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- Discussion Type
- discussion topicspecial functions, elementary, hypergeometric
- Category Latest Changes
- Started by zskoda
- Comments 12
- Last comment by Todd_Trimble
- Last Active Jan 23rd 2014

New entry special function, extensions to hypergeometric function, Selberg integral. New entries gamma function, recently also Euler beta function.

Stub for elementary function.

- Discussion Type
- discussion topicworld sheets for world sheets
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jan 22nd 2014

brief entry on the idea of

*world sheets for world sheets*.In the course of creating this I also started a stub for 2d quantum gravity, but clearly more needs to go there.

- Discussion Type
- discussion topic\bar{f}-cofibration.
- Category Latest Changes
- Started by Tim_Porter
- Comments 2
- Last comment by Mike Shulman
- Last Active Jan 20th 2014

I was looking at simnplicial topological group and found mention of $\bar{f}$-cofibration. This is not provided with a link, and a search for the term did not find anything. What is one of these and where is that explained? (It occured to me that it related to the Strom model category structure on $Top/B$, but I could not find it on the relevant page.)

- Discussion Type
- discussion topicproof net
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Todd_Trimble
- Last Active Jan 20th 2014

created some minimum at

*proof net*(long requested by*string diagram*)

- Discussion Type
- discussion topicSeparator
- Category Latest Changes
- Started by Tim Campion
- Comments 1
- Last comment by Tim Campion
- Last Active Jan 20th 2014

There was some confusion on the separator page in the section on strengthened sorts of separator. I’ve attempted to sort it out.

- Discussion Type
- discussion topicamplimorphism
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by David_Corfield
- Last Active Jan 15th 2014

brief entry for

*amplimorphism*

- Discussion Type
- discussion topicfoundations of mathematics
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Jan 15th 2014

Bas Spitters had mentioned the following article on the HoTT list. While I suppose the conclusion has to be taken with several grains of salt, I found this discussion interesting and illuminating, and have added it now to the references at

*foundations of mathematics*:Freek Wiedijk,

*Is ZF a hack? Comparing the complexity of some (formalist interpretations of) foundational systems for mathematics*(pdf)**Abstract**This paper presents Automath encodings (which also are valid in LF/P) of various kinds of foundations of mathematics. Then it compares these encodings according to their size, to find out which foundation is the simplest.The systems analyzed in this way are two kinds of set theory (ZFC and NF), two systems based on Church’s higher order logic (Isabelle/Pure and HOL), three kinds of type theory (the calculus of constructions, Luo’s extended calculus of constructions, and Martin-Löf predicative type theory) and one foundation based on category theory. The conclusions of this paper are that the simplest system is type theory (the calculus of constructions) but that type theories that know about serious mathematics are not simple at all. Set theory is one of the simpler systems too. Higher order logic is the simplest if one looks at the number of concepts (twenty-five) needed to explain the system. On the other side of the scale, category theory is relatively complex, as is Martin-Löf’s type theory.

- Discussion Type
- discussion topic(oo,1)-monad
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by David_Corfield
- Last Active Jan 14th 2014

- created stub for (infinity,1)-monad, to go with <a href="http://golem.ph.utexas.edu/category/2010/01/generalized_multicategories.html#c030778">this</a> blog comment

- Discussion Type
- discussion topiccomputational type theory
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jan 13th 2014

Is this here an accurate description of what is meant by the words “computational type theory”:

The term

*computational type theory*is often used generally for intuitionistic type theory, referring to its computational content in view of the propositions-as-types and proofs-as-programs interpretation (e.g. Scholarpedia). More specifically it is used for type theory with inductive types and even more specifically (Fairtlough-Mendler 02) for modal type theory, hence for type theory equipped with a monad (in computer science) which exhibits a kind of computation.?

- Discussion Type
- discussion topicS-matrix, amplitudes and Atiyah-Segal FQFT
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 12th 2014

While writing this reply on Physics.SE I thought to myself that it is curious that this tight relation between the topics in the title here is rarely made explict in introductions.

