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2-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology combinatorics complex-geometry computable-mathematics computer-science connection constructive constructive-mathematics cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality education elliptic-cohomology enriched fibration foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity grothendieck group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory kan lie lie-theory limit limits linear linear-algebra locale localization logic mathematics measure-theory modal-logic model model-category-theory monad monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories newpage noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

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- Discussion Type
- discussion topicexperiments -- contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 10th 2019

- Discussion Type
- discussion topicdense subcategory
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Marc
- Last Active Apr 10th 2019

I have tried to brush up the entry dense subcategory a little

(moved the references to the References, moved the part that alluded to the application with nerves to its own section and expanded slightly, added the relevant back-links).

- Discussion Type
- discussion topicblack hole
- Category Latest Changes
- Started by zskoda
- Comments 11
- Last comment by Urs
- Last Active Apr 10th 2019

I am at the Croatian black hole school organized by Jarah Evslin, and I am partially taking care of Croatia related issues (visa, trasnportation advice. communication to the owners of the housing). Lots of interesting things here about star formation, black hole formation, making massive black holes from lighter ones and so on. And some string theory mechanisms related to black hole entropy and similar issues. Most of people are postdocs and students here. Among seniors, Holger Nielsen and Mina Aganagić are present to our benefit.

By the way, started a stub black hole. Please contribute.

- Discussion Type
- discussion topichigher category theory
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Todd_Trimble
- Last Active Apr 9th 2019

started to (re)structure the entry higher category theory roughly along the lines of the new structure at category theory. But for the moment many sections just contain link lists.

- Discussion Type
- discussion topicclosed category
- Category Latest Changes
- Started by Tobias Fritz
- Comments 48
- Last comment by Keith Harbaugh
- Last Active Apr 9th 2019

A query box has been added:

I suspect there is a variant of the definition involving a transformation $R^Z_{X Y} \colon [X,Y] \to [[Y,Z],[X,Z]]$ rather than $L$. Is this correct? If so, how do these two definitions relate? Can one of them be expressed in terms of the other? Or is there a refined definition which comprises both $L$ and $R$?

- Discussion Type
- discussion topicadjunction
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Mike Shulman
- Last Active Apr 9th 2019

A comparatively long and technical section “From hom-functors to units and counits” (on adjoint functors) was sitting inside the

*Idea-section*of*adjunction*. It seemed plainly misplaced there, and distracting attention from what should be the content of this entry, as opposed to the entry*adjoint functor*. So I have moved it now to where it seems to belong: inside the Examples-section.

- Discussion Type
- discussion topicIntroduction to Topology -- 2
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by maxsnew
- Last Active Apr 9th 2019

- Discussion Type
- discussion topicT-duality
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by David_Corfield
- Last Active Apr 9th 2019

Added to T-duality a section with the discussion of the usual path-integral heuristics for why the two sigma-models on T-dual backgrounds yield equivalent quantum field theories.

- Discussion Type
- discussion topicmonadic adjunction
- Category Latest Changes
- Started by Mike Shulman
- Comments 8
- Last comment by maxsnew
- Last Active Apr 9th 2019

- Discussion Type
- discussion topicGonzalo E. Reyes
- Category Latest Changes
- Started by Marc
- Comments 1
- Last comment by Marc
- Last Active Apr 9th 2019

- Discussion Type
- discussion topicrepresentable functor
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by RodMcGuire
- Last Active Apr 9th 2019

while adding to

*representable functor*a pointer to*representable morphism of stacks*I noticed a leftover discussion box that had still be sitting there. So hereby I am moving that from there to here:

[ begin forwarded discussion ]

+–{+ .query} I am pretty unhappy that all entries related to limits, colimits and representable things at nlab say that the limit, colimit and representing functors are what normally in strict treatment are just the vertices of the corresponding universal construction. A representable functor is not a functor which is naturally isomorphic to Hom(-,c) but a

*pair*of an object and such isomorphism! Similarly limit is the synonym for*limiting cone*(= universal cone), not just its vertex. Because if it were most of usages and theorems would not be true. For example, the notion and usage of creating limits under a functor, includes the words about the behaviour of the arrow under the functor, not only of the vertex. Definitions should be the collections of the data and one has to distinguish if the existence is really existence or in fact a**part of the structure**.–ZoranMike: I disagree (partly). First of all, a functor $F$

