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- 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 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.

- Discussion Type
- discussion topicMore quantum channel
- Category Latest Changes
- Started by DavidRoberts
- Comments 2
- Last comment by Ian_Durham
- Last Active Mar 4th 2010

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

- Discussion Type
- discussion topicHamilton operator
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Ian_Durham
- Last Active Mar 4th 2010

started something at Hamilton operator

- Discussion Type
- discussion topicenrichment through lax monoidal functors
- Category Latest Changes
- Started by Todd_Trimble
- Comments 3
- Last comment by Mike Shulman
- Last Active Mar 2nd 2010

I added a small subsection to the definition of an enriched category over which describes them as lax monoidal functors where the codomain is the monoidal category of endospans on in the bicategory of spans.

- Discussion Type
- discussion topicSibe Mardesic
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by TobyBartels
- Last Active Mar 2nd 2010

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

- Discussion Type
- discussion topicrational homotopy theory
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Tim_Porter
- Last Active Mar 1st 2010

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".

- Discussion Type
- discussion topicDefinitions
- Category Latest Changes
- Started by Ian_Durham
- Comments 4
- Last comment by Andrew Stacey
- Last Active Feb 27th 2010

- 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.

- Discussion Type
- discussion topicmore type theory
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by Urs
- Last Active Feb 27th 2010

Wrote identity type and display map and dependent type.

Also, I have a question/correction at internal logic in a presheaf topos

- Discussion Type
- discussion topicEquilogical space
- Category Latest Changes
- Started by SridharRamesh
- Comments 3
- Last comment by SridharRamesh
- Last Active Feb 26th 2010

I've created a stub article for equilogical spaces. I couldn't quite figure out how to make 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

- Discussion Type
- discussion topiccategory fibered in groupoids
- Category Latest Changes
- Started by Urs
- Comments 42
- Last comment by SridharRamesh
- Last Active Feb 25th 2010

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"?

- Discussion Type
- discussion topiccoskeleton
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 25th 2010

started expanding simplicial skeleton. But more needs to eventually go here.

- Discussion Type
- discussion topichomotopy group of an oo-stack
- Category Latest Changes
- Started by Urs
- Comments 76
- Last comment by Urs
- Last Active Feb 24th 2010

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 of an object is, then we know that a "locally constant oo-stack" with finite fibers is nothing but a flat oo-bundle, namely a morphism (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)For each point this comes with the evident forgetful funtor

that picks the object that we are representing on.

Now, Tannaka-Krein reconstruction suggests that we can reconstruct as the automorphisms of the functor.

And that's precisely what happens. This way we can

*find*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.

- Discussion Type
- discussion topicLie theory for stacky Lie groupoids
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 24th 2010

Chenchang Zhu updated the reference to her work at Lie theory for stacky Lie groupoids

- Discussion Type
- discussion topicFinSet as a topos
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 24th 2010

- Discussion Type
- discussion topicObject classifiers for Top
- Category Latest Changes
- Started by David_Corfield
- Comments 4
- Last comment by Urs
- Last Active Feb 24th 2010

- Asked a question at object classifier.

- Discussion Type
- discussion topicderived loop space and infinitesimal paths
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 23rd 2010

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

- Discussion Type
- discussion topicglobal section
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 23rd 2010

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

- Discussion Type
- discussion topicformal dg-algebra
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Feb 22nd 2010

started formal dg-algebra

- Discussion Type
- discussion topicBott periodicity theorem
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Feb 22nd 2010

- Discussion Type
- discussion topicNatural transformations
- Category Latest Changes
- Started by SridharRamesh
- Comments 9
- Last comment by SridharRamesh
- Last Active Feb 21st 2010

- 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.

- Discussion Type
- discussion topicstructures in a gros (oo,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Feb 19th 2010

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 and .

- Discussion Type
- discussion topiclocally contractible (oo,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Feb 19th 2010

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

- Discussion Type
- discussion topicTim : New entries
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Feb 19th 2010

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.

- Discussion Type
- discussion topicdecalage
- Category Latest Changes
- Started by Tim_Porter
- Comments 14
- Last comment by Tim_Porter
- Last Active Feb 19th 2010

I have started an entry on the decalage functor.