Then next occurred to me the observation that, unfortunately, not even the $n$Lab did seem to say this. So therefore I have now briefly copied my reply there also to

*S-matrix – Formalization*and to*FQFT – Idea – General*.This deserves to be expanded on much further, of course, but at least it’s a start now.

- Discussion Type
- discussion topicBub-Clifton theorem
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 12th 2014

entry for

*Bub-Clifton theorem*with a rough idea section and some references

- Discussion Type
- discussion topicvirtual fundamental class
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Jan 11th 2014

gave

*virtual fundamental class*an Idea-section (feel free to improve) and added a bunch of pointers to the literature in the References-section

- Discussion Type
- discussion topicintegral transform
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jan 6th 2014

stub for integral transform (also a highly stubby stub for Fourier-Mukai transform)

- Discussion Type
- discussion topicsheaf of L-infinity algebras
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 4th 2014

started stub entry

*sheaf of L-infinity algebras*, but it is still lacking some evident referencesOriginally I was going to add a comment on how to axiomatize in differential cohesion a sheaf of $L_\infty$-algebras over $X$ as a pointed object in $\mathbf{H}_{/X}$ which is sent by the reduction modality to an identity. But maybe I’ll better do this tomorrow, when I am more awake (or else whenever that happens again).

- Discussion Type
- discussion topicRewriting
- Category Latest Changes
- Started by Tim_Porter
- Comments 7
- Last comment by Tim_Porter
- Last Active Jan 2nd 2014

I have started an entry on rewriting. It is just a stub for the moment.

- Discussion Type
- discussion topicnatural numbers
- Category Latest Changes
- Started by IngoBlechschmidt
- Comments 7
- Last comment by IngoBlechschmidt
- Last Active Dec 30th 2013

Added to

*natural number*a discussion about the fact that constructively, the natural numbers may fail to be (order) complete, as highlighted by Andrej Bauer in a very nice blog post. I quite like this example, because by interpreting a related lemma in the internal language of a certain sheaf topos one obtains a well-known proposition in algebraic geometry almost for free (see entry); but please let me know if stuff like this is too localized for the nLab.

- Discussion Type
- discussion topicfocal point
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by IngoBlechschmidt
- Last Active Dec 30th 2013

created focal point

- Discussion Type
- discussion topiclocal topos
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by IngoBlechschmidt
- Last Active Dec 30th 2013

I typed at local topos in the section Local over-toposes statement and poof that sufficient for a slice topos $\mathcal{E}/X$ to be local is that $X$ is

*tiny*.What are necessary conditions? Is this already necessary?

- Discussion Type
- discussion topicseveral complex variables
- Category Latest Changes
- Started by Todd_Trimble
- Comments 3
- Last comment by Todd_Trimble
- Last Active Dec 30th 2013

I added a few words to several complex variables, even though I am out of my depth here. If we have analysts popping by here, hopefully they will get an urge to add more.

- Discussion Type
- discussion topicNous
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 28th 2013

stub for Anaxagoras’s

*Nous*.

- Discussion Type
- discussion topicAnalytic functions
- Category Latest Changes
- Started by TobyBartels
- Comments 12
- Last comment by Todd_Trimble
- Last Active Dec 28th 2013

I wrote analytic function, mostly just a definition. I found a reference that treated the infinite-dimensional case in pretty fair generality (slightly more than I actually did) without making the definition any more complicated (well, except one place where one must insert the word ‘continuous’), so I did that.

- Discussion Type
- discussion topicIntermediate value theorem
- Category Latest Changes
- Started by TobyBartels
- Comments 3
- Last comment by TobyBartels
- Last Active Dec 28th 2013

I mentioned the intermediate value theorem at pentagon decagon hexagon identity and then began an article on it.

- Discussion Type
- discussion topicdifferentiation / chain rule
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active Dec 27th 2013

created differentiation and chain rule

- Discussion Type
- discussion topiccomplemented lattice
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 24th 2013

brief entry

*complemented lattice*, just to satisfy the link at*quantum logic*

- Discussion Type
- discussion topic$C^\ast$-dynamical system
- Category Latest Changes
- Started by zskoda
- Comments 4
- Last comment by zskoda
- Last Active Dec 23rd 2013

I added a reference to C-star-system. I propose that we change the name of the page to the C-star dynamical system; this is the standard full term, jargon which is skipping dynamical is confusing for an outsider and not explicative. I can imagine many other things which deserve that name.

