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- Discussion Type
- discussion topicgeneralized multicategories
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Apr 6th 2010

I have been greatly expanding the entry generalized multicategory, to work up gradually from the most explicit and easy-to-understand Leinster definition through profunctors, double categories, and up to the most general situation considered in my paper with Geoff, with examples. There's lots more left to do, but I have to run to a seminar now.

- Discussion Type
- discussion topicGrothendieck's existence theorem
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Apr 6th 2010

New entry Grothendieck existence along with many redirects. New person entry Steven Kleiman. Pdf link to write up of Gerd Faltings' 1995 Bonn lectures at moduli space of bundles and some new reference here and there.

- Discussion Type
- discussion topicDold fibrations
- Category Latest Changes
- Started by DavidRoberts
- Comments 1
- Last comment by DavidRoberts
- Last Active Apr 6th 2010

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.

- Discussion Type
- discussion topichom-object in a quasi-category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 5th 2010

created hom-object in a quasi-category

- Discussion Type
- discussion topicGrothendieck fibration
- Category Latest Changes
- Started by Mike Shulman
- Comments 5
- Last comment by Tim_Porter
- Last Active Apr 5th 2010

Did some reorganizing at Grothendieck fibration.

- Discussion Type
- discussion topicSVG Editor HowTo
- Category Latest Changes
- Started by Andrew Stacey
- Comments 12
- Last comment by Andrew Stacey
- Last Active Apr 4th 2010

Created SVG Editor HowTo whilst trying to convert the codecogs monstrosity at exercise in groupoidification - the path integral to SVG (conversion happening in the Sandbox for the moment, and definitely not finished yet).

- Discussion Type
- discussion topicA partial addition and some added details
- Category Latest Changes
- Started by Ian_Durham
- Comments 3
- Last comment by Ian_Durham
- Last Active Apr 3rd 2010

- 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).

- Discussion Type
- discussion topicquantum state
- Category Latest Changes
- Started by Ian_Durham
- Comments 12
- Last comment by Ian_Durham
- Last Active Apr 3rd 2010

- Added page quantum state.

- Discussion Type
- discussion topic‘folk’
- Category Latest Changes
- Started by TobyBartels
- Comments 4
- Last comment by Mike Shulman
- Last Active Apr 2nd 2010

I added a brief note to folk model structure to reflect the other terms suggested on another thread.

- Discussion Type
- discussion topicmonoidal bicategories
- Category Latest Changes
- Started by Mike Shulman
- Comments 4
- Last comment by Mike Shulman
- Last Active Apr 2nd 2010

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.

- Discussion Type
- discussion topicSome updates
- Category Latest Changes
- Started by Ian_Durham
- Comments 7
- Last comment by Ian_Durham
- Last Active Apr 2nd 2010

- 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.

- Discussion Type
- discussion topicGroup Presentation
- Category Latest Changes
- Started by Tim_Porter
- Comments 2
- Last comment by zskoda
- Last Active Apr 1st 2010

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.

- Discussion Type
- discussion topicNew page: [[proper subset]]
- Category Latest Changes
- Started by TobyBartels
- Comments 41
- Last comment by Mike Shulman
- Last Active Mar 31st 2010

Not much there, but I created it: proper subset.

- Discussion Type
- discussion topicClassical field theory
- Category Latest Changes
- Started by Ian_Durham
- Comments 3
- Last comment by Ian_Durham
- Last Active Mar 29th 2010

- I added a query to classical field theory.

- Discussion Type
- discussion topic"Graphical" quantum channels
- Category Latest Changes
- Started by Ian_Durham
- Comments 9
- Last comment by Eric
- Last Active Mar 29th 2010

- 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.

- Discussion Type
- discussion topicmultisymplectic geometry
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by zskoda
- Last Active Mar 29th 2010

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

- Discussion Type
- discussion topicMenagerie
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Tim_Porter
- Last Active Mar 27th 2010

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

- Discussion Type
- discussion topicpartially ordered dagger category
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Mar 27th 2010

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?

- Discussion Type
- discussion topiccartesian morphism
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Mar 26th 2010

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

- Discussion Type
- discussion topicregular singular point
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Mar 25th 2010

The basic notion (though in fact nontrivial) regular singular point needed at Fuchsian equation. Redirects irregular singular point.