*equipped with*an isomorphism $F\cong hom_C(-,c)$ is not a represent**able**functor, it is a represent**ed**functor, or a functor equipped with a representation. A representable functor is one that is “able” to be represented, or*admits*a representation.Second, the page limit says “a limit of a diagram $F : D \to C$ … is an object $lim F$ of $C$

*equipped with*morphisms to the objects $F(d)$ for all $d \in D$…” (emphasis added). It doesn’t say “such that there exist” morphisms. (Prior to today, it defined a limit to be a universal cone.) It is true that one frequently speaks of “the limit” as being the vertex, but this is an abuse of language no worse than other abuses that are common and convenient throughout mathematics (e.g. “let $G$ be a group” rather than “let $(G,\cdot,e)$ be a group”). If there are any*definitions*you find that are wrong (e.g. that say “such that there exists” rather than “equipped with”), please correct them! (Thanks to your post, I just discovered that Kan extension was wrong, and corrected it.)Zoran Skoda I fully agree, Mike that “equipped with” is just a synonym of a “pair”. But look at entry for limit for example, and it is clear there that the limiting cone/universal cone and limit are clearly distinguished there and the term limit is used just for the vertex there. Unlike for limits where up to economy nobody doubt that it is a pair, you are right that many including the very MacLane representable take as existence, but then they really use term “representation” for the whole pair. Practical mathematicians are either sloppy in writing or really mean a pair for representable. Australians and MacLane use indeed word representation for the whole thing, but practical mathematicians (example: algebraic geometers) are not even aware of term “representation” in that sense, and I would side with them. Let us leave as it is for representable, but I do not believe I will ever use term “representation” in such a sense. For limit, colimit let us talk about pairs: I am perfectly happy with word “equipped” as you suggest.

Mike: I’m not sure what your point is about limits. The definition at the beginning very clearly uses the words “equipped with.” Later on in the page, the word “limit” is used to refer to the vertex, but this is just the common abuse of language.

Regarding representable functors, since representations are unique up to unique isomorphism when they exist, it really doesn’t matter whether “representable functor” means “functor such that there exists an isomorphism $F\cong hom_C(-,c)$” or “functor equipped with an isomorphism $F\cong hom_C(-,c)$.” (As long as it doesn’t mean something stupid like “functor equipped with an object $c$ such that there exists an isomorphism $F\cong hom_C(-,c)$.”) In the language of stuff, structure, property, we can say that the Yoneda embedding is fully faithful, so that “being representable” is really a property, rather than structure, on a functor.

[ continued in next comment ]

- Discussion Type
- discussion topicequivariant bordism homology theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 9th 2019

am giving this its own entry in order to record some reference which used to be at

*equivariant cohomotopy*but didn’t really belong there. Thanks to David R. for pointing to Theorem 5 in- Tammo tom Dieck,
*Orbittypen und äquivariante Homologie II*, Arch. Math. (Basel)26(1975), no. 6, 650–662 (pdf)

which much improves in readability over theorem 3.11 in

- Arthur Wasserman, section 3 of
*Equivariant differential topology*, Topology Vol. 8, pp. 127-150, 1969 (pdf)

- Tammo tom Dieck,

- Discussion Type
- discussion topicblack holes in string theory
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Apr 9th 2019

created

*black holes in string theory*, since somebody asked me: a brief paragraph explaining how the entropy-counting works and some references.

- Discussion Type
- discussion topiccategory of simplices
- Category Latest Changes
- Started by Mike Shulman
- Comments 8
- Last comment by Tim_Porter
- Last Active Apr 9th 2019

I added a bit to category of simplices, including the fact that the category of nondegenerate simplices is final and thus colimits can be computed using only that, and that the nerve of the category of simplices itself is colimit-preserving.