- Discussion Type
- discussion topicde Morgan duality
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 23rd 2013

I have cross-linked

$\prod_f \circ \mathbb{D} \simeq \mathbb{D} \circ \sum_f \,.$*de Morgan duality*with*Wirthmüller context*for the statement that in linear logicAlso I have tried to make more of the links in the tables at

*de Morgan duality*point to something.

- Discussion Type
- discussion topiccontinuous-trace algebra
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Dec 21st 2013

- Discussion Type
- discussion topicalgebraic "localized": "at" vs. "away from"
- Category Latest Changes
- Started by jcmckeown
- Comments 7
- Last comment by TobyBartels
- Last Active Dec 21st 2013

The convention, when describing ring extensions, everywhere I’ve seen a convention, is that

- for $S$ a set of primes, “localize at $S$” means “invert what is not divisible by $S$”; so for $p$ prime, localizing “at $p$” means considering only $p$-torsion.
- adjoining inverses $[S^{-1}]$ is pronounced “localized away from $S$”. Inverting a prime $p$ is localizing away from $p$, which means
*ignoring*$p$-torsion.

I have adjusted four instances of former “at” on three pages that would be, algebraicwise, “away from” (and so they now appear).

Evidently, this conflicts with more-categorial uses of “localized”; “inverting weak equivalences” is called localization, by obvious analogy, and is written as “localizing

**at**weak equivalences”. This is confusing! It’s also weird: since a ring is a one-object $Ab$-enriched category with morphisms “multiply-by”, the localization-of-the-category $R$ “at $p$” (or its $Ab$-enriched version, if saying that is necessary)*really*means the localization-of-the-ring $R$ “away from $p$”.You all can sort out that contravariance as/if you like, but don’t break the old algebra papers!

- Discussion Type
- discussion topicmembrane matrix model
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 11th 2013

created a brief entry

*membrane matrix model*with some commented pointers to the literature

- Discussion Type
- discussion topicfields internal to toposes
- Category Latest Changes
- Started by IngoBlechschmidt
- Comments 2
- Last comment by Urs
- Last Active Dec 9th 2013

Added to

*field*examples of internal fields: the canonical ring objects of the petit resp. gros Zariski toposes of a scheme.

- Discussion Type
- discussion topicclosed morphism
- Category Latest Changes
- Started by IngoBlechschmidt
- Comments 4
- Last comment by Urs
- Last Active Dec 9th 2013

Started an entry on closed morphisms, containing examples and characterizations using the internal language. Then I noticed that an entry on closed map already exists, but at the moment the nLab is too slow for proper browsing and editing. Will finish later and maybe merge the entries.

- Discussion Type
- discussion topicUnramified morphism
- Category Latest Changes
- Started by Charles Rezk
- Comments 2
- Last comment by Urs
- Last Active Dec 9th 2013

I repaired the definition of “unramified morphism” of schemes.

- Discussion Type
- discussion topicjoins of quasi-categories
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Tim_Porter
- Last Active Dec 7th 2013

I noticed that the two links : André Joyal, The theory of quasicategories and its applications lectures at Simplicial Methods in Higher Categories, (pdf), near the bottom of the entry join of quasi-categories are dead. Does anyone have a more recent link? or if not an alternative reference?

- Discussion Type
- discussion topicpreconditioner
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Dec 5th 2013

- Discussion Type
- discussion topicBousfield-Kan spectral sequence
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 5th 2013

have added some brief Idea-section and lists of references to

- Discussion Type
- discussion topicentire function
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Dec 4th 2013

- Discussion Type
- discussion topicFredholm determinant
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Dec 4th 2013

Starting a stub Fredholm determinant.

