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• 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?

• I spent quite a bit of time now working on the entry Kähler differential

The motivation was that I had I pointed out the general idea of Kähler differentials at MathOverflow to somebody here. But when I then checked the entry, i found it didn't actually convey the right general picture.

Have a look at what I did, I did invest quite a bit of time in order to bring both the fully general nonsense nPOV, but lead up to it gently and understandably.

So there is a big "Idea and definition" section now that is meant to explain what is really going on, in the large and in the small.

Then the previous content of the entry, on Kähler differentials over ordinary rings and over smooth rings regarded as ordinary rings, I made subsections in a section titled "Specific deifnitions". I added more subsections to this. A stubby one with a pointer to the C^oo-ring case that is discussed in detail at Fermat theory, and then a bit on modules over monoids in general abelian categories, which is what the MO question had been about.

(They are funny at MO. I think this is quite a deep question. Also Tim Porter pointed out rather non-trivial literature on it. But the poster is being verbally abused for asking a question

that asks more of the respondent than you have contributed yourself.

:-o And now they closed down the thread!

Boy, I am just glad that among us we allow us to ask questions even before we are experts on something. Everything else is unhealthy. )

Anyway, the section on C^oo rings is a bit short. When you read the entry, you'll notice that there is an obvious question now: "So is it also true for C^oo-rings that the overcat. etc. pp-" The answer is YES, we checked this and it works all very nicely and reproduces the stuff reviewed at Fermat theory precisely. But this has been found / thought about by two graduate students, and I don't want to publicize this too much right now. But later.

• 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.
• Just added a page on partial trace that is presently linked from quantum operations and channels which I also added to. However, note that the partial trace is not specific to physics so it needs embellishing by the mathematicians among us.
• 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.

• At locally contractible space David Roberts had written a conjecture relating local contractibility to local triviality of constant n-stacks. I have added to that a converse proposition.

• 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.

• I expanded the beginning of physicscontents and edited a bit to bring out more structure.

• Created solid functor with an SVG graphic. The SVG editor is awesome! Even if it is still a little buggy.

• added to derivation at the very end in the exampls section a discussion of derivations on smooth functions (and how they are vector fields) and f derivations on continuous functions (and how they are trivial).

• Jim Stasheff edited a bit at A-infinity category -- but did not yet work in the comments that he posted to the blog

• as part of a reply in my blog discussion with Mike here I created a section

• I have been editing the entry homotopy equivalence to include a brief discussion of strong homotopy equivalences and Vogt's lemma. In so doing, I have followed my nose and found various other entries to edit. For instance that for Hans Baues, that for cylinder functor, etc. I am thinking that the general area of Henry Whitehead's idea of algebraic homotopy, may be a useful intermediate one between the infinity category ideas (which could be seen as just a 'souped up' version of Kan complexes), (I am not saying they are just that a cynic might make them out to be!) and the algebraic topologists desire to perform calculations. Note the quotes at algebraic homotopy. Of course, they d not say what 'compute' means in this context. (Note we do not have an entry on Whitehead as yet.)

• I worked on improving (hopefully) and further expanding (a bit) the discussion of geometric homotopy groups at

I give at

• Idea now just the brief reminder of the simple situation for 1-toposes as described at locally connected topos;

• and then at Defintion I state the very obvious and simple generalization of this to homotopy oo-groupoids of objects in a locally contractible (oo,1)-topos.

Then I say something like: while this definition is very obvious and simple, it seems it has not quite been stated in the literature (except possibly in the thesis by Richad Williamson), but that there are old well-known results in the literature that essentially, with only slight modification of language, already do say precisely this.

Then I go through this claim in detail. I list three subsections with three different methods of how to construct that left adjoint $\Pi(-)$ to the constant oo-stack functor, and then discuss in some detail how old and new references do already -- if slightly implicitly -- discuss precisely this. The three sections are

You'll notice that I also link to the discussioon of the absgtract oo-adjunction on my personal web. Currently I am thinking of the entry on my personal web as talking about the abstract notion of a path oo-groupoid, and of this page here on the nLab as providing all the "well-known" aspects of it (in that these are in the literature).

Please complain if somehow this doesn't look like the right thing to do. I am currently a bit undecided as to what bits of this discussion should be on my personal web and which on the nLab.

• At category with weak equivalences we say that it is unclear whether every (oo,1)-category arises as the simplicial localization of a cat with weak equivalences, but that it seems plausible.

At (infinity,1)-category we say that indeed every (oo,1)-category arises as the simplicial localization of a homotopical category.

I had put in the paragraph that says this based on a message that Andre Joyal recently posted to the CatTheory mailing list.

It would be good to harmonize this with the discussion at category with weak equivalences and maybe to add some references.

• I think I understand locally contractible (oo,1)-toposes now, with their left-adjoint to the constant oo-sheaf functor.

The (simple) observation is here on my personal web.

• (edit: typo in the headline: meant is "bare" path oo-groupoid)

I think I have the proof that when the structured path oo-groupoid of an oo-stack oo-topos exists, as I use on my pages for differential nonabelian cohomology, then its global sections/evaluation on the point yields the bare path oo-groupoid functor, left adjoint to the formation of constant oo-stacks.

A sketch of the proof is now here.

Recall that this goes along with the discussion at locally constant infinity-stack and homotopy group of an infinity-stack.

P.S.

Am in a rush, will get back to the other discussion here that are waiting for my replies a little later. Just wanted to et this here out of the way

• For the record, all I did at geometric morphism#sheaftopoi was to add a paragraph at the beginning of the example, substitute ‘sober’ for ‘Hausdorff’ in appropriate places, and add to the query box there. I mention this because the diff thinks that I did much more than that, and I don't want anybody to waste time looking for such changes!

I still to make the proof apply directly to sober spaces; the part that used that the space was Hausdorff is still in those terms.

• I fiddled a bit with direct image, but maybe didn't end up doing anything of real value...