- Discussion Type
- discussion topicAQFT
- Category Latest Changes
- Started by zskoda
- Comments 15
- Last comment by Urs
- Last Active Apr 9th 2019

I did not change anything, I would not like to do it without Urs’s consent and some opinion. The entry AQFT equates algebraic QFT and axiomatic QFT. In the traditional circle, algebraic quantum field theory meant being based on local nets – local approach of Haag and Araki. This is what the entry now describes. The Weightman axioms are somewhat different, they are based on fields belonging some spaces of distributions, and 30 years ago it was called field axiomatics, unlike the algebraic axiomatics. But these differences are not that important for the main entry on AQFT. What is a bigger drawback is that the third approach to axiomatic QFT if very different and was very strong few decades ago and still has some followers. That is the

**S-matrix axiomatics**which does not believe in physical existence of observables at finite distance, but only in the asymptotic values given by the S-matrix. The first such axiomatics was due Bogoliubov, I think. (Of course he later worked on other approaches, especially on Wightman’s. Both the Wightman’s and Bogoliubov’s formalisms are earlier than the algebraic QFT.)I would like to say that axiomatic QFT has 3 groups of approaches, and especially to distinguish S-matrix axiomatics from the “algebraic QFT”. Is this disputable ?

- Discussion Type
- discussion topiccartesian product
- Category Latest Changes
- Started by Todd_Trimble
- Comments 2
- Last comment by Urs
- Last Active Apr 8th 2019

- Discussion Type
- discussion topicbiproduct
- Category Latest Changes
- Started by Colin Tan
- Comments 12
- Last comment by Mike Shulman
- Last Active Apr 8th 2019

- Discussion Type
- discussion topicAgda
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Apr 8th 2019

- Discussion Type
- discussion topicSO(4)
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by DavidRoberts
- Last Active Apr 8th 2019

in order to have a good place to record the diagram:

$\array{ ( q_1, q_2 ) &\mapsto& (x \mapsto q_1 \cdot x \cdot \overline{q}_2) \\ Sp(1) \times Sp(1) &\overset{\simeq}{\longrightarrow}& Spin(4) \\ \big\downarrow && \big\downarrow \\ Sp(1)\cdot Sp(1) &\overset{\simeq}{\longrightarrow}& SO(4) }$

- Discussion Type
- discussion topicDirac operator
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by nLab edit announcer
- Last Active Apr 8th 2019

- Discussion Type
- discussion topicRyszard Engelking
- Category Latest Changes
- Started by Daniel Luckhardt
- Comments 1
- Last comment by Daniel Luckhardt
- Last Active Apr 7th 2019

- Discussion Type
- discussion topicbilinear map
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Bartek
- Last Active Apr 6th 2019

I have expanded a bit at

*bilinear map*, trying to add a pedagigical comment on the difference to a group homomorphism $A \times B \to C$

- Discussion Type
- discussion topicmetrization theorem
- Category Latest Changes
- Started by Daniel Luckhardt
- Comments 4
- Last comment by Urs
- Last Active Apr 6th 2019

- Discussion Type
- discussion topiccountably tight space
- Category Latest Changes
- Started by Daniel Luckhardt
- Comments 1
- Last comment by Daniel Luckhardt
- Last Active Apr 5th 2019

created this page in connection with topology - countability axioms.

- Discussion Type
- discussion topiclocally discrete set of subsets
- Category Latest Changes
- Started by Daniel Luckhardt
- Comments 1
- Last comment by Daniel Luckhardt
- Last Active Apr 5th 2019

- Discussion Type
- discussion topiccountably locally discrete set of subsets
- Category Latest Changes
- Started by Daniel Luckhardt
- Comments 1
- Last comment by Daniel Luckhardt
- Last Active Apr 5th 2019

Created page in connection with open covers of metric spaces have open countably locally discrete refinements.

- Discussion Type
- discussion topicopen covers of metric spaces have open countably locally discrete refinements
- Category Latest Changes
- Started by Daniel Luckhardt
- Comments 1
- Last comment by Daniel Luckhardt
- Last Active Apr 5th 2019

Created this page in connection with topology - countability axioms.

- Discussion Type
- discussion topictopology - countability axioms
- Category Latest Changes
- Started by Daniel Luckhardt
- Comments 3
- Last comment by Daniel Luckhardt
- Last Active Apr 5th 2019

Working on page to be included in several pages like second-countable space, etc.

- Discussion Type
- discussion topicflavour anomaly
- Category Latest Changes
- Started by Urs
- Comments 60
- Last comment by Urs
- Last Active Apr 5th 2019

- Discussion Type
- discussion topicDaciberg Lima Gonçalves
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Tim_Porter
- Last Active Apr 5th 2019