- Discussion Type
- discussion topicBerkovich analytic geometry
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by David_Corfield
- Last Active Dec 4th 2013

I have added the following reference to

*Berkovich space*. Judging from the abstract this sounds like I nice unifying perspective. But I haven’t studied it yet- Oren Ben-Bassat, Kobi Kremnizer,
*Non-Archimedean analytic geometry as relative algebraic geometry*(arXiv:1312.0338)

We show that Berkovich analytic geometry can be viewed as algebraic geometry in the sense of Toën-Vaquié-Vezzosi over various categories. The objects in these categories are vector spaces over complete valued fields which are equipped with additional structure. The categories themselves will be quasi-abelian and this is needed to define certain topologies on the categories of affine schemes. We give new definitions of categories of Berkovich analytic spaces and in this way we also define (higher) analytic stacks. We characterize in a categorical way the G-topology or the topology of admissible subsets used in analytic geometry. We demonstrate that the category of Berkovich analytic spaces embeds fully faithfully into the categories which we introduce. We also include a treatment of quasi-coherent sheaf theory in analytic geometry proving Tate’s acyclicity theorem for quasi-coherent sheaves. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders.

- Oren Ben-Bassat, Kobi Kremnizer,

- Discussion Type
- discussion topic"J-homomorphism and chromatic homotopy"
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Dec 4th 2013

I wanted to collect some of the stuff recently added to a bunch of chromatic entries in a way that forms an at least semi-coherent story, so I made an entry

This is built mostly from copy-and-pasting stuff that I had added to dedicated entries, equipped with a bit of glue to make it stick together and form a story.

(Special thanks to Marc Hoyois for general discussion and in particular for working on the text on the Lurie spectral sequence.)

I want to further fine-tune this. But not tonight.

- Discussion Type
- discussion topicsimplicial object in an (infinity,1)-category
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Dec 3rd 2013

created an entry

*simplicial object in an (infinity,1)-category*and interlinked it a bit. Nothing much there yet, for the moment this is mostly a reminder for me to get back to it later.

- Discussion Type
- discussion topicspectral sequence of a simplicial stable homotopy type
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 2nd 2013

am starting

*spectral sequence of a simplicial stable homotopy type*, but right now it’s just a stub.Have expanded the

*Lurie spectral sequences – table*further:

- Discussion Type
- discussion topicinfinity-Dold-Kan correspondence
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 2nd 2013

added a tad more content to

*infinity-Dold-Kan correspondence*

- Discussion Type
- discussion topicEinstein-Hilbert action
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by zskoda
- Last Active Dec 2nd 2013

stub for Einstein-Hilbert action

- Discussion Type
- discussion topicHigher toposes of laws of motion
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Nov 30th 2013

as you may have seen in the logs, I am working on an entry

*Higher toposes of laws of motion*, something like extended talk notes.I am running a bit out of time, and so the entry is unpolished and turns into just a list of keywords towards the end, for the moment. But in case anyone is wondering about the logs, here is the announcement.

Don’t look at this yet if you feel like just reading. Of course if you feel like joining in with the editing a bit, that’s welcome, as usual.

- Discussion Type
- discussion topicformally étale morphism of schemes via adjoint duality
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Marc Hoyois
- Last Active Nov 29th 2013

The characterization of formally étale morphisms of schemes by the infinitesimal shape modality had been scattered a bit through the nLab (at

*Q-category*, at*formally etale morphism*a bit, at*differential cohesion*a little).To make the statement more recognizable, I put it into this new entry here:

- Discussion Type
- discussion topicdg-geometry
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by adeelkh
- Last Active Nov 29th 2013

I split off an entry dg-geometry from the entry on Hochschild geometry, since it really deserves a stand-alone discussion.

Eventually somebody should add the references by Kapranov et al on dg-schemes etc. And much more.

- Discussion Type
- discussion topictopos of laws of motion
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 28th 2013

started

*topos of laws of motion*(lower case!) on the actual notion as such.