- Discussion Type
- discussion topicKapranov's NCG
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by zskoda
- Last Active Mar 25th 2010

New entries Kapranov's noncommutative geometry and Dmitry Kaledin. New references and typographic improvements at noncommutative algebraic geometry.

- Discussion Type
- discussion topicFuchsian equation
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by zskoda
- Last Active Mar 25th 2010

- Discussion Type
- discussion topictype theory -- help!
- Category Latest Changes
- Started by Urs
- Comments 35
- Last comment by Urs
- Last Active Mar 25th 2010

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" we say "type" . Instead of morphism we say " is of type " 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?

- Discussion Type
- discussion topicgeometrical optics
- Category Latest Changes
- Started by zskoda
- Comments 6
- Last comment by zskoda
- Last Active Mar 24th 2010

I wrote a stub geometrical optics with redirect geometric optics (maybe it was better other way around, I don't have the feeling which is more used). And created stub optics. Geometric optics is of course in my present program of semiclassical approximation, equivariant localization, wall crossing, (edit typo:) Stokes phenomenon and related notions. Note that in physicscontents we still do have it listed but not written entry deformation quantization. So I just starting a stub.

- Discussion Type
- discussion topicslope filtration
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by zskoda
- Last Active Mar 24th 2010

- Discussion Type
- discussion topicequivalence in a quasi-category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 24th 2010

created stub for equivalence in a quasi-category. Just because i needed to link to something.

- Discussion Type
- discussion topicFundamental Bigroupoids and 2-Covering Spaces
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Mar 24th 2010

created reference-entry on David Roberts' thesis:

and linked to it from Whitehead tower in an (infinity,1)-topos

(David, I slightly edited your nLab-page, accordingly)

- Discussion Type
- discussion topicWhitehead tower in an (oo,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 23rd 2010

`<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>`

- Discussion Type
- discussion topicExtremal quantum channels & open systems
- Category Latest Changes
- Started by Ian_Durham
- Comments 2
- Last comment by Urs
- Last Active Mar 23rd 2010

- Since the original quantum channel entry was getting long, I broke out open quantum systems into its own page and added a page for extremal quantum channels. Both could greatly benefit from some category-theoretic input.

- Discussion Type
- discussion topicholonomic D-module
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by zskoda
- Last Active Mar 22nd 2010

A bit of additions to holonomic D-module, a new reference at D-module and new stub characteristic variety.

- Discussion Type
- discussion topicMarkov chains, part deux
- Category Latest Changes
- Started by Ian_Durham
- Comments 2
- Last comment by Eric
- Last Active Mar 22nd 2010

- I posted a quick query on Markov chains.

- Discussion Type
- discussion topicTalk on categorical quantum channels
- Category Latest Changes
- Started by Ian_Durham
- Comments 1
- Last comment by Ian_Durham
- Last Active Mar 22nd 2010

- 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.

- Discussion Type
- discussion topicquantum mechanics in terms of †-compact categories
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Ian_Durham
- Last Active Mar 21st 2010

started stub for quantum mechanics in terms of dagger-compact categories

this is going to have some overlap with the material that is or is planned at quantum information, but I thought it would be worthwhile to have a page explicitly dedicated to the description using technology from dagger-compact categories.

- Discussion Type
- discussion topicEntanglement
- Category Latest Changes
- Started by Ian_Durham
- Comments 1
- Last comment by Ian_Durham
- Last Active Mar 20th 2010

- 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.

- Discussion Type
- discussion topicAQFTpolishe
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Eric
- Last Active Mar 20th 2010

polished and expanded the Idea-section at AQFT

- Discussion Type
- discussion topicDurov, Skoda et al
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by zskoda
- Last Active Mar 19th 2010

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?

- Discussion Type
- discussion topiccochains on simplicial sets
- Category Latest Changes
- Started by zskoda
- Comments 5
- Last comment by Urs
- Last Active Mar 17th 2010

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...

- Discussion Type
- discussion topicsmooth structure of path groupoid
- Category Latest Changes
- Started by Andrew Stacey
- Comments 3
- Last comment by Andrew Stacey
- Last Active Mar 17th 2010

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.

