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    • 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="">latest comment</a> I copied the material on Whitehead towers in (oo,1)-toposes from the end of <a href="">universal covering space</a> into a dedicated entry:</p> <ul> <li><a href="">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.

    • This is really just for Zoran although anyone else is welcome to help. I felt there needed to be a little more here, but you are also closely involved with this so please, check that what I have added is alright. Thanks. Tim

    • I wanted to add to rational homotopy theory a section that gives a summary overview of the two Lie theoretic approaches, Sullivan's and Quillen's, indicating the main ingredients and listing the relevant references, by collecting some of the information accumulated in the blog discussion.

      But, due to my connection problem discused in another thread, even after trying repeatedly for about 45 minutes, the nLab software still regards me as a spammer and won't let me edit the entry.

      I'll try again tomorrow. Meanwhile, in case a good soul here can help me out, I post the text that I wanted to add to the entry in the next message. It's supposed to go right after the section ""Rational homotopy type".

    • When Urs cleaned up my quantum channel entry he included some empty links to things that needed defining. I created an entry for one (density matrices and operators) but, before I do anymore, wanted to make sure that what I did was appropriate and conforms to the general format for definitions, particularly since it is an applied context and may be somewhat unfamiliar to some people.
    • I've created a stub article for equilogical spaces. I couldn't quite figure out how to make T_0 a link while preserving the subscripting; I guess I could rewrite that to avoid the formatting problem, but presumably someone else knows how to do it anyway

    • started category fibered in groupoids as I think this deserves a page of its own separated from Grothendieck fibration

      I understand that there was some terminological opposition voiced at Grothendieck fibration concerning the term "category cofibered in groupoids", but am I right that this does not imply opposition against "category fibered in groupoids", only that the right term for the arrow-reversed situation should be "opfibration in groupoids"?

    • I am expanding the entry homotopy group (of an infinity-stack) by putting in one previously missing aspect:

      there are two different notions of homotopy groups of oo-stacks, or of objects in an (oo,1)-topos, in general

      • the "categorical" homotopy groups

      • the "geometric" homotopy groups.

      See there for details. This can be seen by hand in same cases That this follows from very general nonsense was pointed out to me by Richard Williamson, a PhD student from Oxford (see credits given there). The basic idea for 1-sheaves is Grothendieck's, for oo-stacks on topological spaces it has been clarified by Toen.

      While writing what I have so far (which I will probably rewrite now) I noticed that the whole story here is actually nothing but an incarnation of Tannak-Krein reconstruction! I think.

      It boils down to this statement, I think:

      IF we already know what the fundamental oo-groupoid  \Pi(X) of an object  X is, then we know that a "locally constant oo-stack" with finite fibers is nothing but a flat oo-bundle, namely a morphism  \Pi(X) \to \infty FinGrpd (think about it for n=1, where it is a very familiar statement). The collectin of all these is nothing but the representation category (on finite o-groupoids)

       Rep( \Pi(X)) := Func(\Pi(X), Fin \infty Grpd)

      For each point x \in X this comes with the evident forgetful funtor

       x_* : Rep(\Pi(X)) \to Fin \infty Grpd

      that picks the object that we are representing on.

      Now, Tannaka-Krein reconstruction suggests that we can reconstruct  \Pi(X) as the automorphisms of the functor.

      And that's precisely what happens. This way we can find  \Pi(X) from just knowing "locally constant oo-stacks" on X, i.e. from known flat oo-bundles with finite fibers on X.

      And this is exactly what is well known for the n=1 case, and what Toen shows for oo-stacks on Top.

    • added comments on  FinSet being a topos to FinSet and to the examples section at topos.

    • (need to rethink what I said here, sorry)

    • added to global section the statement that ooGrpd is indeed the terminal (oo,1)-topos.

    • Added a minor alternative rendition of the definition of natural transformation. I'm not sure that it's particularly useful for anything, but it is at least true, with some aesthetically pleasing qualities.
    • I thought I'd amuse myself with creating a succinct list of all the useful structures that we have canonically in an (oo,1)-topos without any intervention by hand:

      • principal oo-bundles, covering oo-bundles, oo-vector-bundles, fundamental groupoid, flat cohomology, deRham cohomology, Chern character, differential cohomology.

      I started typing that at structures in a gros (oo,1)-topos on my personal web.

      I think this gives a quite remarkable story of pure abstract nonsense. None of this is created "by man" in a way. It all just exists.

      Certainly my list needs lots of improvements. But I am too tired now. I thought I'd share this anyway now. Comments are welcome.

      Main point missing in the list currently is the free loop space object, Hochschild cohomology and Domenico's proposal to define the Chern character along that route. I am still puzzled by how exactly the derived loop space should interact with  \Pi^{inf}(X) and \Pi(X).

    • I am re-reading Simpson/Teleman's "de Rham theorem for oo-stacks" and realize what I missed on first reading:

      they have essentually the statement that I produced recently, that oo-stacks on a site with "geometrically contractible" objects form a localy contractible (infinity,1)-topos in that the terminal geometric morphism is essential.

      They say this (slightly implicitly) at the level of homotopy cateories on page 5.

      I briefly recorded this in a somewhat stubby way now at locally contracible (oo,1)-topos - Examples

    • I have created a brief entry for Phil Higgins. Clearly more could be added for instance his work on using groupoids to prove group theoretic results.

    • I have started an entry on the decalage functor.