- Discussion Type
- discussion topicrational thermodynamics
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Nov 28th 2013

created a brief entry

*rational thermodynamics*.I haven’t actually seen yet the actual detail of this axiomatics (but see the citations given at the above link). What I currently care about is this historical fact, which I added to the Idea section:

What is called

*rational thermodynamics*is a proposal (Truesdell 72) to base the physics of irreversible thermodynamics on a system of axioms and derive the theory from these formally.The success of the axioms of rational thermodynamics as a theory of physical phenomena has been subject of debate. But the idea as such that continuum physics can be and should be given a clear axiomatic foundation seems to have inspired William Lawvere (see there for more), once an undergraduate student of Clifford Truesdell, to base continuum mechanics on constructions in topos theory, such as synthetic differential geometry and cohesion.

- Discussion Type
- discussion topicinfinitesimal extension
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 26th 2013

stub for

*infinitesimal extension*(and cross-linked a bit)Zoran: sorry, I know I should cite that article of yours/your colleagues, could you please add it? Thanks.

- Discussion Type
- discussion topicbasics of étale cohomology
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Nov 26th 2013

as last week, I have created an entry that collects some of the recent edits scattered over the nLab supposedly in one coherent story, it’s

Should be expanded a bit more. But not tonight.

- Discussion Type
- discussion topicclassifying morphism
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by jim_stasheff
- Last Active Nov 26th 2013

At the entry classifying morphism, there is a query (not displayed as one):

Where can I find a construction of the classifying morphism to a classifying space for a G-bundle with connection - using the connections as a 1-form?

Given the subject matter and the name ‘jim’ plus the location of 96.245.205.76 (Pennsylvania …)????

- Discussion Type
- discussion topicSweedler coring
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Nov 26th 2013

With Zoran I am working on entries related to monadic descent. While that is underway, I have added to

a section

and

- Discussion Type
- discussion topicArtin-Schreier sequence
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 25th 2013

at

*Artin-Schreier sequence*I spelled out the existence proof. This derserves/demands to be further polished and streamlined, but i’ll leave it at that for the moment.

- Discussion Type
- discussion topicopen immersion of schemes
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Nov 25th 2013

added to

*open immersion of schemes*brief mentioning of the relation to etale morphisms and to Zariski opens

- Discussion Type
- discussion topicAcyclic Assembly
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Tim_Porter
- Last Active Nov 25th 2013

I have created a stub for the acyclic assembly lemma as there was a grey link in another entry.

- Discussion Type
- discussion topicTamme's "Introduction to Étale Cohomology"
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 24th 2013

in my search for a good way of introducing basics of étale cohomology I switched from Milne to Tamme, and started some hyperlinked index for the latter’s

*Introduction to Étale Cohomology*. As before, in the course of this I created some brief entries for keywords there, if they didn’t exist yet.Hope to expand this now…

- Discussion Type
- discussion topichypercomplete (infinity,1)-topos
- Category Latest Changes
- Started by Marc Hoyois
- Comments 5
- Last comment by Urs
- Last Active Nov 24th 2013

Fixed a couple incorrect statements at hypercomplete (infinity,1)-topos:

- Remark 1 claimed that having enough points in the 1-topos sense implies having enough points in the ∞-sense. I replaced it with a counterexample from HTT.
- In Proposition 1 I replaced “finite homotopy dimension” by “locally of homotopy dimension $\leq n$”.

- Discussion Type
- discussion topicétale ∞-site
- Category Latest Changes
- Started by Urs
- Comments 33
- Last comment by Marc Hoyois
- Last Active Nov 23rd 2013

some trivial/stubby edits, announced here just in case anyone is wondering about edit activity:

added more references to étale (∞,1)-site

added a tad more text to the stubs Weil cohomology, étale cohomology

created stub l-adic cohomology

started splitting off formally étale morphism of schemes from formally étale morphism

- Discussion Type
- discussion topicMilne's "Lectures on Étale Cohomology"
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 22nd 2013

I have started a hyperlinked index for Milne’s

*Lectures on Étale Cohomology*(Up to and including section 27, where the proof of the Weil conjectures starts. )

In the course I have created a bunch of brief entries, if the corresponding keyword didn’t yet have one. Also cross-linked vigorously.

- Discussion Type
- discussion topicétale morphism of schemes
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Nov 22nd 2013

added to the list of equivalent conditions in the definition at

*étale morphism of schemes*the pair “smooth+unramified”. Added a remark after the definition on how to read these pairs of conditions.