- Discussion Type
- discussion topicUniversal Homotopy Theories
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 16th 2010

created a reference-entry on Dugger's article Universal Homotopy Theories

- Discussion Type
- discussion topicglobal model structure on simplicial presheaves
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 16th 2010

polished/expanded the Idea-section at global model structure on simplicial presheaves

- Discussion Type
- discussion topicCech model structure on sheaves
- Category Latest Changes
- Started by Mike Shulman
- Comments 3
- Last comment by Urs
- Last Active Mar 16th 2010

I have a couple of questions at Cech model structure on simplicial presheaves.

- Discussion Type
- discussion topic(oo,1)-category of (oo,1)-functors
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Mike Shulman
- Last Active Mar 16th 2010

expanded (infinity,1)-category of (infinity,1)-functors : more details, more statements, more proofs. In particular concerning the model by the global model structure on functors.

- Discussion Type
- discussion topicspin structure as quantum anomaly cancellation
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Mar 15th 2010

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

- Discussion Type
- discussion topicind-scheme
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by zskoda
- Last Active Mar 15th 2010

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.

- Discussion Type
- discussion topicReedy categories
- Category Latest Changes
- Started by Mike Shulman
- Comments 21
- Last comment by Eric
- Last Active Mar 15th 2010

Created direct category and generalized Reedy category (well, un-stubbified the latter) as spinoffs from Reedy category.

- Discussion Type
- discussion topicfibration sequence
- Category Latest Changes
- Started by domenico_fiorenza
- Comments 4
- Last comment by domenico_fiorenza
- Last Active Mar 12th 2010

- 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.

- Discussion Type
- discussion topicoo-groupoid
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 11th 2010

expanded the entry infinity-groupoid

- Discussion Type
- discussion topicomega-nerve
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 11th 2010

expanded omega-nerve

added to Kan complex a section on how to think of them as oo-groupoids.

- Discussion Type
- discussion topicMore on quantum channels
- Category Latest Changes
- Started by Ian_Durham
- Comments 1
- Last comment by Ian_Durham
- Last Active Mar 11th 2010

- 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.

- Discussion Type
- discussion topicquantum channel
- Category Latest Changes
- Started by Urs
- Comments 37
- Last comment by Urs
- Last Active Mar 9th 2010

I see that Ian Durham created quantum channel

- Discussion Type
- discussion topicWhitehead tower in an (oo,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Tim_Porter
- Last Active Mar 9th 2010

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

So the homotopy fibers of the morphism 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?

- Discussion Type
- discussion topictrace + partial trace
- Category Latest Changes
- Started by Ian_Durham
- Comments 1
- Last comment by Ian_Durham
- Last Active Mar 9th 2010

- 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.

- Discussion Type
- discussion topicrational homotopy equivalence
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Mar 8th 2010

created rational homotopy equivalence

- Discussion Type
- discussion topicpartial trace
- Category Latest Changes
- Started by Ian_Durham
- Comments 4
- Last comment by Ian_Durham
- Last Active Mar 8th 2010

- 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.

- Discussion Type
- discussion topicmodel structures on cosimplicial rings and dg-rings
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Mar 8th 2010

started model structure on cosimplicial rings and model structure on dg-algebras

want to eventually expand on this: does anyone happen have an electronic copy of the article by Jardine referenced at model structure on dg-algebras?

(in the process I also created a quick stub for differential graded ring)

- Discussion Type
- discussion topicQuantum operations
- Category Latest Changes
- Started by Ian_Durham
- Comments 4
- Last comment by Urs
- Last Active Mar 6th 2010

- 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?

- Discussion Type
- discussion topichomotopy coherent nerve
- Category Latest Changes
- Started by Urs
- Comments 23
- Last comment by Urs
- Last Active Mar 5th 2010

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.

- Discussion Type
- discussion topicChapman complement theorem
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Mar 5th 2010

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.

- Discussion Type
- discussion topicUnitary operators
- Category Latest Changes
- Started by Ian_Durham
- Comments 28
- Last comment by DavidRoberts
- Last Active Mar 5th 2010

- 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.

- Discussion Type
- discussion topicWick rotation
- Category Latest Changes
- Started by Ian_Durham
- Comments 2
- Last comment by DavidRoberts
- Last Active Mar 5th 2010

- I was hunting around for things a newbie could contribute to and noticed an empty link to Wick rotation so I filled it in.