether unitary operators on a given Hilbert space form a category.
    • I was hunting around for things a newbie could contribute to and noticed an empty link to Wick rotation so I filled it in.

    • Some more discussion (Ian and myself) at quantum channel about the definition of QChan when taking into account classical information.

    • I added a small subsection to the definition of an enriched category X over M which describes them as lax monoidal functors M^{op} \to Span(X) where the codomain is the monoidal category of endospans on X in the bicategory of spans.

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